Površ – Wikipedija/Википедија

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id="mw-content-text" class="mw-body-content"> <script>function mfTempOpenSection(id){var block=document.getElementById("mf-section-"+id);block.className+=" open-block";block.previousSibling.className+=" open-block";}</script> <div class="mw-content-ltr mw-parser-output" lang="sh-Latn" dir="ltr"> <section class="mf-section-0" id="mf-section-0"> Površ je dvoparametarski skup tačaka u <a href="https://sh-m-wikipedia-org.translate.goog/wiki/Prostor?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Prostor">prostoru</a>, tj. skup tačaka prostora čije su koordinate funkcije dva parametra u i v. Naprimjer, funkcije krivolinijskih koordinata tačke na površi. U ovom se pretpostavlja da ove funkcije imaju <a href="https://sh-m-wikipedia-org.translate.goog/wiki/Izvod?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Izvod">izvode</a> do nekog reda. Ako su u i v krivolinijske koordinate na površi, onda se površ može odrediti <a href="https://sh-m-wikipedia-org.translate.goog/wiki/Jedna%C4%8Dina?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Jednačina">jednačinama</a>: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=x(u,v),\;y=y(u,v),\;z=z(u,v),}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> x </mi> <mo> = </mo> <mi> x </mi> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mo> , </mo> <mspace width="thickmathspace"></mspace> <mi> y </mi> <mo> = </mo> <mi> y </mi> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mo> , </mo> <mspace width="thickmathspace"></mspace> <mi> z </mi> <mo> = </mo> <mi> z </mi> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle x=x(u,v),\;y=y(u,v),\;z=z(u,v),} </annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa8a1656973ee4810dff44f2e18c3660c3411bd5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.349ex; height:2.843ex;" alt="{\displaystyle x=x(u,v),\;y=y(u,v),\;z=z(u,v),}"> </dd> </dl> gdje su <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> x </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle x} </annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> y </mi> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle y,} </annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99bd9829c9ef4adcb0f9f5d53b27463a873a8e88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.802ex; height:2.009ex;" alt="{\displaystyle y,}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> z </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle z} </annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"> diferencijabilne skalarne funkcije. odnosno <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r(u,v)=x(u,v)i+y(u,v)j+z(u,v)k}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> r </mi> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mi> x </mi> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mi> i </mi> <mo> + </mo> <mi> y </mi> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mi> j </mi> <mo> + </mo> <mi> z </mi> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mi> k </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r(u,v)=x(u,v)i+y(u,v)j+z(u,v)k} </annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34070a624d52d7b2b396cec883401b52de9e6812" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.575ex; height:2.843ex;" alt="{\displaystyle r(u,v)=x(u,v)i+y(u,v)j+z(u,v)k}"> gdje su <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> x </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle x} </annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> y </mi> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle y,} </annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99bd9829c9ef4adcb0f9f5d53b27463a873a8e88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.802ex; height:2.009ex;" alt="{\displaystyle y,}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> z </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle z} </annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"> realne funkcije klase <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{1}(U)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> C </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msup> <mo stretchy="false"> ( </mo> <mi> U </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle C^{1}(U)} </annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e14c03a15fa07db857bf734188a60efa6136184b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.444ex; height:3.176ex;" alt="{\displaystyle C^{1}(U)}"> tj. imaju neprekidne prve parcijalne derivacije na <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> U </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle U} </annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}">. koje se nazivaju parametarske jednačine površi. Površ drugog reda je skup svih tacaka trodimenzionalnog prostora koje zadovoljavaju jednačinu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Ax^{2}+By^{2}+Cz^{2}+Dxy+Exz+Fyz+Gx+Hy+Iz+K=0}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> A </mi> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <mi> B </mi> <msup> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <mi> C </mi> <msup> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <mi> D </mi> <mi> x </mi> <mi> y </mi> <mo> + </mo> <mi> E </mi> <mi> x </mi> <mi> z </mi> <mo> + </mo> <mi> F </mi> <mi> y </mi> <mi> z </mi> <mo> + </mo> <mi> G </mi> <mi> x </mi> <mo> + </mo> <mi> H </mi> <mi> y </mi> <mo> + </mo> <mi> I </mi> <mi> z </mi> <mo> + </mo> <mi> K </mi> <mo> = </mo> <mn> 0 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle Ax^{2}+By^{2}+Cz^{2}+Dxy+Exz+Fyz+Gx+Hy+Iz+K=0} </annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8b215d3817aa4067193b65e71cb034ff297df1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:65.13ex; height:3.009ex;" alt="{\displaystyle Ax^{2}+By^{2}+Cz^{2}+Dxy+Exz+Fyz+Gx+Hy+Iz+K=0}"> za bar jedan <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,B,C,D,E\neq 0}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> A </mi> <mo> , </mo> <mi> B </mi> <mo> , </mo> <mi> C </mi> <mo> , </mo> <mi> D </mi> <mo> , </mo> <mi> E </mi> <mo> ≠ </mo> <mn> 0 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle A,B,C,D,E\neq 0} </annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5484c4b8196f5ca81ac54c435c08a55d889a7e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.37ex; height:2.676ex;" alt="{\displaystyle A,B,C,D,E\neq 0}"> tj. u formuli postoji barem jedan netrivijalni nelinearni član. <dl> <dt> primjer </dt> </dl> <a href="https://sh-m-wikipedia-org.translate.goog/wiki/Sfera?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Sfera">sfera</a> O(R) se može odrediti parametarskim jednačinama: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=R\cos u\cos v,\quad y=R\cos u\sin v,\quad z=R\sin u,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> x </mi> <mo> = </mo> <mi> R </mi> <mi> cos </mi> <mo> ⁡ </mo> <mi> u </mi> <mi> cos </mi> <mo> ⁡ </mo> <mi> v </mi> <mo> , </mo> <mspace width="1em"></mspace> <mi> y </mi> <mo> = </mo> <mi> R </mi> <mi> cos </mi> <mo> ⁡ </mo> <mi> u </mi> <mi> sin </mi> <mo> ⁡ </mo> <mi> v </mi> <mo> , </mo> <mspace width="1em"></mspace> <mi> z </mi> <mo> = </mo> <mi> R </mi> <mi> sin </mi> <mo> ⁡ </mo> <mi> u </mi> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle x=R\cos u\cos v,\quad y=R\cos u\sin v,\quad z=R\sin u,} </annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ddffbd51a22fb1644bcab638be41cad3c54bd63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:50.68ex; height:2.509ex;" alt="{\displaystyle x=R\cos u\cos v,\quad y=R\cos u\sin v,\quad z=R\sin u,}"> </dd> </dl> gdje je u širina, v dužina tačke na sferi. Eliminisanjem (isključenjem) u i v iz ovih jednačina dobija se poznata jednačina sfere: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+y^{2}+z^{2}=R^{2}.\,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <msup> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <msup> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> = </mo> <msup> <mi> R </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> . </mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle x^{2}+y^{2}+z^{2}=R^{2}.\,} </annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33f588b25794b1cb9eb9a3079c75864ceb21afdf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.375ex; height:3.009ex;" alt="{\displaystyle x^{2}+y^{2}+z^{2}=R^{2}.\,}"> </dd> </dl> Jednačina<a href="https://sh-m-wikipedia-org.translate.goog/wiki/Lopta?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Lopta"> sfere </a>(loptine površi) radijusa <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> r </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r} </annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"> s centrom u tački <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{0},y_{0},z_{0})}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> , </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> , </mo> <msub> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (x_{0},y_{0},z_{0})} </annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39177ddeeb9f9a393b664e522bc8e3bf0face153" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.59ex; height:2.843ex;" alt="{\displaystyle (x_{0},y_{0},z_{0})}"> data je sa <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \displaystyle (x-x_{0})^{2}+(y-y_{0})^{2}+(z-z_{0})^{2}=r^{2}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> − </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <mo stretchy="false"> ( </mo> <mi> y </mi> <mo> − </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <mo stretchy="false"> ( </mo> <mi> z </mi> <mo> − </mo> <msub> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> = </mo> <msup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \displaystyle (x-x_{0})^{2}+(y-y_{0})^{2}+(z-z_{0})^{2}=r^{2}} </annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fe3c1006772c8f083c1a2676562608cb46ee51c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.28ex; height:3.176ex;" alt="{\displaystyle \displaystyle (x-x_{0})^{2}+(y-y_{0})^{2}+(z-z_{0})^{2}=r^{2}}"> Ovom formulom su zadane dvije<a href="https://sh-m-wikipedia-org.translate.goog/wiki/Funkcija?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Funkcija"> funkcije</a> dvije varijable: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=f_{1}(x,y)=z_{0}+{\sqrt {r^{2}-(x-x_{0})^{2}-(y-y_{0})^{2}}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> z </mi> <mo> = </mo> <msub> <mi> f </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> , </mo> <mi> y </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <msub> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> − </mo> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> − </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> − </mo> <mo stretchy="false"> ( </mo> <mi> y </mi> <mo> − </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle z=f_{1}(x,y)=z_{0}+{\sqrt {r^{2}-(x-x_{0})^{2}-(y-y_{0})^{2}}}} </annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a80c3bc197a20d04748a7bcf5601cd67f1c8bffa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:48.361ex; height:4.843ex;" alt="{\displaystyle z=f_{1}(x,y)=z_{0}+{\sqrt {r^{2}-(x-x_{0})^{2}-(y-y_{0})^{2}}}}"> </dd> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=f_{2}(x,y)=z_{0}-{\sqrt {r^{2}-(x-x_{0})^{2}-(y-y_{0})^{2}}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> z </mi> <mo> = </mo> <msub> <mi> f </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> , </mo> <mi> y </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <msub> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> − </mo> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> − </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> − </mo> <mo stretchy="false"> ( </mo> <mi> y </mi> <mo> − </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle z=f_{2}(x,y)=z_{0}-{\sqrt {r^{2}-(x-x_{0})^{2}-(y-y_{0})^{2}}}} </annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da236ce5a206e0bdbf6f8994b66dc3a28bb640a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:48.361ex; height:4.843ex;" alt="{\displaystyle z=f_{2}(x,y)=z_{0}-{\sqrt {r^{2}-(x-x_{0})^{2}-(y-y_{0})^{2}}}}"> </dd> </dl> Nivo-površi sfere (presjeci s <a href="https://sh-m-wikipedia-org.translate.goog/wiki/Ravan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ravan">ravnima paralelnim</a> s <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xy}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> x </mi> <mi> y </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle xy} </annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c72eb345e496513fb8b2fa4aa8c4d89b855f9a01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.485ex; height:2.009ex;" alt="{\displaystyle xy}"> ravni) i presjeci s ravnima paralelnim s <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xz}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> x </mi> <mi> z </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle xz} </annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97005bee6e83614cf6ce64d4e68e5ab2ac280709" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.418ex; height:1.676ex;" alt="{\displaystyle xz}"> i <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle yz}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> y </mi> <mi> z </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle yz} </annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9c7cbe4344a9d72acfc98a97e2b3ec44bb48d1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.244ex; height:2.009ex;" alt="{\displaystyle yz}"> ravnima su <a href="https://sh-m-wikipedia-org.translate.goog/w/index.php?title=Kruznica&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Kruznica (stranica ne postoji)">kruznice</a>. Jednačina površi se može zadati i u drugim oblicima, naprimjer, u obliku: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x,y,z)=0,\,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> f </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> , </mo> <mi> y </mi> <mo> , </mo> <mi> z </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mn> 0 </mn> <mo> , </mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle f(x,y,z)=0,\,} </annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c05e392ae3113b911e829e715be0da5cffb01897" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.024ex; height:2.843ex;" alt="{\displaystyle f(x,y,z)=0,\,}"> ili <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=f(x,y).\,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> z </mi> <mo> = </mo> <mi> f </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> , </mo> <mi> y </mi> <mo stretchy="false"> ) </mo> <mo> . </mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle z=f(x,y).\,} </annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ccae6558d4e7bda59eb34e2014e934e4e049d70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.828ex; height:2.843ex;" alt="{\displaystyle z=f(x,y).\,}"> </dd> </dl> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"> <input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none"> <div class="toctitle" lang="sh-Latn" dir="ltr"> <h2 id="mw-toc-heading">Sadržaj</h2><label class="toctogglelabel" for="toctogglecheckbox"></label> </div> <ul> <li class="toclevel-1 tocsection-1"><a href="https://sh-m-wikipedia-org.translate.goog/wiki/Povr%C5%A1?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Regularne_i_singularne_ta%C4%8Dke_povr%C5%A1i">1 Regularne i singularne tačke površi</a></li> <li class="toclevel-1 tocsection-2"><a href="https://sh-m-wikipedia-org.translate.goog/wiki/Povr%C5%A1?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Krivolinijski_ili_Gaussov_koordinatni_sistem_na_povr%C5%A1">2 Krivolinijski ili Gaussov koordinatni sistem na površ</a></li> <li class="toclevel-1 tocsection-3"><a href="https://sh-m-wikipedia-org.translate.goog/wiki/Povr%C5%A1?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Eksplicitna_jedna%C4%8Dina_povr%C5%A1i">3 Eksplicitna jednačina površi</a></li> <li class="toclevel-1 tocsection-4"><a href="https://sh-m-wikipedia-org.translate.goog/wiki/Povr%C5%A1?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Implicitna_jedna%C4%8Dina_povr%C5%A1i">4 Implicitna jednačina površi</a></li> <li class="toclevel-1 tocsection-5"><a href="https://sh-m-wikipedia-org.translate.goog/wiki/Povr%C5%A1?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Tangentna_ravan_i_normala_na_povr%C5%A1">5 Tangentna ravan i normala na površ</a></li> <li class="toclevel-1 tocsection-6"><a href="https://sh-m-wikipedia-org.translate.goog/wiki/Povr%C5%A1?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Linijske_povr%C5%A1i">6 Linijske površi</a></li> <li class="toclevel-1 tocsection-7"><a href="https://sh-m-wikipedia-org.translate.goog/wiki/Povr%C5%A1?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Elipsoid">7 Elipsoid</a></li> <li class="toclevel-1 tocsection-8"><a href="https://sh-m-wikipedia-org.translate.goog/wiki/Povr%C5%A1?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Hiperboloid">8 Hiperboloid</a></li> <li class="toclevel-1 tocsection-9"><a href="https://sh-m-wikipedia-org.translate.goog/wiki/Povr%C5%A1?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Konusne_povr%C5%A1i">9 Konusne površi</a></li> <li class="toclevel-1 tocsection-10"><a href="https://sh-m-wikipedia-org.translate.goog/wiki/Povr%C5%A1?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Valjkaste_povr%C5%A1i">10 Valjkaste površi</a></li> <li class="toclevel-1 tocsection-11"><a href="https://sh-m-wikipedia-org.translate.goog/wiki/Povr%C5%A1?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Rotacione_povr%C5%A1i">11 Rotacione površi</a></li> <li class="toclevel-1 tocsection-12"><a href="https://sh-m-wikipedia-org.translate.goog/wiki/Povr%C5%A1?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Izvori">12 Izvori</a></li> </ul> </div> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(1)"> <h2 id="Regularne_i_singularne_tačke_površi">Regularne i singularne tačke površi</h2> <a role="button" href="https://sh-m-wikipedia-org.translate.goog/w/index.php?title=Povr%C5%A1&action=edit&section=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Uredi odjeljak Regularne i singularne tačke površi" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> uredi </a> </div> <section class="mf-section-1 collapsible-block" id="mf-section-1"> Parcijalne derivacije vektorske funkcije <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r(u,v)=x(u,v)i+y(u,v)j+z(u,v)k}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> r </mi> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mi> x </mi> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mi> i </mi> <mo> + </mo> <mi> y </mi> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mi> j </mi> <mo> + </mo> <mi> z </mi> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mi> k </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r(u,v)=x(u,v)i+y(u,v)j+z(u,v)k} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34070a624d52d7b2b396cec883401b52de9e6812" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.575ex; height:2.843ex;" alt="{\displaystyle r(u,v)=x(u,v)i+y(u,v)j+z(u,v)k}"> </noscript><span class="lazy-image-placeholder" style="width: 37.575ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34070a624d52d7b2b396cec883401b52de9e6812" data-alt="{\displaystyle r(u,v)=x(u,v)i+y(u,v)j+z(u,v)k}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> ) su, prema pretpostavci, neprekidne vektorske funkcije <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{u},r_{v}:U\to R^{3}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> u </mi> </mrow> </msub> <mo> , </mo> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mo> : </mo> <mi> U </mi> <mo stretchy="false"> → </mo> <msup> <mi> R </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r_{u},r_{v}:U\to R^{3}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4de6aebc3c87dbec52290aa5417a00b7c62caf16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.485ex; height:3.009ex;" alt="{\displaystyle r_{u},r_{v}:U\to R^{3}}"> </noscript><span class="lazy-image-placeholder" style="width: 15.485ex;height: 3.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4de6aebc3c87dbec52290aa5417a00b7c62caf16" data-alt="{\displaystyle r_{u},r_{v}:U\to R^{3}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  date formulama: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{u}(u,v)=x_{u}(u,v)i+y_{u}(u,v)j+z_{u}(u,v)k}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> u </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> u </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mi> i </mi> <mo> + </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> u </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mi> j </mi> <mo> + </mo> <msub> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> u </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mi> k </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r_{u}(u,v)=x_{u}(u,v)i+y_{u}(u,v)j+z_{u}(u,v)k} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/093f87b79571885798afe072afb224c68ec2f3c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.242ex; height:2.843ex;" alt="{\displaystyle r_{u}(u,v)=x_{u}(u,v)i+y_{u}(u,v)j+z_{u}(u,v)k}"> </noscript><span class="lazy-image-placeholder" style="width: 42.242ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/093f87b79571885798afe072afb224c68ec2f3c0" data-alt="{\displaystyle r_{u}(u,v)=x_{u}(u,v)i+y_{u}(u,v)j+z_{u}(u,v)k}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{v}(u,v)=x_{v}(u,v)i+y_{v}(u,v)j+z_{v}(u,v)k}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mi> i </mi> <mo> + </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mi> j </mi> <mo> + </mo> <msub> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mi> k </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r_{v}(u,v)=x_{v}(u,v)i+y_{v}(u,v)j+z_{v}(u,v)k} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b04a3ce7076946689e61c92605bb84721c110da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.67ex; height:2.843ex;" alt="{\displaystyle r_{v}(u,v)=x_{v}(u,v)i+y_{v}(u,v)j+z_{v}(u,v)k}"> </noscript><span class="lazy-image-placeholder" style="width: 41.67ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b04a3ce7076946689e61c92605bb84721c110da" data-alt="{\displaystyle r_{v}(u,v)=x_{v}(u,v)i+y_{v}(u,v)j+z_{v}(u,v)k}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Jacobijeva matrica parametrizacije <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (U,r)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <mi> U </mi> <mo> , </mo> <mi> r </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (U,r)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5532c8f8d85bb9de3cf6ac04619290d6bc122b3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.674ex; height:2.843ex;" alt="{\displaystyle (U,r)}"> </noscript><span class="lazy-image-placeholder" style="width: 5.674ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5532c8f8d85bb9de3cf6ac04619290d6bc122b3e" data-alt="{\displaystyle (U,r)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  je matrica oblika: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {J}}(r)={\begin{bmatrix}r_{u}(u,v\\r_{v}(u,v\\\end{bmatrix}}={\begin{bmatrix}x_{u}(u,v)\ y_{u}(u,v)\ z_{u}(u,v)\\x_{v}(u,v)\ y_{v}(u,v)\ z_{v}(u,v)\end{bmatrix}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur"> J </mi> </mrow> </mrow> <mo stretchy="false"> ( </mo> <mi> r </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> [ </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> u </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> </mtd> </mtr> </mtable> <mo> ] </mo> </mrow> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> [ </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> u </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mtext>   </mtext> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> u </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mtext>   </mtext> <msub> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> u </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mtext>   </mtext> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mtext>   </mtext> <msub> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> </mtd> </mtr> </mtable> <mo> ] </mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\mathfrak {J}}(r)={\begin{bmatrix}r_{u}(u,v\\r_{v}(u,v\\\end{bmatrix}}={\begin{bmatrix}x_{u}(u,v)\ y_{u}(u,v)\ z_{u}(u,v)\\x_{v}(u,v)\ y_{v}(u,v)\ z_{v}(u,v)\end{bmatrix}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/645f9c6083db4685f13ce05ab34becacff16868a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; margin-left: -0.022ex; width:47.521ex; height:6.176ex;" alt="{\displaystyle {\mathfrak {J}}(r)={\begin{bmatrix}r_{u}(u,v\\r_{v}(u,v\\\end{bmatrix}}={\begin{bmatrix}x_{u}(u,v)\ y_{u}(u,v)\ z_{u}(u,v)\\x_{v}(u,v)\ y_{v}(u,v)\ z_{v}(u,v)\end{bmatrix}}}"> </noscript><span class="lazy-image-placeholder" style="width: 47.521ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/645f9c6083db4685f13ce05ab34becacff16868a" data-alt="{\displaystyle {\mathfrak {J}}(r)={\begin{bmatrix}r_{u}(u,v\\r_{v}(u,v\\\end{bmatrix}}={\begin{bmatrix}x_{u}(u,v)\ y_{u}(u,v)\ z_{u}(u,v)\\x_{v}(u,v)\ y_{v}(u,v)\ z_{v}(u,v)\end{bmatrix}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Sljedeće četiri tvrdnje su ekvivalentne: <ol> <li>Vektori <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{u}(u,v)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> u </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r_{u}(u,v)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cce536584293b18b52da4849d7d58964ab4f0832" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.522ex; height:2.843ex;" alt="{\displaystyle r_{u}(u,v)}"> </noscript><span class="lazy-image-placeholder" style="width: 7.522ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cce536584293b18b52da4849d7d58964ab4f0832" data-alt="{\displaystyle r_{u}(u,v)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  i <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{v}(u,v)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r_{v}(u,v)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7a92f3512403a37e6fe3b9e3a83efffef36a899" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.379ex; height:2.843ex;" alt="{\displaystyle r_{v}(u,v)}"> </noscript><span class="lazy-image-placeholder" style="width: 7.379ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7a92f3512403a37e6fe3b9e3a83efffef36a899" data-alt="{\displaystyle r_{v}(u,v)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  linearno su nezavisni.</li> <li>r_u(u, v) × r_v(u, v)≠ 0</li> <li>Matrica <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {J}}(r)(u,v)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur"> J </mi> </mrow> </mrow> <mo stretchy="false"> ( </mo> <mi> r </mi> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\mathfrak {J}}(r)(u,v)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c08aaacd2eb3dcd8a3e16f9f7a37a9c7aa6715a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.022ex; width:9.464ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {J}}(r)(u,v)}"> </noscript><span class="lazy-image-placeholder" style="width: 9.464ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c08aaacd2eb3dcd8a3e16f9f7a37a9c7aa6715a2" data-alt="{\displaystyle {\mathfrak {J}}(r)(u,v)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  je ranga 2.</li> <li>Barem jedna od funkcijskih determinanti</li> </ol> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{vmatrix}x_{u}(u,v)\ y_{u}(u,v)\\x_{v}(u,v)\ y_{v}(u,v\end{vmatrix}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> | </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> u </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mtext>   </mtext> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> u </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mtext>   </mtext> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> </mtd> </mtr> </mtable> <mo> | </mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{vmatrix}x_{u}(u,v)\ y_{u}(u,v)\\x_{v}(u,v)\ y_{v}(u,v\end{vmatrix}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/336b70291d51a67a1e2f45cbebd5458690f42b9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.041ex; height:6.176ex;" alt="{\displaystyle {\begin{vmatrix}x_{u}(u,v)\ y_{u}(u,v)\\x_{v}(u,v)\ y_{v}(u,v\end{vmatrix}}}"> </noscript><span class="lazy-image-placeholder" style="width: 18.041ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/336b70291d51a67a1e2f45cbebd5458690f42b9f" data-alt="{\displaystyle {\begin{vmatrix}x_{u}(u,v)\ y_{u}(u,v)\\x_{v}(u,v)\ y_{v}(u,v\end{vmatrix}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{vmatrix}x_{u}(u,v)\ z_{u}(u,v)\\x_{v}(u,v)\ z_{v}(u,v\end{vmatrix}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> | </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> u </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mtext>   </mtext> <msub> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> u </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mtext>   </mtext> <msub> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> </mtd> </mtr> </mtable> <mo> | </mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{vmatrix}x_{u}(u,v)\ z_{u}(u,v)\\x_{v}(u,v)\ z_{v}(u,v\end{vmatrix}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de5bca78b58c3b3d2d2289d5e805844d29cf8c4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.983ex; height:6.176ex;" alt="{\displaystyle {\begin{vmatrix}x_{u}(u,v)\ z_{u}(u,v)\\x_{v}(u,v)\ z_{v}(u,v\end{vmatrix}}}"> </noscript><span class="lazy-image-placeholder" style="width: 17.983ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de5bca78b58c3b3d2d2289d5e805844d29cf8c4c" data-alt="{\displaystyle {\begin{vmatrix}x_{u}(u,v)\ z_{u}(u,v)\\x_{v}(u,v)\ z_{v}(u,v\end{vmatrix}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{vmatrix}y_{u}(u,v)\ z_{u}(u,v)\\y_{v}(u,v)\ z_{v}(u,v\end{vmatrix}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> | </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> u </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mtext>   </mtext> <msub> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> u </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mtext>   </mtext> <msub> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> </mtd> </mtr> </mtable> <mo> | </mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{vmatrix}y_{u}(u,v)\ z_{u}(u,v)\\y_{v}(u,v)\ z_{v}(u,v\end{vmatrix}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d094443c125e5f190eff4af87affc11b6557513b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.792ex; height:6.176ex;" alt="{\displaystyle {\begin{vmatrix}y_{u}(u,v)\ z_{u}(u,v)\\y_{v}(u,v)\ z_{v}(u,v\end{vmatrix}}}"> </noscript><span class="lazy-image-placeholder" style="width: 17.792ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d094443c125e5f190eff4af87affc11b6557513b" data-alt="{\displaystyle {\begin{vmatrix}y_{u}(u,v)\ z_{u}(u,v)\\y_{v}(u,v)\ z_{v}(u,v\end{vmatrix}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  je različita od nule. Za tačku T površi F koja odgovara uređenom paru <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (u_{0},v_{0})}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> , </mo> <msub> <mi> v </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (u_{0},v_{0})} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b729d790fa6e280ed610be112ba3348172aec62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.409ex; height:2.843ex;" alt="{\displaystyle (u_{0},v_{0})}"> </noscript><span class="lazy-image-placeholder" style="width: 7.409ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b729d790fa6e280ed610be112ba3348172aec62" data-alt="{\displaystyle (u_{0},v_{0})}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  kažemo da je regularna tačka parametrizacije <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (U,r)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <mi> U </mi> <mo> , </mo> <mi> r </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (U,r)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5532c8f8d85bb9de3cf6ac04619290d6bc122b3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.674ex; height:2.843ex;" alt="{\displaystyle (U,r)}"> </noscript><span class="lazy-image-placeholder" style="width: 5.674ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5532c8f8d85bb9de3cf6ac04619290d6bc122b3e" data-alt="{\displaystyle (U,r)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  ako je r_u(u_0, v_0) × r_v(u_0, v_0)≠ 0 Za tačku T površi F koja odgovara uređenom paru <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (u_{0},v_{0})}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> , </mo> <msub> <mi> v </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (u_{0},v_{0})} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b729d790fa6e280ed610be112ba3348172aec62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.409ex; height:2.843ex;" alt="{\displaystyle (u_{0},v_{0})}"> </noscript><span class="lazy-image-placeholder" style="width: 7.409ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b729d790fa6e280ed610be112ba3348172aec62" data-alt="{\displaystyle (u_{0},v_{0})}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  kažemo da je singularna tačka parametrizacije <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (U,r)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <mi> U </mi> <mo> , </mo> <mi> r </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (U,r)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5532c8f8d85bb9de3cf6ac04619290d6bc122b3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.674ex; height:2.843ex;" alt="{\displaystyle (U,r)}"> </noscript><span class="lazy-image-placeholder" style="width: 5.674ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5532c8f8d85bb9de3cf6ac04619290d6bc122b3e" data-alt="{\displaystyle (U,r)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  ako je r_u(u_0, v_0) × r_v(u_0, v_0)= 0 Neka površ <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {F}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur"> F </mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\mathfrak {F}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec863025da8f3844272a146530208eb001c3dddb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.423ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {F}}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.423ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec863025da8f3844272a146530208eb001c3dddb" data-alt="{\displaystyle {\mathfrak {F}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  može imati više različitih parametrizacija. Tačka površi koja je singularna za jednu parametrizaciju nemora biti singularna i za ostale njezine parametrizacije. Za površ <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {F}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur"> F </mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\mathfrak {F}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec863025da8f3844272a146530208eb001c3dddb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.423ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {F}}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.423ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec863025da8f3844272a146530208eb001c3dddb" data-alt="{\displaystyle {\mathfrak {F}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  kažemo da je regularna ako svaka njezina tačka ima u <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {F}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur"> F </mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\mathfrak {F}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec863025da8f3844272a146530208eb001c3dddb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.423ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {F}}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.423ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec863025da8f3844272a146530208eb001c3dddb" data-alt="{\displaystyle {\mathfrak {F}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  okolinu s regularnom parametrizacijom. Za tačku <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\in {\mathfrak {F}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> S </mi> <mo> ∈ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur"> F </mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle S\in {\mathfrak {F}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5d4ed3ab55239781a2f72c597466bd938f9d611" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.762ex; height:2.509ex;" alt="{\displaystyle S\in {\mathfrak {F}}}"> </noscript><span class="lazy-image-placeholder" style="width: 5.762ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5d4ed3ab55239781a2f72c597466bd938f9d611" data-alt="{\displaystyle S\in {\mathfrak {F}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  kažemo da je singularna tačka površi ako je ona singularna tačka svake njene parametrizacije. Sfera je primjer regularne površi koja se ne može pokriti jednom regularnom parametrizacijom. Standardna parametrizacija sfere poluprečnika <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> r </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"> </noscript><span class="lazy-image-placeholder" style="width: 1.049ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" data-alt="{\displaystyle r}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  je <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (u,v)\to r(cosusinv,sinusinv,cosv)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mo stretchy="false"> → </mo> <mi> r </mi> <mo stretchy="false"> ( </mo> <mi> c </mi> <mi> o </mi> <mi> s </mi> <mi> u </mi> <mi> s </mi> <mi> i </mi> <mi> n </mi> <mi> v </mi> <mo> , </mo> <mi> s </mi> <mi> i </mi> <mi> n </mi> <mi> u </mi> <mi> s </mi> <mi> i </mi> <mi> n </mi> <mi> v </mi> <mo> , </mo> <mi> c </mi> <mi> o </mi> <mi> s </mi> <mi> v </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (u,v)\to r(cosusinv,sinusinv,cosv)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f98cb4eebac14784a1e22fd073571f21d1d46ec0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.195ex; height:2.843ex;" alt="{\displaystyle (u,v)\to r(cosusinv,sinusinv,cosv)}"> </noscript><span class="lazy-image-placeholder" style="width: 36.195ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f98cb4eebac14784a1e22fd073571f21d1d46ec0" data-alt="{\displaystyle (u,v)\to r(cosusinv,sinusinv,cosv)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  gdje je <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (u,v)\in [0,2\pi ]}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mo> ∈ </mo> <mo stretchy="false"> [ </mo> <mn> 0 </mn> <mo> , </mo> <mn> 2 </mn> <mi> π </mi> <mo stretchy="false"> ] </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (u,v)\in [0,2\pi ]} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2bf31652846f3049a2337f007836f8449ec0b0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.126ex; height:2.843ex;" alt="{\displaystyle (u,v)\in [0,2\pi ]}"> </noscript><span class="lazy-image-placeholder" style="width: 14.126ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2bf31652846f3049a2337f007836f8449ec0b0f" data-alt="{\displaystyle (u,v)\in [0,2\pi ]}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  × <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [-\pi /2,\pi /2]}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> [ </mo> <mo> − </mo> <mi> π </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 2 </mn> <mo> , </mo> <mi> π </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 2 </mn> <mo stretchy="false"> ] </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle [-\pi /2,\pi /2]} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd702a5a7041be010f870c0e23750d98ba9919f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.45ex; height:2.843ex;" alt="{\displaystyle [-\pi /2,\pi /2]}"> </noscript><span class="lazy-image-placeholder" style="width: 11.45ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd702a5a7041be010f870c0e23750d98ba9919f5" data-alt="{\displaystyle [-\pi /2,\pi /2]}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Pri toj parametrizaciji u-krive (v je konstanta) nazivamo paralelama, a v-krive (u je konstanta) meridijanima. Polovi, tj. tačke <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,0,\pm r)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <mn> 0 </mn> <mo> , </mo> <mn> 0 </mn> <mo> , </mo> <mo> ± </mo> <mi> r </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (0,0,\pm r)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8893c496c6b0a6e50aa3838ad140490cadd38f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.059ex; height:2.843ex;" alt="{\displaystyle (0,0,\pm r)}"> </noscript><span class="lazy-image-placeholder" style="width: 9.059ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8893c496c6b0a6e50aa3838ad140490cadd38f6" data-alt="{\displaystyle (0,0,\pm r)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , singularne su tačke te parametrizacije. Međutim, svaka se sfera može pokriti već s dvije regularne parametrizacije. U sigularnoj tački površ samu sebe siječe, dodiruje i sl. Ako su sve toačke neke krive na površi singularne, onda takvu liniju nazivamo singularnom linijom površi. </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(2)"> <h2 id="Krivolinijski_ili_Gaussov_koordinatni_sistem_na_površ">Krivolinijski ili Gaussov koordinatni sistem na površ</h2> <a role="button" href="https://sh-m-wikipedia-org.translate.goog/w/index.php?title=Povr%C5%A1&action=edit&section=2&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Uredi odjeljak Krivolinijski ili Gaussov koordinatni sistem na površ" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> uredi </a> </div> <section class="mf-section-2 collapsible-block" id="mf-section-2"> Ako se u jednačinama <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=x(u,v)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> x </mi> <mo> = </mo> <mi> x </mi> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle x=x(u,v)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fdd605515e00cc0a45af42ba3df16046defac9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.058ex; height:2.843ex;" alt="{\displaystyle x=x(u,v)}"> </noscript><span class="lazy-image-placeholder" style="width: 11.058ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fdd605515e00cc0a45af42ba3df16046defac9b" data-alt="{\displaystyle x=x(u,v)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=y(u,v)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> y </mi> <mo> = </mo> <mi> y </mi> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle y=y(u,v)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dc932826f9f3a969aee6da3cf579aad8966ffb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.71ex; height:2.843ex;" alt="{\displaystyle y=y(u,v)}"> </noscript><span class="lazy-image-placeholder" style="width: 10.71ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dc932826f9f3a969aee6da3cf579aad8966ffb8" data-alt="{\displaystyle y=y(u,v)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=z(u,v)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> z </mi> <mo> = </mo> <mi> z </mi> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle z=z(u,v)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e263aebd2eeba9e55fd1072eb87e251125d01faf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.575ex; height:2.843ex;" alt="{\displaystyle z=z(u,v)}"> </noscript><span class="lazy-image-placeholder" style="width: 10.575ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e263aebd2eeba9e55fd1072eb87e251125d01faf" data-alt="{\displaystyle z=z(u,v)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  za jedan parametar uzme konstanta, dok drugi mijenja vrijednosti unutar područja <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> U </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle U} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"> </noscript><span class="lazy-image-placeholder" style="width: 1.783ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" data-alt="{\displaystyle U}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , parametarski je zadana prostorna kriva koja leži na zadanoj površi. Tako je za <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v=v_{0}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> v </mi> <mo> = </mo> <msub> <mi> v </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle v=v_{0}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/600f1fced4041372dcb68ef2f69706559e45c024" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.408ex; height:2.009ex;" alt="{\displaystyle v=v_{0}}"> </noscript><span class="lazy-image-placeholder" style="width: 6.408ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/600f1fced4041372dcb68ef2f69706559e45c024" data-alt="{\displaystyle v=v_{0}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  jednačina <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=x(u,v_{0})}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> x </mi> <mo> = </mo> <mi> x </mi> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <msub> <mi> v </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle x=x(u,v_{0})} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbdd7f12cb71063563742a47262536c3a03bb1a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.113ex; height:2.843ex;" alt="{\displaystyle x=x(u,v_{0})}"> </noscript><span class="lazy-image-placeholder" style="width: 12.113ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbdd7f12cb71063563742a47262536c3a03bb1a8" data-alt="{\displaystyle x=x(u,v_{0})}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=y(u,v_{0})}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> y </mi> <mo> = </mo> <mi> y </mi> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <msub> <mi> v </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle y=y(u,v_{0})} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae161bdd3192edb6fec661f97eeb805a424efe91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.764ex; height:2.843ex;" alt="{\displaystyle y=y(u,v_{0})}"> </noscript><span class="lazy-image-placeholder" style="width: 11.764ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae161bdd3192edb6fec661f97eeb805a424efe91" data-alt="{\displaystyle y=y(u,v_{0})}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=z(u,v_{0})}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> z </mi> <mo> = </mo> <mi> z </mi> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <msub> <mi> v </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle z=z(u,v_{0})} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/665755d407fee5cfcddf45a317ee8f846cddbbc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.63ex; height:2.843ex;" alt="{\displaystyle z=z(u,v_{0})}"> </noscript><span class="lazy-image-placeholder" style="width: 11.63ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/665755d407fee5cfcddf45a317ee8f846cddbbc3" data-alt="{\displaystyle z=z(u,v_{0})}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  parametarski zadana tzv. u − kriva površi, a za <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u=u_{0}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> u </mi> <mo> = </mo> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle u=u_{0}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2ffea3ef298f4e8ccd41b4c77b931c7e9ad356f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.812ex; height:2.009ex;" alt="{\displaystyle u=u_{0}}"> </noscript><span class="lazy-image-placeholder" style="width: 6.812ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2ffea3ef298f4e8ccd41b4c77b931c7e9ad356f" data-alt="{\displaystyle u=u_{0}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  jednačina <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=x(u_{0},v)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> x </mi> <mo> = </mo> <mi> x </mi> <mo stretchy="false"> ( </mo> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle x=x(u_{0},v)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b501b1ef3e2b5132608ee59fd7124684bd8105f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.113ex; height:2.843ex;" alt="{\displaystyle x=x(u_{0},v)}"> </noscript><span class="lazy-image-placeholder" style="width: 12.113ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b501b1ef3e2b5132608ee59fd7124684bd8105f" data-alt="{\displaystyle x=x(u_{0},v)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=y(u_{0},v)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> y </mi> <mo> = </mo> <mi> y </mi> <mo stretchy="false"> ( </mo> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle y=y(u_{0},v)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b676fee647c42706d9e8f19a62d7a4f743fcbec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.764ex; height:2.843ex;" alt="{\displaystyle y=y(u_{0},v)}"> </noscript><span class="lazy-image-placeholder" style="width: 11.764ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b676fee647c42706d9e8f19a62d7a4f743fcbec" data-alt="{\displaystyle y=y(u_{0},v)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=z(u_{0},v)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> z </mi> <mo> = </mo> <mi> z </mi> <mo stretchy="false"> ( </mo> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle z=z(u_{0},v)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93df0f101e659fba9abc8b7f9bc77b2423d6554a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.63ex; height:2.843ex;" alt="{\displaystyle z=z(u_{0},v)}"> </noscript><span class="lazy-image-placeholder" style="width: 11.63ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93df0f101e659fba9abc8b7f9bc77b2423d6554a" data-alt="{\displaystyle z=z(u_{0},v)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  parametarski je zadana tzv. v − kriva površi. Na taj način će za različite konstante <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u=u_{i}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> u </mi> <mo> = </mo> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle u=u_{i}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9c54f001c21f6e35b360a81ee2105cf274da807" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.558ex; height:2.009ex;" alt="{\displaystyle u=u_{i}}"> </noscript><span class="lazy-image-placeholder" style="width: 6.558ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9c54f001c21f6e35b360a81ee2105cf274da807" data-alt="{\displaystyle u=u_{i}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v=v_{k}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> v </mi> <mo> = </mo> <msub> <mi> v </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle v=v_{k}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88d98846e052707be81e4a35af17d91d6a3f6da7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.442ex; height:2.009ex;" alt="{\displaystyle v=v_{k}}"> </noscript><span class="lazy-image-placeholder" style="width: 6.442ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88d98846e052707be81e4a35af17d91d6a3f6da7" data-alt="{\displaystyle v=v_{k}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , ( <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i,k\in R}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> i </mi> <mo> , </mo> <mi> k </mi> <mo> ∈ </mo> <mi> R </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle i,k\in R} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4097cdebd11c4ce06ad07644e397b5c912600dcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.652ex; height:2.509ex;" alt="{\displaystyle i,k\in R}"> </noscript><span class="lazy-image-placeholder" style="width: 7.652ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4097cdebd11c4ce06ad07644e397b5c912600dcf" data-alt="{\displaystyle i,k\in R}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> ) na zadanoj površi nastati dva sistema prostornih krivi pri čemu svaka kriva jednog sistema siječe svaku krivu drugog sistema u jednoj i samo jednoj toački. Svaka tačka na površii biće određena sjecištem dviju prostornih kriviiz različitih sistema. Takve krive nazivamo koordinatnim ili parametarskim krivama površi. Odabirom po jedne krive iz svakog sistema za koordinantne ose, a njihovog sjecišta za ishodište, uspostavlja se krivolinijski ili Gaussov koordinatni sistem na površi. Svakoj tački površi pridružena su dva realna broja <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{0}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle u_{0}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c7425f9c7ab645587060423c0af62f8a61fbc65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle u_{0}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.384ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c7425f9c7ab645587060423c0af62f8a61fbc65" data-alt="{\displaystyle u_{0}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  i <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{0}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> v </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle v_{0}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60faad24775635f4722ccc438093dbbfe05f34ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.182ex; height:2.009ex;" alt="{\displaystyle v_{0}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.182ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60faad24775635f4722ccc438093dbbfe05f34ae" data-alt="{\displaystyle v_{0}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , tzv. krivolinijske ili Gaussove koordinate tačke, koje određuju krive prvog i drugog sistema koje se sijeku u toj tački. Prema pretpostavci, funkcije iz jednačine <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=x(u,v)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> x </mi> <mo> = </mo> <mi> x </mi> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle x=x(u,v)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fdd605515e00cc0a45af42ba3df16046defac9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.058ex; height:2.843ex;" alt="{\displaystyle x=x(u,v)}"> </noscript><span class="lazy-image-placeholder" style="width: 11.058ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fdd605515e00cc0a45af42ba3df16046defac9b" data-alt="{\displaystyle x=x(u,v)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=y(u,v)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> y </mi> <mo> = </mo> <mi> y </mi> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle y=y(u,v)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dc932826f9f3a969aee6da3cf579aad8966ffb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.71ex; height:2.843ex;" alt="{\displaystyle y=y(u,v)}"> </noscript><span class="lazy-image-placeholder" style="width: 10.71ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dc932826f9f3a969aee6da3cf579aad8966ffb8" data-alt="{\displaystyle y=y(u,v)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=z(u,v)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> z </mi> <mo> = </mo> <mi> z </mi> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle z=z(u,v)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e263aebd2eeba9e55fd1072eb87e251125d01faf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.575ex; height:2.843ex;" alt="{\displaystyle z=z(u,v)}"> </noscript><span class="lazy-image-placeholder" style="width: 10.575ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e263aebd2eeba9e55fd1072eb87e251125d01faf" data-alt="{\displaystyle z=z(u,v)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  imaju neprekidne prve parcijalne derivacije po <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> u </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle u} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}"> </noscript><span class="lazy-image-placeholder" style="width: 1.33ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" data-alt="{\displaystyle u}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  i po <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> v </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle v} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"> </noscript><span class="lazy-image-placeholder" style="width: 1.128ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" data-alt="{\displaystyle v}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , koordinatne krive u svakoj svojoj tački imaju tangentu. Vektori <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{u}(u,v)=x_{u}(u,v)i+y_{u}(u,v)j+z_{u}(u,v)k}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> u </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> u </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mi> i </mi> <mo> + </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> u </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mi> j </mi> <mo> + </mo> <msub> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> u </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mi> k </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r_{u}(u,v)=x_{u}(u,v)i+y_{u}(u,v)j+z_{u}(u,v)k} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/093f87b79571885798afe072afb224c68ec2f3c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.242ex; height:2.843ex;" alt="{\displaystyle r_{u}(u,v)=x_{u}(u,v)i+y_{u}(u,v)j+z_{u}(u,v)k}"> </noscript><span class="lazy-image-placeholder" style="width: 42.242ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/093f87b79571885798afe072afb224c68ec2f3c0" data-alt="{\displaystyle r_{u}(u,v)=x_{u}(u,v)i+y_{u}(u,v)j+z_{u}(u,v)k}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{v}(u,v)=x_{v}(u,v)i+y_{v}(u,v)j+z_{v}(u,v)k}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mi> i </mi> <mo> + </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mi> j </mi> <mo> + </mo> <msub> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mi> k </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r_{v}(u,v)=x_{v}(u,v)i+y_{v}(u,v)j+z_{v}(u,v)k} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b04a3ce7076946689e61c92605bb84721c110da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.67ex; height:2.843ex;" alt="{\displaystyle r_{v}(u,v)=x_{v}(u,v)i+y_{v}(u,v)j+z_{v}(u,v)k}"> </noscript><span class="lazy-image-placeholder" style="width: 41.67ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b04a3ce7076946689e61c92605bb84721c110da" data-alt="{\displaystyle r_{v}(u,v)=x_{v}(u,v)i+y_{v}(u,v)j+z_{v}(u,v)k}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  vektori su tangenata koordinatnih krivi. Njihove su dužine: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |ru|={\sqrt {(x_{u})^{2}+(y_{u})^{2}+(z_{u})^{2}}}={\sqrt {(x_{v})^{2}+(y_{v})^{2}+(z_{v})^{2}}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mi> r </mi> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> u </mi> </mrow> </msub> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <mo stretchy="false"> ( </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> u </mi> </mrow> </msub> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <mo stretchy="false"> ( </mo> <msub> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> u </mi> </mrow> </msub> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </msqrt> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <mo stretchy="false"> ( </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <mo stretchy="false"> ( </mo> <msub> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle |ru|={\sqrt {(x_{u})^{2}+(y_{u})^{2}+(z_{u})^{2}}}={\sqrt {(x_{v})^{2}+(y_{v})^{2}+(z_{v})^{2}}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e0ef6e6ad8d26f0ccc350053984b2a18ee8c806" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:56.766ex; height:4.843ex;" alt="{\displaystyle |ru|={\sqrt {(x_{u})^{2}+(y_{u})^{2}+(z_{u})^{2}}}={\sqrt {(x_{v})^{2}+(y_{v})^{2}+(z_{v})^{2}}}}"> </noscript><span class="lazy-image-placeholder" style="width: 56.766ex;height: 4.843ex;vertical-align: -1.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e0ef6e6ad8d26f0ccc350053984b2a18ee8c806" data-alt="{\displaystyle |ru|={\sqrt {(x_{u})^{2}+(y_{u})^{2}+(z_{u})^{2}}}={\sqrt {(x_{v})^{2}+(y_{v})^{2}+(z_{v})^{2}}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(3)"> <h2 id="Eksplicitna_jednačina_površi">Eksplicitna jednačina površi</h2> <a role="button" href="https://sh-m-wikipedia-org.translate.goog/w/index.php?title=Povr%C5%A1&action=edit&section=3&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Uredi odjeljak Eksplicitna jednačina površi" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> uredi </a> </div> <section class="mf-section-3 collapsible-block" id="mf-section-3"> Neka je <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {U}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur"> U </mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\mathfrak {U}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/042ea9f1b7b1e435446133c8cc8bc5a204766da5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-left: -0.057ex; width:1.603ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {U}}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.603ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/042ea9f1b7b1e435446133c8cc8bc5a204766da5" data-alt="{\displaystyle {\mathfrak {U}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  područje (otvoren i povezan skup) u <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{2}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> R </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle R^{2}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ce07e278be3e058a6303de8359f8b4a4288264a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.818ex; height:2.676ex;" alt="{\displaystyle R^{2}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.818ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ce07e278be3e058a6303de8359f8b4a4288264a" data-alt="{\displaystyle R^{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  i neka <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:U\to R}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> f </mi> <mo> : </mo> <mi> U </mi> <mo stretchy="false"> → </mo> <mi> R </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle f:U\to R} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85805459a88397440c2a4cf468c64237081018f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.376ex; height:2.509ex;" alt="{\displaystyle f:U\to R}"> </noscript><span class="lazy-image-placeholder" style="width: 10.376ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85805459a88397440c2a4cf468c64237081018f4" data-alt="{\displaystyle f:U\to R}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  ima na <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {U}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur"> U </mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\mathfrak {U}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/042ea9f1b7b1e435446133c8cc8bc5a204766da5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-left: -0.057ex; width:1.603ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {U}}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.603ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/042ea9f1b7b1e435446133c8cc8bc5a204766da5" data-alt="{\displaystyle {\mathfrak {U}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  neprekidne prve parcijalne derivacije po <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> x </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle x} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> </noscript><span class="lazy-image-placeholder" style="width: 1.33ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" data-alt="{\displaystyle x}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  i <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> y </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle y} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"> </noscript><span class="lazy-image-placeholder" style="width: 1.155ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" data-alt="{\displaystyle y}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  . Graf funkcije <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> f </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle f} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> </noscript><span class="lazy-image-placeholder" style="width: 1.279ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" data-alt="{\displaystyle f}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  nazivamo regularnom (glatkom) površi. Jednačinu takve površi nazivamo eksplicitnom i ona glasi <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=f(x,y)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> z </mi> <mo> = </mo> <mi> f </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> , </mo> <mi> y </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle z=f(x,y)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1eefb2840000f404c8c0f3f5d6d72f2624854591" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.794ex; height:2.843ex;" alt="{\displaystyle z=f(x,y)}"> </noscript><span class="lazy-image-placeholder" style="width: 10.794ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1eefb2840000f404c8c0f3f5d6d72f2624854591" data-alt="{\displaystyle z=f(x,y)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Da bi se s parametarskog oblika zadavanja površi moglo preči na eksplicitan oblik barem jedna od funkcijskih determinanti (iv) mora biti različita od nule. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{vmatrix}x_{u}(u_{0},v_{0})&y_{u}(u_{0},v_{0})\\x_{v}(u_{0},v_{0})&x_{v}(u_{0},v_{0})\end{vmatrix}}\neq 0}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> | </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> u </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> , </mo> <msub> <mi> v </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> </mtd> <mtd> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> u </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> , </mo> <msub> <mi> v </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> , </mo> <msub> <mi> v </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> </mtd> <mtd> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> , </mo> <msub> <mi> v </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> </mtd> </mtr> </mtable> <mo> | </mo> </mrow> </mrow> <mo> ≠ </mo> <mn> 0 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{vmatrix}x_{u}(u_{0},v_{0})&y_{u}(u_{0},v_{0})\\x_{v}(u_{0},v_{0})&x_{v}(u_{0},v_{0})\end{vmatrix}}\neq 0} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a75e2d3508cad8e32a9defe294d056a4eadddf39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.308ex; height:6.176ex;" alt="{\displaystyle {\begin{vmatrix}x_{u}(u_{0},v_{0})&y_{u}(u_{0},v_{0})\\x_{v}(u_{0},v_{0})&x_{v}(u_{0},v_{0})\end{vmatrix}}\neq 0}"> </noscript><span class="lazy-image-placeholder" style="width: 28.308ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a75e2d3508cad8e32a9defe294d056a4eadddf39" data-alt="{\displaystyle {\begin{vmatrix}x_{u}(u_{0},v_{0})&y_{u}(u_{0},v_{0})\\x_{v}(u_{0},v_{0})&x_{v}(u_{0},v_{0})\end{vmatrix}}\neq 0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Možemo izvršiti inverziju prvih dviju jednačina od <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=x(u,v),\;y=y(u,v),\;z=z(u,v),}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> x </mi> <mo> = </mo> <mi> x </mi> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mo> , </mo> <mspace width="thickmathspace"></mspace> <mi> y </mi> <mo> = </mo> <mi> y </mi> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mo> , </mo> <mspace width="thickmathspace"></mspace> <mi> z </mi> <mo> = </mo> <mi> z </mi> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle x=x(u,v),\;y=y(u,v),\;z=z(u,v),} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa8a1656973ee4810dff44f2e18c3660c3411bd5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.349ex; height:2.843ex;" alt="{\displaystyle x=x(u,v),\;y=y(u,v),\;z=z(u,v),}"> </noscript><span class="lazy-image-placeholder" style="width: 36.349ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa8a1656973ee4810dff44f2e18c3660c3411bd5" data-alt="{\displaystyle x=x(u,v),\;y=y(u,v),\;z=z(u,v),}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  i postaviti dvije nove, jednoznačine, neprekidne funkcije <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(x,y)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> u </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> , </mo> <mi> y </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle u(x,y)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b57461332078df873c81bca2a395fd13f78bef50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.658ex; height:2.843ex;" alt="{\displaystyle u(x,y)}"> </noscript><span class="lazy-image-placeholder" style="width: 6.658ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b57461332078df873c81bca2a395fd13f78bef50" data-alt="{\displaystyle u(x,y)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  i <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(x,y)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> v </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> , </mo> <mi> y </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle v(x,y)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca1cc47d643640c35af2867bd47f907af79574d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.456ex; height:2.843ex;" alt="{\displaystyle v(x,y)}"> </noscript><span class="lazy-image-placeholder" style="width: 6.456ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca1cc47d643640c35af2867bd47f907af79574d4" data-alt="{\displaystyle v(x,y)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> koje imaju neprekidne prve parcijalne derivacije u okolini tačke <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{0},y_{0}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> , </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (x_{0},y_{0}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/383c2fc56c722e2619ac09ed579e3275c56730fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.516ex; height:2.843ex;" alt="{\displaystyle (x_{0},y_{0}}"> </noscript><span class="lazy-image-placeholder" style="width: 6.516ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/383c2fc56c722e2619ac09ed579e3275c56730fc" data-alt="{\displaystyle (x_{0},y_{0}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  koja odgovara tački <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (u_{0},v_{0})}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> , </mo> <msub> <mi> v </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (u_{0},v_{0})} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b729d790fa6e280ed610be112ba3348172aec62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.409ex; height:2.843ex;" alt="{\displaystyle (u_{0},v_{0})}"> </noscript><span class="lazy-image-placeholder" style="width: 7.409ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b729d790fa6e280ed610be112ba3348172aec62" data-alt="{\displaystyle (u_{0},v_{0})}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Pri tome vrijedi <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(x_{0},y_{0})=u_{0}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> u </mi> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> , </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> = </mo> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle u(x_{0},y_{0})=u_{0}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a49f2c38a774164912ac964ceceb037a5dd18c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.233ex; height:2.843ex;" alt="{\displaystyle u(x_{0},y_{0})=u_{0}}"> </noscript><span class="lazy-image-placeholder" style="width: 14.233ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a49f2c38a774164912ac964ceceb037a5dd18c1" data-alt="{\displaystyle u(x_{0},y_{0})=u_{0}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  i <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(x_{0},y_{0})=v_{0}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> v </mi> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> , </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> = </mo> <msub> <mi> v </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle v(x_{0},y_{0})=v_{0}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83e17dc6719d5851a24cfb5c776e5dbca187d0c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.829ex; height:2.843ex;" alt="{\displaystyle v(x_{0},y_{0})=v_{0}}"> </noscript><span class="lazy-image-placeholder" style="width: 13.829ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83e17dc6719d5851a24cfb5c776e5dbca187d0c2" data-alt="{\displaystyle v(x_{0},y_{0})=v_{0}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Nakon uvrštavanja tih dviju funkcija u jednačinu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=z(u,v)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> z </mi> <mo> = </mo> <mi> z </mi> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle z=z(u,v)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e263aebd2eeba9e55fd1072eb87e251125d01faf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.575ex; height:2.843ex;" alt="{\displaystyle z=z(u,v)}"> </noscript><span class="lazy-image-placeholder" style="width: 10.575ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e263aebd2eeba9e55fd1072eb87e251125d01faf" data-alt="{\displaystyle z=z(u,v)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  nastaje jednoznačnu, složena i neprekidna funkcija <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{1}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle z_{1}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3621e468231ab352b7caa30bcf0ce9b452241a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.135ex; height:2.009ex;" alt="{\displaystyle z_{1}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.135ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3621e468231ab352b7caa30bcf0ce9b452241a6" data-alt="{\displaystyle z_{1}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  od <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> x </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle x} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> </noscript><span class="lazy-image-placeholder" style="width: 1.33ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" data-alt="{\displaystyle x}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  i <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> y </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle y} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"> </noscript><span class="lazy-image-placeholder" style="width: 1.155ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" data-alt="{\displaystyle y}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , a jednačina <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=z(u(x,y),v(x,y))=z_{1}(x,y)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> z </mi> <mo> = </mo> <mi> z </mi> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> , </mo> <mi> y </mi> <mo stretchy="false"> ) </mo> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> , </mo> <mi> y </mi> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ) </mo> <mo> = </mo> <msub> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> , </mo> <mi> y </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle z=z(u(x,y),v(x,y))=z_{1}(x,y)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c70708e7c5244db59424b62a7fd834a4c185d94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.794ex; height:2.843ex;" alt="{\displaystyle z=z(u(x,y),v(x,y))=z_{1}(x,y)}"> </noscript><span class="lazy-image-placeholder" style="width: 31.794ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c70708e7c5244db59424b62a7fd834a4c185d94" data-alt="{\displaystyle z=z(u(x,y),v(x,y))=z_{1}(x,y)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  predstavlja eksplicitan oblik zadavanja površi. Ako su uvažene sve pretpostavke, funkcija <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{1}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle z_{1}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3621e468231ab352b7caa30bcf0ce9b452241a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.135ex; height:2.009ex;" alt="{\displaystyle z_{1}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.135ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3621e468231ab352b7caa30bcf0ce9b452241a6" data-alt="{\displaystyle z_{1}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  mora imati neprekidne prve parcijalne derivacije po <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> x </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle x} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> </noscript><span class="lazy-image-placeholder" style="width: 1.33ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" data-alt="{\displaystyle x}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  i <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> y </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle y} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"> </noscript><span class="lazy-image-placeholder" style="width: 1.155ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" data-alt="{\displaystyle y}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(4)"> <h2 id="Implicitna_jednačina_površi">Implicitna jednačina površi</h2> <a role="button" href="https://sh-m-wikipedia-org.translate.goog/w/index.php?title=Povr%C5%A1&action=edit&section=4&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Uredi odjeljak Implicitna jednačina površi" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> uredi </a> </div> <section class="mf-section-4 collapsible-block" id="mf-section-4"> Neka je <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {U}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur"> U </mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\mathfrak {U}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/042ea9f1b7b1e435446133c8cc8bc5a204766da5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-left: -0.057ex; width:1.603ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {U}}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.603ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/042ea9f1b7b1e435446133c8cc8bc5a204766da5" data-alt="{\displaystyle {\mathfrak {U}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  područje u <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{3}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> R </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle R^{3}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ab2376f6a3a3b77ddc94ade4f6fbc96e85ca29b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.818ex; height:2.676ex;" alt="{\displaystyle R^{3}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.818ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ab2376f6a3a3b77ddc94ade4f6fbc96e85ca29b" data-alt="{\displaystyle R^{3}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  i neka je funkcija <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F:U\to R}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> F </mi> <mo> : </mo> <mi> U </mi> <mo stretchy="false"> → </mo> <mi> R </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle F:U\to R} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7d159d22f658e248676e5d003659c9ae685011d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.839ex; height:2.176ex;" alt="{\displaystyle F:U\to R}"> </noscript><span class="lazy-image-placeholder" style="width: 10.839ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7d159d22f658e248676e5d003659c9ae685011d" data-alt="{\displaystyle F:U\to R}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  klase <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{1}({\mathfrak {U}})}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> C </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msup> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur"> U </mi> </mrow> </mrow> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle C^{1}({\mathfrak {U}})} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3331451a22c135d37d090db055793cd8b7dff50a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.207ex; height:3.176ex;" alt="{\displaystyle C^{1}({\mathfrak {U}})}"> </noscript><span class="lazy-image-placeholder" style="width: 6.207ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3331451a22c135d37d090db055793cd8b7dff50a" data-alt="{\displaystyle C^{1}({\mathfrak {U}})}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  tj. prve parcijalne derivacije <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{x},F_{y},F_{z}:{\mathfrak {U}}\to R}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> F </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> </mrow> </msub> <mo> , </mo> <msub> <mi> F </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> y </mi> </mrow> </msub> <mo> , </mo> <msub> <mi> F </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> z </mi> </mrow> </msub> <mo> : </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur"> U </mi> </mrow> </mrow> <mo stretchy="false"> → </mo> <mi> R </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle F_{x},F_{y},F_{z}:{\mathfrak {U}}\to R} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a9d8e5a3d18d1662f40463acb09cc836fbf2912" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.636ex; height:2.843ex;" alt="{\displaystyle F_{x},F_{y},F_{z}:{\mathfrak {U}}\to R}"> </noscript><span class="lazy-image-placeholder" style="width: 18.636ex;height: 2.843ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a9d8e5a3d18d1662f40463acb09cc836fbf2912" data-alt="{\displaystyle F_{x},F_{y},F_{z}:{\mathfrak {U}}\to R}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  su neprekidne funkcije na <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {U}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur"> U </mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\mathfrak {U}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/042ea9f1b7b1e435446133c8cc8bc5a204766da5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-left: -0.057ex; width:1.603ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {U}}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.603ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/042ea9f1b7b1e435446133c8cc8bc5a204766da5" data-alt="{\displaystyle {\mathfrak {U}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Jednačinu <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x,y,z)=0}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> F </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> , </mo> <mi> y </mi> <mo> , </mo> <mi> z </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mn> 0 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle F(x,y,z)=0} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0796f292c93e738b5c7c46a9aea6023ceb2d8ec7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.452ex; height:2.843ex;" alt="{\displaystyle F(x,y,z)=0}"> </noscript><span class="lazy-image-placeholder" style="width: 13.452ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0796f292c93e738b5c7c46a9aea6023ceb2d8ec7" data-alt="{\displaystyle F(x,y,z)=0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> nazivamo implicitnom površi, ako postoji barem jedna tačka <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{,}y_{0},z_{0})}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> , </mo> </mrow> </msub> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> , </mo> <msub> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (x_{,}y_{0},z_{0})} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17426410689d0322f6368dedf8565aa7e651e669" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.191ex; height:3.009ex;" alt="{\displaystyle (x_{,}y_{0},z_{0})}"> </noscript><span class="lazy-image-placeholder" style="width: 9.191ex;height: 3.009ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17426410689d0322f6368dedf8565aa7e651e669" data-alt="{\displaystyle (x_{,}y_{0},z_{0})}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  takva da zadovoljava jednačinu i da je u njoj barem jedna od parcijalnih derivacija <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{x},F_{y},F_{z}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> F </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> </mrow> </msub> <mo> , </mo> <msub> <mi> F </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> y </mi> </mrow> </msub> <mo> , </mo> <msub> <mi> F </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> z </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle F_{x},F_{y},F_{z}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3979f4e2545f2e8a099d283c4555353e62805175" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.775ex; height:2.843ex;" alt="{\displaystyle F_{x},F_{y},F_{z}}"> </noscript><span class="lazy-image-placeholder" style="width: 9.775ex;height: 2.843ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3979f4e2545f2e8a099d283c4555353e62805175" data-alt="{\displaystyle F_{x},F_{y},F_{z}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  različita od <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn> 0 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle 0} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"> </noscript><span class="lazy-image-placeholder" style="width: 1.162ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" data-alt="{\displaystyle 0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Ovaj uslov osigurava egzistenciju regularnog dijela površi. Ako je <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{z}(x_{0},y_{0},z_{0})\to 0}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> F </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> z </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> , </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> , </mo> <msub> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo stretchy="false"> → </mo> <mn> 0 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle F_{z}(x_{0},y_{0},z_{0})\to 0} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f5c47679a6923378e39a89c20da52c0c2b16ee7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.863ex; height:2.843ex;" alt="{\displaystyle F_{z}(x_{0},y_{0},z_{0})\to 0}"> </noscript><span class="lazy-image-placeholder" style="width: 17.863ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f5c47679a6923378e39a89c20da52c0c2b16ee7" data-alt="{\displaystyle F_{z}(x_{0},y_{0},z_{0})\to 0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , postoji jednoznačna, neprekidna funkcija <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=z(x,y)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> z </mi> <mo> = </mo> <mi> z </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> , </mo> <mi> y </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle z=z(x,y)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/593eac2c3078f3f74c5a535817f028f139eafeb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.603ex; height:2.843ex;" alt="{\displaystyle z=z(x,y)}"> </noscript><span class="lazy-image-placeholder" style="width: 10.603ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/593eac2c3078f3f74c5a535817f028f139eafeb2" data-alt="{\displaystyle z=z(x,y)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  koja u okolini tačke <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{0},y_{0})}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> , </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (x_{0},y_{0})} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29c296094af9a1c665425debeac5eaab99a37a04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.421ex; height:2.843ex;" alt="{\displaystyle (x_{0},y_{0})}"> </noscript><span class="lazy-image-placeholder" style="width: 7.421ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29c296094af9a1c665425debeac5eaab99a37a04" data-alt="{\displaystyle (x_{0},y_{0})}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  identički zadovoljava vezu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x,y,z(x,y))=0}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> F </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> , </mo> <mi> y </mi> <mo> , </mo> <mi> z </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> , </mo> <mi> y </mi> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ) </mo> <mo> = </mo> <mn> 0 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle F(x,y,z(x,y))=0} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7c7ac4e49a710c29211d8390b95c85a5947d48b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.781ex; height:2.843ex;" alt="{\displaystyle F(x,y,z(x,y))=0}"> </noscript><span class="lazy-image-placeholder" style="width: 18.781ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7c7ac4e49a710c29211d8390b95c85a5947d48b" data-alt="{\displaystyle F(x,y,z(x,y))=0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  i u toj tački funkcija <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> z </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle z} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"> </noscript><span class="lazy-image-placeholder" style="width: 1.088ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" data-alt="{\displaystyle z}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  ima neprekidne prve parcijalne derivacije po <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> x </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle x} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> </noscript><span class="lazy-image-placeholder" style="width: 1.33ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" data-alt="{\displaystyle x}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  i <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> y </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle y} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"> </noscript><span class="lazy-image-placeholder" style="width: 1.155ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" data-alt="{\displaystyle y}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Tačku <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{0},y_{0},z_{0})}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> , </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> , </mo> <msub> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (x_{0},y_{0},z_{0})} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39177ddeeb9f9a393b664e522bc8e3bf0face153" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.59ex; height:2.843ex;" alt="{\displaystyle (x_{0},y_{0},z_{0})}"> </noscript><span class="lazy-image-placeholder" style="width: 10.59ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39177ddeeb9f9a393b664e522bc8e3bf0face153" data-alt="{\displaystyle (x_{0},y_{0},z_{0})}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  u kojoj su ispunjeni navedeni uslovi zovemo običnom ili regularnom tačkom površi. Kako bi barem jedna od parcijalnih derivacija funkcije <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> F </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle F} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"> </noscript><span class="lazy-image-placeholder" style="width: 1.741ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" data-alt="{\displaystyle F}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  bila različita od nule, za regularnu tačku površi mora biti zadovoljen uslov <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{x}^{2}(x_{0},y_{0},z_{0})+F_{y}^{2}(x_{0},y_{0},z_{0})+F_{z}^{2}(x_{0},y_{0},z_{0})\to 0}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi> F </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> , </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> , </mo> <msub> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> + </mo> <msubsup> <mi> F </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> y </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> , </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> , </mo> <msub> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> + </mo> <msubsup> <mi> F </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> z </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> , </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> , </mo> <msub> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo stretchy="false"> → </mo> <mn> 0 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle F_{x}^{2}(x_{0},y_{0},z_{0})+F_{y}^{2}(x_{0},y_{0},z_{0})+F_{z}^{2}(x_{0},y_{0},z_{0})\to 0} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cac42951d43a5d14615467bbc88960d24f7a3c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:50.834ex; height:3.176ex;" alt="{\displaystyle F_{x}^{2}(x_{0},y_{0},z_{0})+F_{y}^{2}(x_{0},y_{0},z_{0})+F_{z}^{2}(x_{0},y_{0},z_{0})\to 0}"> </noscript><span class="lazy-image-placeholder" style="width: 50.834ex;height: 3.176ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cac42951d43a5d14615467bbc88960d24f7a3c1" data-alt="{\displaystyle F_{x}^{2}(x_{0},y_{0},z_{0})+F_{y}^{2}(x_{0},y_{0},z_{0})+F_{z}^{2}(x_{0},y_{0},z_{0})\to 0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Kako je <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{0},y_{0},z_{0})}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> , </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> , </mo> <msub> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (x_{0},y_{0},z_{0})} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39177ddeeb9f9a393b664e522bc8e3bf0face153" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.59ex; height:2.843ex;" alt="{\displaystyle (x_{0},y_{0},z_{0})}"> </noscript><span class="lazy-image-placeholder" style="width: 10.59ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39177ddeeb9f9a393b664e522bc8e3bf0face153" data-alt="{\displaystyle (x_{0},y_{0},z_{0})}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  singularna toačka implicitno zadane površi ako ona zadovoljava jednačinu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x,y,z)=0}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> F </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> , </mo> <mi> y </mi> <mo> , </mo> <mi> z </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mn> 0 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle F(x,y,z)=0} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0796f292c93e738b5c7c46a9aea6023ceb2d8ec7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.452ex; height:2.843ex;" alt="{\displaystyle F(x,y,z)=0}"> </noscript><span class="lazy-image-placeholder" style="width: 13.452ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0796f292c93e738b5c7c46a9aea6023ceb2d8ec7" data-alt="{\displaystyle F(x,y,z)=0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  i ako vrijedi <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{x}(x_{0},y_{0},z_{0})+F_{y}(x_{0},y_{0},z_{0})+F_{z}(x_{0},y_{0},z_{0})=0}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> F </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> , </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> , </mo> <msub> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> + </mo> <msub> <mi> F </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> y </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> , </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> , </mo> <msub> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> + </mo> <msub> <mi> F </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> z </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> , </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> , </mo> <msub> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> = </mo> <mn> 0 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle F_{x}(x_{0},y_{0},z_{0})+F_{y}(x_{0},y_{0},z_{0})+F_{z}(x_{0},y_{0},z_{0})=0} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae99f99fa0c033fd890f857810da15b41dce2afb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:49.419ex; height:3.009ex;" alt="{\displaystyle F_{x}(x_{0},y_{0},z_{0})+F_{y}(x_{0},y_{0},z_{0})+F_{z}(x_{0},y_{0},z_{0})=0}"> </noscript><span class="lazy-image-placeholder" style="width: 49.419ex;height: 3.009ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae99f99fa0c033fd890f857810da15b41dce2afb" data-alt="{\displaystyle F_{x}(x_{0},y_{0},z_{0})+F_{y}(x_{0},y_{0},z_{0})+F_{z}(x_{0},y_{0},z_{0})=0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(5)"> <h2 id="Tangentna_ravan_i_normala_na_površ">Tangentna ravan i normala na površ</h2> <a role="button" href="https://sh-m-wikipedia-org.translate.goog/w/index.php?title=Povr%C5%A1&action=edit&section=5&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Uredi odjeljak Tangentna ravan i normala na površ" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> uredi </a> </div> <section class="mf-section-5 collapsible-block" id="mf-section-5"> Bilo koja kriva na regularnoj površi F zadanoj vektorskom jednačinom <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r(u,v)=x(u,v)i+y(u,v)j+z(u,v)k}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> r </mi> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mi> x </mi> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mi> i </mi> <mo> + </mo> <mi> y </mi> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mi> j </mi> <mo> + </mo> <mi> z </mi> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mi> k </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r(u,v)=x(u,v)i+y(u,v)j+z(u,v)k} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34070a624d52d7b2b396cec883401b52de9e6812" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.575ex; height:2.843ex;" alt="{\displaystyle r(u,v)=x(u,v)i+y(u,v)j+z(u,v)k}"> </noscript><span class="lazy-image-placeholder" style="width: 37.575ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34070a624d52d7b2b396cec883401b52de9e6812" data-alt="{\displaystyle r(u,v)=x(u,v)i+y(u,v)j+z(u,v)k}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  može biti zadana parametarskom jednačinom <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u=u(t)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> u </mi> <mo> = </mo> <mi> u </mi> <mo stretchy="false"> ( </mo> <mi> t </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle u=u(t)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/188f5066b4620c7cd5b44eb4ca51b383881d00d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.407ex; height:2.843ex;" alt="{\displaystyle u=u(t)}"> </noscript><span class="lazy-image-placeholder" style="width: 8.407ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/188f5066b4620c7cd5b44eb4ca51b383881d00d6" data-alt="{\displaystyle u=u(t)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v=v(t)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> v </mi> <mo> = </mo> <mi> v </mi> <mo stretchy="false"> ( </mo> <mi> t </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle v=v(t)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0744b609436f4a6e2fde78abc352e6488dfa779c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.003ex; height:2.843ex;" alt="{\displaystyle v=v(t)}"> </noscript><span class="lazy-image-placeholder" style="width: 8.003ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0744b609436f4a6e2fde78abc352e6488dfa779c" data-alt="{\displaystyle v=v(t)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  gdje za <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall t\in (a,b)\notin R}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal"> ∀ </mi> <mi> t </mi> <mo> ∈ </mo> <mo stretchy="false"> ( </mo> <mi> a </mi> <mo> , </mo> <mi> b </mi> <mo stretchy="false"> ) </mo> <mo> ∉ </mo> <mi> R </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \forall t\in (a,b)\notin R} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99dcb8c33ba71b206916fb39923fda37188f3aa4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.648ex; height:2.843ex;" alt="{\displaystyle \forall t\in (a,b)\notin R}"> </noscript><span class="lazy-image-placeholder" style="width: 14.648ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99dcb8c33ba71b206916fb39923fda37188f3aa4" data-alt="{\displaystyle \forall t\in (a,b)\notin R}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  vrijedi da se <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (u(t),v(t))}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo stretchy="false"> ( </mo> <mi> t </mi> <mo stretchy="false"> ) </mo> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ( </mo> <mi> t </mi> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (u(t),v(t))} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58cb77418556384cfa3c34cf9a0cefb3bfe3c229" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.598ex; height:2.843ex;" alt="{\displaystyle (u(t),v(t))}"> </noscript><span class="lazy-image-placeholder" style="width: 10.598ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58cb77418556384cfa3c34cf9a0cefb3bfe3c229" data-alt="{\displaystyle (u(t),v(t))}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  nalazi u području <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {U}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur"> U </mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\mathfrak {U}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/042ea9f1b7b1e435446133c8cc8bc5a204766da5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-left: -0.057ex; width:1.603ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {U}}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.603ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/042ea9f1b7b1e435446133c8cc8bc5a204766da5" data-alt="{\displaystyle {\mathfrak {U}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , a funkcije <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(t}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> u </mi> <mo stretchy="false"> ( </mo> <mi> t </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle u(t} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/108533b21a3eef3087c98ff54b46a83977373f2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.074ex; height:2.843ex;" alt="{\displaystyle u(t}"> </noscript><span class="lazy-image-placeholder" style="width: 3.074ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/108533b21a3eef3087c98ff54b46a83977373f2c" data-alt="{\displaystyle u(t}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> ) i <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(t)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> v </mi> <mo stretchy="false"> ( </mo> <mi> t </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle v(t)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/243a0bf98a12f48552ba6a70302122d81b237b3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.777ex; height:2.843ex;" alt="{\displaystyle v(t)}"> </noscript><span class="lazy-image-placeholder" style="width: 3.777ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/243a0bf98a12f48552ba6a70302122d81b237b3d" data-alt="{\displaystyle v(t)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  neprekidne su funkcije od <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> t </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle t} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"> </noscript><span class="lazy-image-placeholder" style="width: 0.84ex;height: 2.009ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" data-alt="{\displaystyle t}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Ako kriva u svakoj tački ima tangentu moraju i derivacije <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u'(t)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> u </mi> <mo> ′ </mo> </msup> <mo stretchy="false"> ( </mo> <mi> t </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle u'(t)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1530c44c813400c03a673a8dcd8bdb61368c9c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.663ex; height:3.009ex;" alt="{\displaystyle u'(t)}"> </noscript><span class="lazy-image-placeholder" style="width: 4.663ex;height: 3.009ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1530c44c813400c03a673a8dcd8bdb61368c9c8" data-alt="{\displaystyle u'(t)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  i <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v'(t)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> v </mi> <mo> ′ </mo> </msup> <mo stretchy="false"> ( </mo> <mi> t </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle v'(t)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f8a024ea25190c68ccbcf96ccef94f9dc38b475" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.461ex; height:3.009ex;" alt="{\displaystyle v'(t)}"> </noscript><span class="lazy-image-placeholder" style="width: 4.461ex;height: 3.009ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f8a024ea25190c68ccbcf96ccef94f9dc38b475" data-alt="{\displaystyle v'(t)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  biti neprekidne. Kriva mora zadovoljavati jednačinu površi, vektori tačaka na krivoj dati su izrazom <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=r((u(t),v(t))}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> r </mi> <mo> = </mo> <mi> r </mi> <mo stretchy="false"> ( </mo> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo stretchy="false"> ( </mo> <mi> t </mi> <mo stretchy="false"> ) </mo> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ( </mo> <mi> t </mi> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r=r((u(t),v(t))} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b424e28769dbb4bbc86d2830c8e8bddce7975db2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.699ex; height:2.843ex;" alt="{\displaystyle r=r((u(t),v(t))}"> </noscript><span class="lazy-image-placeholder" style="width: 16.699ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b424e28769dbb4bbc86d2830c8e8bddce7975db2" data-alt="{\displaystyle r=r((u(t),v(t))}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  <a href="https://sh-m-wikipedia-org.translate.goog/wiki/Vektor?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Vektor">Vektor</a> tangente na tu krivu je <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r'(t)=r_{u}(u,v)(t)+r_{v}(u,v)v'(t)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> r </mi> <mo> ′ </mo> </msup> <mo stretchy="false"> ( </mo> <mi> t </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> u </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ( </mo> <mi> t </mi> <mo stretchy="false"> ) </mo> <mo> + </mo> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <msup> <mi> v </mi> <mo> ′ </mo> </msup> <mo stretchy="false"> ( </mo> <mi> t </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r'(t)=r_{u}(u,v)(t)+r_{v}(u,v)v'(t)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46fbdeeade9c4386030c70362ea363146a97a3de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.332ex; height:3.009ex;" alt="{\displaystyle r'(t)=r_{u}(u,v)(t)+r_{v}(u,v)v'(t)}"> </noscript><span class="lazy-image-placeholder" style="width: 32.332ex;height: 3.009ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46fbdeeade9c4386030c70362ea363146a97a3de" data-alt="{\displaystyle r'(t)=r_{u}(u,v)(t)+r_{v}(u,v)v'(t)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Proizvoljnom čvrstom tačkom <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(u,v)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> T </mi> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle T(u,v)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24348030a9fdca5c0234c2df0915881701e0715d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.937ex; height:2.843ex;" alt="{\displaystyle T(u,v)}"> </noscript><span class="lazy-image-placeholder" style="width: 6.937ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24348030a9fdca5c0234c2df0915881701e0715d" data-alt="{\displaystyle T(u,v)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  površi <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {F}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur"> F </mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\mathfrak {F}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec863025da8f3844272a146530208eb001c3dddb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.423ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {F}}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.423ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec863025da8f3844272a146530208eb001c3dddb" data-alt="{\displaystyle {\mathfrak {F}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  prolazi beskonačno mnogo prostornih krivi koje leže na površi. Za sve takve krive vektori <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{u}(u,v)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> u </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r_{u}(u,v)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cce536584293b18b52da4849d7d58964ab4f0832" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.522ex; height:2.843ex;" alt="{\displaystyle r_{u}(u,v)}"> </noscript><span class="lazy-image-placeholder" style="width: 7.522ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cce536584293b18b52da4849d7d58964ab4f0832" data-alt="{\displaystyle r_{u}(u,v)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  i <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{v}(u,v)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r_{v}(u,v)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7a92f3512403a37e6fe3b9e3a83efffef36a899" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.379ex; height:2.843ex;" alt="{\displaystyle r_{v}(u,v)}"> </noscript><span class="lazy-image-placeholder" style="width: 7.379ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7a92f3512403a37e6fe3b9e3a83efffef36a899" data-alt="{\displaystyle r_{v}(u,v)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  biće jednaki, budući da oni zavise samo o koordinatama <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> u </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle u} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}"> </noscript><span class="lazy-image-placeholder" style="width: 1.33ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" data-alt="{\displaystyle u}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  i <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> v </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle v} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"> </noscript><span class="lazy-image-placeholder" style="width: 1.128ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" data-alt="{\displaystyle v}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  tačke T, dok ́će derivacije <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u'(t)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> u </mi> <mo> ′ </mo> </msup> <mo stretchy="false"> ( </mo> <mi> t </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle u'(t)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1530c44c813400c03a673a8dcd8bdb61368c9c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.663ex; height:3.009ex;" alt="{\displaystyle u'(t)}"> </noscript><span class="lazy-image-placeholder" style="width: 4.663ex;height: 3.009ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1530c44c813400c03a673a8dcd8bdb61368c9c8" data-alt="{\displaystyle u'(t)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  i <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v'(t)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> v </mi> <mo> ′ </mo> </msup> <mo stretchy="false"> ( </mo> <mi> t </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle v'(t)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f8a024ea25190c68ccbcf96ccef94f9dc38b475" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.461ex; height:3.009ex;" alt="{\displaystyle v'(t)}"> </noscript><span class="lazy-image-placeholder" style="width: 4.461ex;height: 3.009ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f8a024ea25190c68ccbcf96ccef94f9dc38b475" data-alt="{\displaystyle v'(t)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  za pojedine krive biti različite. Svi vektori tangenata na krivu koje prolaze tačkom T linearne su kombinacije vektora <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{u}(u,v)=x_{u}(u,v)i+y_{u}(u,v)j+z_{u}(u,v)k}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> u </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> u </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mi> i </mi> <mo> + </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> u </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mi> j </mi> <mo> + </mo> <msub> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> u </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mi> k </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r_{u}(u,v)=x_{u}(u,v)i+y_{u}(u,v)j+z_{u}(u,v)k} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/093f87b79571885798afe072afb224c68ec2f3c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.242ex; height:2.843ex;" alt="{\displaystyle r_{u}(u,v)=x_{u}(u,v)i+y_{u}(u,v)j+z_{u}(u,v)k}"> </noscript><span class="lazy-image-placeholder" style="width: 42.242ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/093f87b79571885798afe072afb224c68ec2f3c0" data-alt="{\displaystyle r_{u}(u,v)=x_{u}(u,v)i+y_{u}(u,v)j+z_{u}(u,v)k}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{v}(u,v)=x_{v}(u,v)i+y_{v}(u,v)j+z_{v}(u,v)k}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mi> i </mi> <mo> + </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mi> j </mi> <mo> + </mo> <msub> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mi> k </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r_{v}(u,v)=x_{v}(u,v)i+y_{v}(u,v)j+z_{v}(u,v)k} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b04a3ce7076946689e61c92605bb84721c110da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.67ex; height:2.843ex;" alt="{\displaystyle r_{v}(u,v)=x_{v}(u,v)i+y_{v}(u,v)j+z_{v}(u,v)k}"> </noscript><span class="lazy-image-placeholder" style="width: 41.67ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b04a3ce7076946689e61c92605bb84721c110da" data-alt="{\displaystyle r_{v}(u,v)=x_{v}(u,v)i+y_{v}(u,v)j+z_{v}(u,v)k}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Tangente prostornih krivi koje su na površi i prolaze tačkom T leže u ravni koju određuju tangentni vektori koordinatnih krivi te tačke. Ta se ravan naziva tangentna ravan na površ u tački T, a tačka T je njeno diralište. Jednadnačina tangentne ravnine u parametarskom obliku je <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=r_{0}(u,v)+\rho _{1}r_{u}+\rho _{1}r_{v}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> r </mi> <mo> = </mo> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mo> + </mo> <msub> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> u </mi> </mrow> </msub> <mo> + </mo> <msub> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r=r_{0}(u,v)+\rho _{1}r_{u}+\rho _{1}r_{v}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d30b9478770340e33d4c698a870d1e241fc7ebaf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.043ex; height:2.843ex;" alt="{\displaystyle r=r_{0}(u,v)+\rho _{1}r_{u}+\rho _{1}r_{v}}"> </noscript><span class="lazy-image-placeholder" style="width: 26.043ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d30b9478770340e33d4c698a870d1e241fc7ebaf" data-alt="{\displaystyle r=r_{0}(u,v)+\rho _{1}r_{u}+\rho _{1}r_{v}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  gdje je <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> r </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"> </noscript><span class="lazy-image-placeholder" style="width: 1.049ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" data-alt="{\displaystyle r}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  radijus-vektor bilo koje tačke tangenne ravni, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{0}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r_{0}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb12fcfddb65e3d1e6a044215f6e833f0cd4337b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{0}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.103ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb12fcfddb65e3d1e6a044215f6e833f0cd4337b" data-alt="{\displaystyle r_{0}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  radijus- vektor dirališta <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> T </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle T} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"> </noscript><span class="lazy-image-placeholder" style="width: 1.636ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" data-alt="{\displaystyle T}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{1}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \rho _{1}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e0f2d347f2a0ed7f7c9808c427a89813b957017" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.256ex; height:2.176ex;" alt="{\displaystyle \rho _{1}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.256ex;height: 2.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e0f2d347f2a0ed7f7c9808c427a89813b957017" data-alt="{\displaystyle \rho _{1}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  i <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{2}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \rho _{2}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/793b211571b3ffe34c4639654d567296d29d7f72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.256ex; height:2.176ex;" alt="{\displaystyle \rho _{2}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.256ex;height: 2.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/793b211571b3ffe34c4639654d567296d29d7f72" data-alt="{\displaystyle \rho _{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  realni parametri koji poprimaju, nezavisno jedan o drugom, vrijednosti između <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\infty }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo> − </mo> <mi mathvariant="normal"> ∞ </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle -\infty } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle -\infty }"> </noscript><span class="lazy-image-placeholder" style="width: 4.132ex;height: 2.176ex;vertical-align: -0.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1" data-alt="{\displaystyle -\infty }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  i <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +\infty }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo> + </mo> <mi mathvariant="normal"> ∞ </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle +\infty } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bddbb0e4420a7e744cf71bd71216e11b0bf88831" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle +\infty }"> </noscript><span class="lazy-image-placeholder" style="width: 4.132ex;height: 2.176ex;vertical-align: -0.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bddbb0e4420a7e744cf71bd71216e11b0bf88831" data-alt="{\displaystyle +\infty }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Vektor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{u}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> u </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r_{u}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1b33f1518fd9b64ad2cf5c1272813b77cab949a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.221ex; height:2.009ex;" alt="{\displaystyle r_{u}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.221ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1b33f1518fd9b64ad2cf5c1272813b77cab949a" data-alt="{\displaystyle r_{u}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> x <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{v}={\begin{vmatrix}i&j&k\\x_{u}&y_{u}&z_{u}\\x_{v}&y_{v}&z_{v}\end{vmatrix}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> | </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi> i </mi> </mtd> <mtd> <mi> j </mi> </mtd> <mtd> <mi> k </mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> u </mi> </mrow> </msub> </mtd> <mtd> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> u </mi> </mrow> </msub> </mtd> <mtd> <msub> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> u </mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> </mtd> <mtd> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> </mtd> <mtd> <msub> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo> | </mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r_{v}={\begin{vmatrix}i&j&k\\x_{u}&y_{u}&z_{u}\\x_{v}&y_{v}&z_{v}\end{vmatrix}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/058d699916ea954ae2b6b17bb9863c3ae77c72cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:18.935ex; height:9.509ex;" alt="{\displaystyle r_{v}={\begin{vmatrix}i&j&k\\x_{u}&y_{u}&z_{u}\\x_{v}&y_{v}&z_{v}\end{vmatrix}}}"> </noscript><span class="lazy-image-placeholder" style="width: 18.935ex;height: 9.509ex;vertical-align: -4.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/058d699916ea954ae2b6b17bb9863c3ae77c72cb" data-alt="{\displaystyle r_{v}={\begin{vmatrix}i&j&k\\x_{u}&y_{u}&z_{u}\\x_{v}&y_{v}&z_{v}\end{vmatrix}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  normalan je na vektore <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{u}(u,v)=x_{u}(u,v)i+y_{u}(u,v)j+z_{u}(u,v)k}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> u </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> u </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mi> i </mi> <mo> + </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> u </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mi> j </mi> <mo> + </mo> <msub> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> u </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mi> k </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r_{u}(u,v)=x_{u}(u,v)i+y_{u}(u,v)j+z_{u}(u,v)k} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/093f87b79571885798afe072afb224c68ec2f3c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.242ex; height:2.843ex;" alt="{\displaystyle r_{u}(u,v)=x_{u}(u,v)i+y_{u}(u,v)j+z_{u}(u,v)k}"> </noscript><span class="lazy-image-placeholder" style="width: 42.242ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/093f87b79571885798afe072afb224c68ec2f3c0" data-alt="{\displaystyle r_{u}(u,v)=x_{u}(u,v)i+y_{u}(u,v)j+z_{u}(u,v)k}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{v}(u,v)=x_{v}(u,v)i+y_{v}(u,v)j+z_{v}(u,v)k}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mi> i </mi> <mo> + </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mi> j </mi> <mo> + </mo> <msub> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mi> k </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r_{v}(u,v)=x_{v}(u,v)i+y_{v}(u,v)j+z_{v}(u,v)k} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b04a3ce7076946689e61c92605bb84721c110da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.67ex; height:2.843ex;" alt="{\displaystyle r_{v}(u,v)=x_{v}(u,v)i+y_{v}(u,v)j+z_{v}(u,v)k}"> </noscript><span class="lazy-image-placeholder" style="width: 41.67ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b04a3ce7076946689e61c92605bb84721c110da" data-alt="{\displaystyle r_{v}(u,v)=x_{v}(u,v)i+y_{v}(u,v)j+z_{v}(u,v)k}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  i prema tome i na tangentnu ravan u tački T. Naziva se vektorom normale površi. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{u}(u,v)=x_{u}(u,v)i+y_{u}(u,v)j+z_{u}(u,v)k}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> u </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> u </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mi> i </mi> <mo> + </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> u </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mi> j </mi> <mo> + </mo> <msub> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> u </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mi> k </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r_{u}(u,v)=x_{u}(u,v)i+y_{u}(u,v)j+z_{u}(u,v)k} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/093f87b79571885798afe072afb224c68ec2f3c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.242ex; height:2.843ex;" alt="{\displaystyle r_{u}(u,v)=x_{u}(u,v)i+y_{u}(u,v)j+z_{u}(u,v)k}"> </noscript><span class="lazy-image-placeholder" style="width: 42.242ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/093f87b79571885798afe072afb224c68ec2f3c0" data-alt="{\displaystyle r_{u}(u,v)=x_{u}(u,v)i+y_{u}(u,v)j+z_{u}(u,v)k}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{v}(u,v)=x_{v}(u,v)i+y_{v}(u,v)j+z_{v}(u,v)k}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mi> i </mi> <mo> + </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mi> j </mi> <mo> + </mo> <msub> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mi> k </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r_{v}(u,v)=x_{v}(u,v)i+y_{v}(u,v)j+z_{v}(u,v)k} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b04a3ce7076946689e61c92605bb84721c110da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.67ex; height:2.843ex;" alt="{\displaystyle r_{v}(u,v)=x_{v}(u,v)i+y_{v}(u,v)j+z_{v}(u,v)k}"> </noscript><span class="lazy-image-placeholder" style="width: 41.67ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b04a3ce7076946689e61c92605bb84721c110da" data-alt="{\displaystyle r_{v}(u,v)=x_{v}(u,v)i+y_{v}(u,v)j+z_{v}(u,v)k}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Vektori svojim međusobnim položajem određuju orjentaciju u tangentnoj ravni te tačke. Ona je pozitivna ako prvi vektor prelazi na drugi vektor vrtnjom za neki ugao u pozitivnom smislu (suprotno smjeru kazaljke na satu). Vektor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{0}=}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> n </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> = </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n_{0}=} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53e346e7b6f2b55741ae4805e3910f8e363a877d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.902ex; height:2.009ex;" alt="{\displaystyle n_{0}=}"> </noscript><span class="lazy-image-placeholder" style="width: 4.902ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53e346e7b6f2b55741ae4805e3910f8e363a877d" data-alt="{\displaystyle n_{0}=}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  ( <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{u}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> u </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r_{u}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1b33f1518fd9b64ad2cf5c1272813b77cab949a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.221ex; height:2.009ex;" alt="{\displaystyle r_{u}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.221ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1b33f1518fd9b64ad2cf5c1272813b77cab949a" data-alt="{\displaystyle r_{u}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> x <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{v}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r_{v}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6351a9c12f3cd743f37050ddb9a0d0adfc33f190" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.078ex; height:2.009ex;" alt="{\displaystyle r_{v}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.078ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6351a9c12f3cd743f37050ddb9a0d0adfc33f190" data-alt="{\displaystyle r_{v}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> )/ <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{Vmatrix}r_{u}xr_{v}\end{Vmatrix}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo symmetric="true"> ‖ </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> u </mi> </mrow> </msub> <mi> x </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo symmetric="true"> ‖ </mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{Vmatrix}r_{u}xr_{v}\end{Vmatrix}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef4da499027084f30bb0656a9b1ed6b83e2ce613" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.705ex; height:2.843ex;" alt="{\displaystyle {\begin{Vmatrix}r_{u}xr_{v}\end{Vmatrix}}}"> </noscript><span class="lazy-image-placeholder" style="width: 8.705ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef4da499027084f30bb0656a9b1ed6b83e2ce613" data-alt="{\displaystyle {\begin{Vmatrix}r_{u}xr_{v}\end{Vmatrix}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  naziva se jediničnim vektorm normale površi. On ima pozitivnu orijentaciju ako s pozitivnim smjerom vrtnje u tangentnoj ravni tačke T čini desni vijak. Kako vektor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r-r_{0}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> r </mi> <mo> − </mo> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r-r_{0}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/670d9aeef0f9b5910502a4bafef0dd25d63a38cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.992ex; height:2.343ex;" alt="{\displaystyle r-r_{0}}"> </noscript><span class="lazy-image-placeholder" style="width: 5.992ex;height: 2.343ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/670d9aeef0f9b5910502a4bafef0dd25d63a38cb" data-alt="{\displaystyle r-r_{0}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  leži u tangentnoj ravni, koja ja normalna na vektor normale. Jednaćina tangentne ravni može se napisati pomoću mješovitog proizvoda <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (r-r_{0})*(r_{u}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <mi> r </mi> <mo> − </mo> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> ∗ </mo> <mo stretchy="false"> ( </mo> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> u </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (r-r_{0})*(r_{u}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8298bc4c4f9b4f7802302c11442f61e1c3586d12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.122ex; height:2.843ex;" alt="{\displaystyle (r-r_{0})*(r_{u}}"> </noscript><span class="lazy-image-placeholder" style="width: 13.122ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8298bc4c4f9b4f7802302c11442f61e1c3586d12" data-alt="{\displaystyle (r-r_{0})*(r_{u}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> x <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{v}=0}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mo> = </mo> <mn> 0 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r_{v}=0} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d5b900acbfbc88f617f99771ee31755abffafbc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.339ex; height:2.509ex;" alt="{\displaystyle r_{v}=0}"> </noscript><span class="lazy-image-placeholder" style="width: 6.339ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d5b900acbfbc88f617f99771ee31755abffafbc" data-alt="{\displaystyle r_{v}=0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Može se napisati i u skalarnim komponentama pomoću determinante <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{vmatrix}x-x_{0}&y-y_{0}&z-z_{0}\\x_{u}&y_{u}&z_{u}\\x_{v}&y_{v}&z_{v}\end{vmatrix}}=0}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> | </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi> x </mi> <mo> − </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> </mtd> <mtd> <mi> y </mi> <mo> − </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> </mtd> <mtd> <mi> z </mi> <mo> − </mo> <msub> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> u </mi> </mrow> </msub> </mtd> <mtd> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> u </mi> </mrow> </msub> </mtd> <mtd> <msub> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> u </mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> </mtd> <mtd> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> </mtd> <mtd> <msub> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo> | </mo> </mrow> </mrow> <mo> = </mo> <mn> 0 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{vmatrix}x-x_{0}&y-y_{0}&z-z_{0}\\x_{u}&y_{u}&z_{u}\\x_{v}&y_{v}&z_{v}\end{vmatrix}}=0} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f21c66d409793a37c47da7e4d34453b11c14781f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:29.759ex; height:9.509ex;" alt="{\displaystyle {\begin{vmatrix}x-x_{0}&y-y_{0}&z-z_{0}\\x_{u}&y_{u}&z_{u}\\x_{v}&y_{v}&z_{v}\end{vmatrix}}=0}"> </noscript><span class="lazy-image-placeholder" style="width: 29.759ex;height: 9.509ex;vertical-align: -4.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f21c66d409793a37c47da7e4d34453b11c14781f" data-alt="{\displaystyle {\begin{vmatrix}x-x_{0}&y-y_{0}&z-z_{0}\\x_{u}&y_{u}&z_{u}\\x_{v}&y_{v}&z_{v}\end{vmatrix}}=0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  gdje su <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> x </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle x} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> </noscript><span class="lazy-image-placeholder" style="width: 1.33ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" data-alt="{\displaystyle x}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> y </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle y} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"> </noscript><span class="lazy-image-placeholder" style="width: 1.155ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" data-alt="{\displaystyle y}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> z </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle z} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"> </noscript><span class="lazy-image-placeholder" style="width: 1.088ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" data-alt="{\displaystyle z}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  koordinate bilo koje tačke tangentne ravni, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x-0}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> x </mi> <mo> − </mo> <mn> 0 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle x-0} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4afac6dc665baba635230a2df9798f826f985379" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.333ex; height:2.343ex;" alt="{\displaystyle x-0}"> </noscript><span class="lazy-image-placeholder" style="width: 5.333ex;height: 2.343ex;vertical-align: -0.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4afac6dc665baba635230a2df9798f826f985379" data-alt="{\displaystyle x-0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{0}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle y_{0}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d943dbbb0b56ca750c4d62c5b54b4ae29a773da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.193ex; height:2.009ex;" alt="{\displaystyle y_{0}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.193ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d943dbbb0b56ca750c4d62c5b54b4ae29a773da" data-alt="{\displaystyle y_{0}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{0}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle z_{0}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e72d1d86e86355892b39b8eb32b964834e113bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.135ex; height:2.009ex;" alt="{\displaystyle z_{0}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.135ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e72d1d86e86355892b39b8eb32b964834e113bf" data-alt="{\displaystyle z_{0}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  koordinate dirališta T, a u derivacije koordinata uvrštavaju se vrijednosti <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> u </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle u} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}"> </noscript><span class="lazy-image-placeholder" style="width: 1.33ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" data-alt="{\displaystyle u}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  i <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> v </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle v} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"> </noscript><span class="lazy-image-placeholder" style="width: 1.128ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" data-alt="{\displaystyle v}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  koje odgovaraju tački T. Jednačina normale površi u tački T je <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=r_{0}+\rho (r_{u}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> r </mi> <mo> = </mo> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> + </mo> <mi> ρ </mi> <mo stretchy="false"> ( </mo> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> u </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r=r_{0}+\rho (r_{u}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b97b1d2814901842d581070e00fd65a4141822e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.418ex; height:2.843ex;" alt="{\displaystyle r=r_{0}+\rho (r_{u}}"> </noscript><span class="lazy-image-placeholder" style="width: 13.418ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b97b1d2814901842d581070e00fd65a4141822e9" data-alt="{\displaystyle r=r_{0}+\rho (r_{u}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  × <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{v})}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r_{v})} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/262810778ae910b5817d9833ed138ce20e029d23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.983ex; height:2.843ex;" alt="{\displaystyle r_{v})}"> </noscript><span class="lazy-image-placeholder" style="width: 2.983ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/262810778ae910b5817d9833ed138ce20e029d23" data-alt="{\displaystyle r_{v})}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  gdje je <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> ρ </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \rho } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"> </noscript><span class="lazy-image-placeholder" style="width: 1.202ex;height: 2.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" data-alt="{\displaystyle \rho }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  realni parametar koji prima vrijednosti između <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\infty }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo> − </mo> <mi mathvariant="normal"> ∞ </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle -\infty } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle -\infty }"> </noscript><span class="lazy-image-placeholder" style="width: 4.132ex;height: 2.176ex;vertical-align: -0.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1" data-alt="{\displaystyle -\infty }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  i <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +\infty }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo> + </mo> <mi mathvariant="normal"> ∞ </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle +\infty } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bddbb0e4420a7e744cf71bd71216e11b0bf88831" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle +\infty }"> </noscript><span class="lazy-image-placeholder" style="width: 4.132ex;height: 2.176ex;vertical-align: -0.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bddbb0e4420a7e744cf71bd71216e11b0bf88831" data-alt="{\displaystyle +\infty }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(6)"> <h2 id="Linijske_površi">Linijske površi</h2> <a role="button" href="https://sh-m-wikipedia-org.translate.goog/w/index.php?title=Povr%C5%A1&action=edit&section=6&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Uredi odjeljak Linijske površi" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> uredi </a> </div> <section class="mf-section-6 collapsible-block" id="mf-section-6"> <a href="https://sh-m-wikipedia-org.translate.goog/w/index.php?title=Linijska_povr%C5%A1&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Linijska površ (stranica ne postoji)">Linijska površ</a> je skup pravih prostora neprekinuto povezanih po nekom zakonu . Nastaju na sljedeći način: <ul> <li>klizanjem prave po nekoj prostornoj <a href="https://sh-m-wikipedia-org.translate.goog/wiki/Kriva?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Kriva">krivoj</a>. Prava koja klizi naziva se izvodnica ili generatrisa, a <a href="https://sh-m-wikipedia-org.translate.goog/w/index.php?title=Krive_drugog_reda&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Krive drugog reda (stranica ne postoji)">kriva </a> po kojoj klize, ravnalica ili greben površi</li> <li>povezivanjem triju krivih (ravnalica) transverzalama.</li> </ul> Ako su za ravnalice odabrane algebarske krive, nastaje algebarska površ. Za ovaj prikaz bitne su samo površi koje nastaju povezivanjem triju algebarskih ravnalica transverzalama. Njihova se konstrukcija može izvesti na sljedeći način: Neka su zadane krive <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{1}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> k </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle k_{1}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/376315fd4983f01dada5ec2f7bebc48455b14a66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.265ex; height:2.509ex;" alt="{\displaystyle k_{1}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.265ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/376315fd4983f01dada5ec2f7bebc48455b14a66" data-alt="{\displaystyle k_{1}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{2}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> k </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle k_{2}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c51b4ba57ee596d8435fc4ed76703ca3a2fc444a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.265ex; height:2.509ex;" alt="{\displaystyle k_{2}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.265ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c51b4ba57ee596d8435fc4ed76703ca3a2fc444a" data-alt="{\displaystyle k_{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  i <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{3}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> k </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle k_{3}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40d32e1c66b85257bfd6ad8be93186742d71a804" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.265ex; height:2.509ex;" alt="{\displaystyle k_{3}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.265ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40d32e1c66b85257bfd6ad8be93186742d71a804" data-alt="{\displaystyle k_{3}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Na krivoj <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{1}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> k </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle k_{1}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/376315fd4983f01dada5ec2f7bebc48455b14a66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.265ex; height:2.509ex;" alt="{\displaystyle k_{1}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.265ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/376315fd4983f01dada5ec2f7bebc48455b14a66" data-alt="{\displaystyle k_{1}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  uoči se tačka A koja pravim spoji sa svim talkama krive <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{2}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> k </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle k_{2}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c51b4ba57ee596d8435fc4ed76703ca3a2fc444a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.265ex; height:2.509ex;" alt="{\displaystyle k_{2}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.265ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c51b4ba57ee596d8435fc4ed76703ca3a2fc444a" data-alt="{\displaystyle k_{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  čime je formirana kupa <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> F </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle F} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"> </noscript><span class="lazy-image-placeholder" style="width: 1.741ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" data-alt="{\displaystyle F}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Kriva <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{3}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> k </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle k_{3}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40d32e1c66b85257bfd6ad8be93186742d71a804" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.265ex; height:2.509ex;" alt="{\displaystyle k_{3}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.265ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40d32e1c66b85257bfd6ad8be93186742d71a804" data-alt="{\displaystyle k_{3}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  probada kupu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> F </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle F} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"> </noscript><span class="lazy-image-placeholder" style="width: 1.741ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" data-alt="{\displaystyle F}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  u konačnom broju tačaka. Jednim tako dobivenim probodištem prolazi izvodnica kupe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> F </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle F} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"> </noscript><span class="lazy-image-placeholder" style="width: 1.741ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" data-alt="{\displaystyle F}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , a to je ujedno i transverzala krivih <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{1}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> k </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle k_{1}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/376315fd4983f01dada5ec2f7bebc48455b14a66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.265ex; height:2.509ex;" alt="{\displaystyle k_{1}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.265ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/376315fd4983f01dada5ec2f7bebc48455b14a66" data-alt="{\displaystyle k_{1}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{2}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> k </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle k_{2}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c51b4ba57ee596d8435fc4ed76703ca3a2fc444a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.265ex; height:2.509ex;" alt="{\displaystyle k_{2}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.265ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c51b4ba57ee596d8435fc4ed76703ca3a2fc444a" data-alt="{\displaystyle k_{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  i <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{3}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> k </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle k_{3}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40d32e1c66b85257bfd6ad8be93186742d71a804" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.265ex; height:2.509ex;" alt="{\displaystyle k_{3}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.265ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40d32e1c66b85257bfd6ad8be93186742d71a804" data-alt="{\displaystyle k_{3}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Taj se postupak ponavlja za ostale tačke krive <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{1}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> k </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle k_{1}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/376315fd4983f01dada5ec2f7bebc48455b14a66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.265ex; height:2.509ex;" alt="{\displaystyle k_{1}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.265ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/376315fd4983f01dada5ec2f7bebc48455b14a66" data-alt="{\displaystyle k_{1}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> čcime je formiran jednoparametarski skup izvodnica <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> i </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle i} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"> </noscript><span class="lazy-image-placeholder" style="width: 0.802ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" data-alt="{\displaystyle i}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Sve takve izvodnice i čine linijsku površ. <dl> <dt> <a href="https://sh-m-wikipedia-org.translate.goog/wiki/Teorem?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Teorem">Teorema</a> (o redu linijske povrsi) </dt> </dl> Ako su algebarske krive <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{1}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> k </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle k_{1}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/376315fd4983f01dada5ec2f7bebc48455b14a66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.265ex; height:2.509ex;" alt="{\displaystyle k_{1}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.265ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/376315fd4983f01dada5ec2f7bebc48455b14a66" data-alt="{\displaystyle k_{1}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{2}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> k </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle k_{2}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c51b4ba57ee596d8435fc4ed76703ca3a2fc444a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.265ex; height:2.509ex;" alt="{\displaystyle k_{2}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.265ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c51b4ba57ee596d8435fc4ed76703ca3a2fc444a" data-alt="{\displaystyle k_{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  i <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{3}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> k </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle k_{3}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40d32e1c66b85257bfd6ad8be93186742d71a804" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.265ex; height:2.509ex;" alt="{\displaystyle k_{3}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.265ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40d32e1c66b85257bfd6ad8be93186742d71a804" data-alt="{\displaystyle k_{3}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  redova <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{1}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> n </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n_{1}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee784b70e772f55ede5e6e0bdc929994bff63413" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.449ex; height:2.009ex;" alt="{\displaystyle n_{1}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.449ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee784b70e772f55ede5e6e0bdc929994bff63413" data-alt="{\displaystyle n_{1}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{2}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> n </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n_{2}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/840e456e3058bc0be28e5cf653b170cdbfcc3be4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.449ex; height:2.009ex;" alt="{\displaystyle n_{2}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.449ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/840e456e3058bc0be28e5cf653b170cdbfcc3be4" data-alt="{\displaystyle n_{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  i <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{3}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> n </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n_{3}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d35ab39fc1af104a61e369bf3b6065e3612a5f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.449ex; height:2.009ex;" alt="{\displaystyle n_{3}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.449ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d35ab39fc1af104a61e369bf3b6065e3612a5f7" data-alt="{\displaystyle n_{3}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . i ako se krive <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{1}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> k </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle k_{1}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/376315fd4983f01dada5ec2f7bebc48455b14a66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.265ex; height:2.509ex;" alt="{\displaystyle k_{1}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.265ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/376315fd4983f01dada5ec2f7bebc48455b14a66" data-alt="{\displaystyle k_{1}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{2}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> k </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle k_{2}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c51b4ba57ee596d8435fc4ed76703ca3a2fc444a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.265ex; height:2.509ex;" alt="{\displaystyle k_{2}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.265ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c51b4ba57ee596d8435fc4ed76703ca3a2fc444a" data-alt="{\displaystyle k_{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  sijeku u <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{3}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> s </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle s_{3}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bd55c668c8fec10ff916bef319c7ff9d3a7de7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.145ex; height:2.009ex;" alt="{\displaystyle s_{3}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.145ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bd55c668c8fec10ff916bef319c7ff9d3a7de7a" data-alt="{\displaystyle s_{3}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  tačaka krive <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{1}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> k </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle k_{1}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/376315fd4983f01dada5ec2f7bebc48455b14a66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.265ex; height:2.509ex;" alt="{\displaystyle k_{1}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.265ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/376315fd4983f01dada5ec2f7bebc48455b14a66" data-alt="{\displaystyle k_{1}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  i <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{3}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> k </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle k_{3}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40d32e1c66b85257bfd6ad8be93186742d71a804" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.265ex; height:2.509ex;" alt="{\displaystyle k_{3}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.265ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40d32e1c66b85257bfd6ad8be93186742d71a804" data-alt="{\displaystyle k_{3}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  u <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{2}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> s </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle s_{2}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4b9a7acc0ae8f54da4b7f4eef2c777d44faecd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.145ex; height:2.009ex;" alt="{\displaystyle s_{2}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.145ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4b9a7acc0ae8f54da4b7f4eef2c777d44faecd4" data-alt="{\displaystyle s_{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , a krive <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{2}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> k </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle k_{2}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c51b4ba57ee596d8435fc4ed76703ca3a2fc444a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.265ex; height:2.509ex;" alt="{\displaystyle k_{2}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.265ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c51b4ba57ee596d8435fc4ed76703ca3a2fc444a" data-alt="{\displaystyle k_{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  i <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{3}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> k </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle k_{3}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40d32e1c66b85257bfd6ad8be93186742d71a804" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.265ex; height:2.509ex;" alt="{\displaystyle k_{3}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.265ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40d32e1c66b85257bfd6ad8be93186742d71a804" data-alt="{\displaystyle k_{3}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  u <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{1}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> s </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle s_{1}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb8baad278d51283e0ef3c99898d583cf2c8a8fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.145ex; height:2.009ex;" alt="{\displaystyle s_{1}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.145ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb8baad278d51283e0ef3c99898d583cf2c8a8fd" data-alt="{\displaystyle s_{1}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  tacaka, tada je linijska površ zadana krivama <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{1}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> k </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle k_{1}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/376315fd4983f01dada5ec2f7bebc48455b14a66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.265ex; height:2.509ex;" alt="{\displaystyle k_{1}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.265ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/376315fd4983f01dada5ec2f7bebc48455b14a66" data-alt="{\displaystyle k_{1}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{2}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> k </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle k_{2}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c51b4ba57ee596d8435fc4ed76703ca3a2fc444a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.265ex; height:2.509ex;" alt="{\displaystyle k_{2}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.265ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c51b4ba57ee596d8435fc4ed76703ca3a2fc444a" data-alt="{\displaystyle k_{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  i <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{3}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> k </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle k_{3}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40d32e1c66b85257bfd6ad8be93186742d71a804" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.265ex; height:2.509ex;" alt="{\displaystyle k_{3}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.265ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40d32e1c66b85257bfd6ad8be93186742d71a804" data-alt="{\displaystyle k_{3}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  reda: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R=2n_{1}n_{2}n_{3}-(s_{3}n_{3}+s_{2}n_{2}+s_{1}n_{1})}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> R </mi> <mo> = </mo> <mn> 2 </mn> <msub> <mi> n </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <msub> <mi> n </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <msub> <mi> n </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> <mo> − </mo> <mo stretchy="false"> ( </mo> <msub> <mi> s </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> <msub> <mi> n </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> <mo> + </mo> <msub> <mi> s </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <msub> <mi> n </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo> + </mo> <msub> <mi> s </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <msub> <mi> n </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle R=2n_{1}n_{2}n_{3}-(s_{3}n_{3}+s_{2}n_{2}+s_{1}n_{1})} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53ea61e44668dcc6687d00ac84368366ac30fd51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.483ex; height:2.843ex;" alt="{\displaystyle R=2n_{1}n_{2}n_{3}-(s_{3}n_{3}+s_{2}n_{2}+s_{1}n_{1})}"> </noscript><span class="lazy-image-placeholder" style="width: 37.483ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53ea61e44668dcc6687d00ac84368366ac30fd51" data-alt="{\displaystyle R=2n_{1}n_{2}n_{3}-(s_{3}n_{3}+s_{2}n_{2}+s_{1}n_{1})}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Svaka algebarska linijska površ ima stepen. Linijske površi mogu biti razmotive i nerazmotive ili vitopere. Vitopere linijske površi ne mogu se razmotati u ravni jer su im svake dvije neizmjerno blize izvodnice mimoilazne prave. </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(7)"> <h2 id="Elipsoid">Elipsoid</h2> <a role="button" href="https://sh-m-wikipedia-org.translate.goog/w/index.php?title=Povr%C5%A1&action=edit&section=7&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Uredi odjeljak Elipsoid" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> uredi </a> </div> <section class="mf-section-7 collapsible-block" id="mf-section-7"> <a href="https://sh-m-wikipedia-org.translate.goog/w/index.php?title=Elipsoid&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Elipsoid (stranica ne postoji)">Elipsoid</a> (troosi) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \displaystyle {\frac {(x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}+{\frac {z^{2}}{c^{2}}}=1}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false"> ( </mo> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mfrac> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <msup> <mi> b </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mfrac> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <msup> <mi> c </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mfrac> </mrow> <mo> = </mo> <mn> 1 </mn> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \displaystyle {\frac {(x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}+{\frac {z^{2}}{c^{2}}}=1} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ee519d1e59d468d46faa2b4c1ec9964ef3769c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:20.098ex; height:6.176ex;" alt="{\displaystyle \displaystyle {\frac {(x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}+{\frac {z^{2}}{c^{2}}}=1}"> </noscript><span class="lazy-image-placeholder" style="width: 20.098ex;height: 6.176ex;vertical-align: -2.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ee519d1e59d468d46faa2b4c1ec9964ef3769c6" data-alt="{\displaystyle \displaystyle {\frac {(x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}+{\frac {z^{2}}{c^{2}}}=1}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Ako je <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a>b>c>0}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> a </mi> <mo> > </mo> <mi> b </mi> <mo> > </mo> <mi> c </mi> <mo> > </mo> <mn> 0 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle a>b>c>0} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/291666363e8b41ff5b645aa60b7bbfe5a136356e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.692ex; height:2.176ex;" alt="{\displaystyle a>b>c>0}"> </noscript><span class="lazy-image-placeholder" style="width: 13.692ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/291666363e8b41ff5b645aa60b7bbfe5a136356e" data-alt="{\displaystyle a>b>c>0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  tada kažemo da je <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> a </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle a} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> </noscript><span class="lazy-image-placeholder" style="width: 1.23ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" data-alt="{\displaystyle a}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  velika poluosa, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> b </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle b} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"> </noscript><span class="lazy-image-placeholder" style="width: 0.998ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" data-alt="{\displaystyle b}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  srednja poluosa i <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> c </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle c} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"> </noscript><span class="lazy-image-placeholder" style="width: 1.007ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" data-alt="{\displaystyle c}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  mala poluosa elipsoida. Ako su dvije poluose jednake, npr. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a>b=c>0}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> a </mi> <mo> > </mo> <mi> b </mi> <mo> = </mo> <mi> c </mi> <mo> > </mo> <mn> 0 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle a>b=c>0} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af740443bff546abe767a00b9a9935107aae7071" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.692ex; height:2.176ex;" alt="{\displaystyle a>b=c>0}"> </noscript><span class="lazy-image-placeholder" style="width: 13.692ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af740443bff546abe767a00b9a9935107aae7071" data-alt="{\displaystyle a>b=c>0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  tada dobijemo rotacioni elipsoid. Ako su sve tri poluose jednake dobijamo sferu ili loptinu površ. <ul class="gallery mw-gallery-traditional"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"> <a href="https://sh-m-wikipedia-org.translate.goog/wiki/Datoteka:Ellipsoid_321.png?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"> <noscript> <img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Ellipsoid_321.png/120px-Ellipsoid_321.png" decoding="async" width="120" height="80" class="mw-file-element" data-file-width="475" data-file-height="315"> </noscript><span class="lazy-image-placeholder" style="width: 120px;height: 80px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Ellipsoid_321.png/120px-Ellipsoid_321.png" data-width="120" data-height="80" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Ellipsoid_321.png/180px-Ellipsoid_321.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Ellipsoid_321.png/240px-Ellipsoid_321.png 2x" data-class="mw-file-element"> </a> </div> <div class="gallerytext"></div></li> </ul> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \displaystyle {\frac {(x-x_{0})2}{a^{2}}}+{\frac {(y-y_{0})2}{b^{2}}}+{\frac {(z-z_{0})2}{c^{2}}}=1}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> − </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> <mn> 2 </mn> </mrow> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mfrac> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false"> ( </mo> <mi> y </mi> <mo> − </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> <mn> 2 </mn> </mrow> <msup> <mi> b </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mfrac> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false"> ( </mo> <mi> z </mi> <mo> − </mo> <msub> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> <mn> 2 </mn> </mrow> <msup> <mi> c </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mfrac> </mrow> <mo> = </mo> <mn> 1 </mn> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \displaystyle {\frac {(x-x_{0})2}{a^{2}}}+{\frac {(y-y_{0})2}{b^{2}}}+{\frac {(z-z_{0})2}{c^{2}}}=1} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ec72c4cbe6522dcf73644179f0d0a98095f2737" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:40.173ex; height:6.009ex;" alt="{\displaystyle \displaystyle {\frac {(x-x_{0})2}{a^{2}}}+{\frac {(y-y_{0})2}{b^{2}}}+{\frac {(z-z_{0})2}{c^{2}}}=1}"> </noscript><span class="lazy-image-placeholder" style="width: 40.173ex;height: 6.009ex;vertical-align: -2.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ec72c4cbe6522dcf73644179f0d0a98095f2737" data-alt="{\displaystyle \displaystyle {\frac {(x-x_{0})2}{a^{2}}}+{\frac {(y-y_{0})2}{b^{2}}}+{\frac {(z-z_{0})2}{c^{2}}}=1}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  je jednačina elipsoida čije su glavne ose paralne s koordinatnim osama <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y,z}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> x </mi> <mo> , </mo> <mi> y </mi> <mo> , </mo> <mi> z </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle x,y,z} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbeca34b28f569a407ef74a955d041df9f360268" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.641ex; height:2.009ex;" alt="{\displaystyle x,y,z}"> </noscript><span class="lazy-image-placeholder" style="width: 5.641ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbeca34b28f569a407ef74a955d041df9f360268" data-alt="{\displaystyle x,y,z}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  , a dužine poluosa su <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b,c}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> a </mi> <mo> , </mo> <mi> b </mi> <mo> , </mo> <mi> c </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle a,b,c} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f13f068df656c1b1911ae9f81628c49a6181194d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.302ex; height:2.509ex;" alt="{\displaystyle a,b,c}"> </noscript><span class="lazy-image-placeholder" style="width: 5.302ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f13f068df656c1b1911ae9f81628c49a6181194d" data-alt="{\displaystyle a,b,c}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  redom. Nivo plohe elipsoida kao i presjeci s ravnima paralelnim s <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xz}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> x </mi> <mi> z </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle xz} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97005bee6e83614cf6ce64d4e68e5ab2ac280709" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.418ex; height:1.676ex;" alt="{\displaystyle xz}"> </noscript><span class="lazy-image-placeholder" style="width: 2.418ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97005bee6e83614cf6ce64d4e68e5ab2ac280709" data-alt="{\displaystyle xz}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  i <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle yz}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> y </mi> <mi> z </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle yz} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9c7cbe4344a9d72acfc98a97e2b3ec44bb48d1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.244ex; height:2.009ex;" alt="{\displaystyle yz}"> </noscript><span class="lazy-image-placeholder" style="width: 2.244ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9c7cbe4344a9d72acfc98a97e2b3ec44bb48d1f" data-alt="{\displaystyle yz}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  ravnima su<a href="https://sh-m-wikipedia-org.translate.goog/wiki/Elipsa?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Elipsa"> elipse</a>. </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(8)"> <h2 id="Hiperboloid">Hiperboloid</h2> <a role="button" href="https://sh-m-wikipedia-org.translate.goog/w/index.php?title=Povr%C5%A1&action=edit&section=8&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Uredi odjeljak Hiperboloid" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> uredi </a> </div> <section class="mf-section-8 collapsible-block" id="mf-section-8"> <ul class="gallery mw-gallery-traditional"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"> <a href="https://sh-m-wikipedia-org.translate.goog/wiki/Datoteka:Hyperboloid_jednodilny_rotacni.png?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"> <noscript> <img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e9/Hyperboloid_jednodilny_rotacni.png/96px-Hyperboloid_jednodilny_rotacni.png" decoding="async" width="96" height="120" class="mw-file-element" data-file-width="346" data-file-height="431"> </noscript><span class="lazy-image-placeholder" style="width: 96px;height: 120px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e9/Hyperboloid_jednodilny_rotacni.png/96px-Hyperboloid_jednodilny_rotacni.png" data-width="96" data-height="120" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e9/Hyperboloid_jednodilny_rotacni.png/144px-Hyperboloid_jednodilny_rotacni.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e9/Hyperboloid_jednodilny_rotacni.png/193px-Hyperboloid_jednodilny_rotacni.png 2x" data-class="mw-file-element"> </a> </div> <div class="gallerytext"></div></li> </ul> Jednokrilni <a href="https://sh-m-wikipedia-org.translate.goog/wiki/Hiperboloid?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Hiperboloid">hiperboloid</a> zadan je formulom <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}-{\frac {z^{2}}{c^{2}}}=1}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mfrac> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <msup> <mi> b </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mfrac> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <msup> <mi> c </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mfrac> </mrow> <mo> = </mo> <mn> 1 </mn> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}-{\frac {z^{2}}{c^{2}}}=1} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39791d8101f971b4f41a5d9e380483d69b5fe295" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:19.193ex; height:6.009ex;" alt="{\displaystyle \displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}-{\frac {z^{2}}{c^{2}}}=1}"> </noscript><span class="lazy-image-placeholder" style="width: 19.193ex;height: 6.009ex;vertical-align: -2.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39791d8101f971b4f41a5d9e380483d69b5fe295" data-alt="{\displaystyle \displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}-{\frac {z^{2}}{c^{2}}}=1}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Dvokrilni hiperboloid zadan je s formulom <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \displaystyle -{\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}+{\frac {z^{2}}{c^{2}}}=1}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="true" scriptlevel="0"> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mfrac> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <msup> <mi> b </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mfrac> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <msup> <mi> c </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mfrac> </mrow> <mo> = </mo> <mn> 1 </mn> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \displaystyle -{\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}+{\frac {z^{2}}{c^{2}}}=1} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/605bf2fd4ff606a8f8b74a4ce05ceb177a6925bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:21.001ex; height:6.009ex;" alt="{\displaystyle \displaystyle -{\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}+{\frac {z^{2}}{c^{2}}}=1}"> </noscript><span class="lazy-image-placeholder" style="width: 21.001ex;height: 6.009ex;vertical-align: -2.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/605bf2fd4ff606a8f8b74a4ce05ceb177a6925bc" data-alt="{\displaystyle \displaystyle -{\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}+{\frac {z^{2}}{c^{2}}}=1}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Nivo površi hiperboloida su <a href="https://sh-m-wikipedia-org.translate.goog/wiki/Elipsa?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Elipsa">elipse</a>, a presjeci s <a href="https://sh-m-wikipedia-org.translate.goog/wiki/Ravan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ravan">ravnima</a> koje su paralelne s <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> z </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle z} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"> </noscript><span class="lazy-image-placeholder" style="width: 1.088ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" data-alt="{\displaystyle z}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  osom su <a href="https://sh-m-wikipedia-org.translate.goog/wiki/Hiperbola?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Hiperbola">hiperbole.</a> Kao i kod ostalih površi, pomoću transformacije <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\to x-x_{0}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> x </mi> <mo stretchy="false"> → </mo> <mi> x </mi> <mo> − </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle x\to x-x_{0}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f3bcc1efa8389d933490e5f692a45701d25c061" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.498ex; height:2.343ex;" alt="{\displaystyle x\to x-x_{0}}"> </noscript><span class="lazy-image-placeholder" style="width: 11.498ex;height: 2.343ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f3bcc1efa8389d933490e5f692a45701d25c061" data-alt="{\displaystyle x\to x-x_{0}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  pomićemo središte hiperboloida, a cikličkom zamjenom varijabli nastaju hiperboloidi koji se protežu u smjeru ostalih koordinatnih osi. </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(9)"> <h2 id="Konusne_površi">Konusne površi</h2> <a role="button" href="https://sh-m-wikipedia-org.translate.goog/w/index.php?title=Povr%C5%A1&action=edit&section=9&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Uredi odjeljak Konusne površi" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> uredi </a> </div> <section class="mf-section-9 collapsible-block" id="mf-section-9"> <ul class="gallery mw-gallery-traditional"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"> <a href="https://sh-m-wikipedia-org.translate.goog/wiki/Datoteka:DoubleCone.png?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"> <noscript> <img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/DoubleCone.png/120px-DoubleCone.png" decoding="async" width="120" height="113" class="mw-file-element" data-file-width="1350" data-file-height="1274"> </noscript><span class="lazy-image-placeholder" style="width: 120px;height: 113px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/DoubleCone.png/120px-DoubleCone.png" data-width="120" data-height="113" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/DoubleCone.png/180px-DoubleCone.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/72/DoubleCone.png/240px-DoubleCone.png 2x" data-class="mw-file-element"> </a> </div> <div class="gallerytext"></div></li> </ul> <a href="https://sh-m-wikipedia-org.translate.goog/wiki/Kupa_(geometrija)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Kupa (geometrija)">Konus</a> (kupa) je zadana formulom <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \displaystyle (z-z_{0})^{2}={\frac {(x-x_{0})^{2}}{a^{2}}}+{\frac {(y-y_{0})^{2}}{b^{2}}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <mi> z </mi> <mo> − </mo> <msub> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> − </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mfrac> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false"> ( </mo> <mi> y </mi> <mo> − </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> <msup> <mi> b </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \displaystyle (z-z_{0})^{2}={\frac {(x-x_{0})^{2}}{a^{2}}}+{\frac {(y-y_{0})^{2}}{b^{2}}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97f6a61a1c088b34a6e17a35b82675a37930a2d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:35.009ex; height:6.176ex;" alt="{\displaystyle \displaystyle (z-z_{0})^{2}={\frac {(x-x_{0})^{2}}{a^{2}}}+{\frac {(y-y_{0})^{2}}{b^{2}}}}"> </noscript><span class="lazy-image-placeholder" style="width: 35.009ex;height: 6.176ex;vertical-align: -2.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97f6a61a1c088b34a6e17a35b82675a37930a2d4" data-alt="{\displaystyle \displaystyle (z-z_{0})^{2}={\frac {(x-x_{0})^{2}}{a^{2}}}+{\frac {(y-y_{0})^{2}}{b^{2}}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Ovim izrazom su zadane dvije funkcije od dvije varijable: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \displaystyle z=z_{0}+{\sqrt {{\frac {(x-x_{0})^{2}}{a^{2}}}+{\frac {(y-y_{0})^{2}}{b^{2}}}}}\quad {\textrm {i}}\quad z=z_{0}-{\sqrt {{\frac {(x-x_{0})^{2}}{a^{2}}}+{\frac {(y-y_{0})^{2}}{b^{2}}}}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="true" scriptlevel="0"> <mi> z </mi> <mo> = </mo> <msub> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> − </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mfrac> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false"> ( </mo> <mi> y </mi> <mo> − </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> <msup> <mi> b </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mfrac> </mrow> </msqrt> </mrow> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext> i </mtext> </mrow> </mrow> <mspace width="1em"></mspace> <mi> z </mi> <mo> = </mo> <msub> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> − </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mfrac> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false"> ( </mo> <mi> y </mi> <mo> − </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> <msup> <mi> b </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mfrac> </mrow> </msqrt> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \displaystyle z=z_{0}+{\sqrt {{\frac {(x-x_{0})^{2}}{a^{2}}}+{\frac {(y-y_{0})^{2}}{b^{2}}}}}\quad {\textrm {i}}\quad z=z_{0}-{\sqrt {{\frac {(x-x_{0})^{2}}{a^{2}}}+{\frac {(y-y_{0})^{2}}{b^{2}}}}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95c7b8e83642727a79754b594b2346e956394cf8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:74.23ex; height:7.676ex;" alt="{\displaystyle \displaystyle z=z_{0}+{\sqrt {{\frac {(x-x_{0})^{2}}{a^{2}}}+{\frac {(y-y_{0})^{2}}{b^{2}}}}}\quad {\textrm {i}}\quad z=z_{0}-{\sqrt {{\frac {(x-x_{0})^{2}}{a^{2}}}+{\frac {(y-y_{0})^{2}}{b^{2}}}}}}"> </noscript><span class="lazy-image-placeholder" style="width: 74.23ex;height: 7.676ex;vertical-align: -3.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95c7b8e83642727a79754b594b2346e956394cf8" data-alt="{\displaystyle \displaystyle z=z_{0}+{\sqrt {{\frac {(x-x_{0})^{2}}{a^{2}}}+{\frac {(y-y_{0})^{2}}{b^{2}}}}}\quad {\textrm {i}}\quad z=z_{0}-{\sqrt {{\frac {(x-x_{0})^{2}}{a^{2}}}+{\frac {(y-y_{0})^{2}}{b^{2}}}}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Želimo pronaći<a href="https://sh-m-wikipedia-org.translate.goog/wiki/Jedna%C4%8Dine?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Jednačine"> jednačinu</a> konusne površi čije izvodnice prolaze kroz koordinantni početak<a href="https://sh-m-wikipedia-org.translate.goog/wiki/Koordinatni_sistem?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-disambig" title="Koordinatni sistem"> koordinatnog sistema i</a> kroz tačke krive <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x,y)=0}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> F </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> , </mo> <mi> y </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mn> 0 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle F(x,y)=0} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c910dd64922e23f212bc46e970af0540811c48f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.33ex; height:2.843ex;" alt="{\displaystyle F(x,y)=0}"> </noscript><span class="lazy-image-placeholder" style="width: 11.33ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c910dd64922e23f212bc46e970af0540811c48f2" data-alt="{\displaystyle F(x,y)=0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=1}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> z </mi> <mo> = </mo> <mn> 1 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle z=1} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/078535cde78d90bfa1d9fbb2446204593a921d57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.349ex; height:2.176ex;" alt="{\displaystyle z=1}"> </noscript><span class="lazy-image-placeholder" style="width: 5.349ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/078535cde78d90bfa1d9fbb2446204593a921d57" data-alt="{\displaystyle z=1}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Na toj krivoj odaberimo proizvoljnu tačku <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{0}(x_{0},y_{0},1)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> T </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> , </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> , </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle T_{0}(x_{0},y_{0},1)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a7a8965c42a0e4fcc580f193699add25e9a6c6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.029ex; height:2.843ex;" alt="{\displaystyle T_{0}(x_{0},y_{0},1)}"> </noscript><span class="lazy-image-placeholder" style="width: 12.029ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a7a8965c42a0e4fcc580f193699add25e9a6c6d" data-alt="{\displaystyle T_{0}(x_{0},y_{0},1)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Jednačina izvodnice (prave) kroz tačke <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(0,0,0)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> O </mi> <mo stretchy="false"> ( </mo> <mn> 0 </mn> <mo> , </mo> <mn> 0 </mn> <mo> , </mo> <mn> 0 </mn> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle O(0,0,0)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07c0a5cba5b86d77b215ff1e4162660220cb0318" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.138ex; height:2.843ex;" alt="{\displaystyle O(0,0,0)}"> </noscript><span class="lazy-image-placeholder" style="width: 9.138ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07c0a5cba5b86d77b215ff1e4162660220cb0318" data-alt="{\displaystyle O(0,0,0)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  i <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{0}(x_{0},y_{0},1)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> T </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> , </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> , </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle T_{0}(x_{0},y_{0},1)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a7a8965c42a0e4fcc580f193699add25e9a6c6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.029ex; height:2.843ex;" alt="{\displaystyle T_{0}(x_{0},y_{0},1)}"> </noscript><span class="lazy-image-placeholder" style="width: 12.029ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a7a8965c42a0e4fcc580f193699add25e9a6c6d" data-alt="{\displaystyle T_{0}(x_{0},y_{0},1)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  glasi <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {x}{x_{0}}}={\frac {x}{y_{0}}}={\frac {z}{1}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> x </mi> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> </mfrac> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> x </mi> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> </mfrac> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> z </mi> <mn> 1 </mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\frac {x}{x_{0}}}={\frac {x}{y_{0}}}={\frac {z}{1}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/504a7e317b89fccea464027bee55c15658d409dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.445ex; height:5.176ex;" alt="{\displaystyle {\frac {x}{x_{0}}}={\frac {x}{y_{0}}}={\frac {z}{1}}}"> </noscript><span class="lazy-image-placeholder" style="width: 14.445ex;height: 5.176ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/504a7e317b89fccea464027bee55c15658d409dc" data-alt="{\displaystyle {\frac {x}{x_{0}}}={\frac {x}{y_{0}}}={\frac {z}{1}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Vrijedi: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=zx_{0}=>x_{0}={\frac {x}{z}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> x </mi> <mo> = </mo> <mi> z </mi> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> => </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> x </mi> <mi> z </mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle x=zx_{0}=>x_{0}={\frac {x}{z}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ab6fe4ae294910107b27ee09a889c2e6533c034" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.455ex; height:4.676ex;" alt="{\displaystyle x=zx_{0}=>x_{0}={\frac {x}{z}}}"> </noscript><span class="lazy-image-placeholder" style="width: 20.455ex;height: 4.676ex;vertical-align: -1.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ab6fe4ae294910107b27ee09a889c2e6533c034" data-alt="{\displaystyle x=zx_{0}=>x_{0}={\frac {x}{z}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=zy_{0}=>y_{0}={\frac {y}{z}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> y </mi> <mo> = </mo> <mi> z </mi> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> => </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> y </mi> <mi> z </mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle y=zy_{0}=>y_{0}={\frac {y}{z}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3172718e7b24503862f916d3f437462d5e4e2b33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.726ex; height:4.843ex;" alt="{\displaystyle y=zy_{0}=>y_{0}={\frac {y}{z}}}"> </noscript><span class="lazy-image-placeholder" style="width: 19.726ex;height: 4.843ex;vertical-align: -1.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3172718e7b24503862f916d3f437462d5e4e2b33" data-alt="{\displaystyle y=zy_{0}=>y_{0}={\frac {y}{z}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Jednačina konusne površi čije izvodnice prolaze kroz koordinantni početak i kroz tačke krive <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x,z)=0}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> F </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> , </mo> <mi> z </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mn> 0 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle F(x,z)=0} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4a543b259a8b6b3a31a52394b27052ced5ae3b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.263ex; height:2.843ex;" alt="{\displaystyle F(x,z)=0}"> </noscript><span class="lazy-image-placeholder" style="width: 11.263ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4a543b259a8b6b3a31a52394b27052ced5ae3b8" data-alt="{\displaystyle F(x,z)=0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=1}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> y </mi> <mo> = </mo> <mn> 1 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle y=1} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10f53b404b1fdd041a589f1f2425e45a2edba110" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.416ex; height:2.509ex;" alt="{\displaystyle y=1}"> </noscript><span class="lazy-image-placeholder" style="width: 5.416ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10f53b404b1fdd041a589f1f2425e45a2edba110" data-alt="{\displaystyle y=1}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Kako tačka <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{0}(x_{0},y_{0},1)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> T </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> , </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> , </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle T_{0}(x_{0},y_{0},1)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a7a8965c42a0e4fcc580f193699add25e9a6c6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.029ex; height:2.843ex;" alt="{\displaystyle T_{0}(x_{0},y_{0},1)}"> </noscript><span class="lazy-image-placeholder" style="width: 12.029ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a7a8965c42a0e4fcc580f193699add25e9a6c6d" data-alt="{\displaystyle T_{0}(x_{0},y_{0},1)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  leži na krivoj mora vrijediti <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x_{0},y_{0})=0}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> F </mi> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> , </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> = </mo> <mn> 0 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle F(x_{0},y_{0})=0} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78f23f124a1b5ff33397b39d81f7683a9862d01d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.422ex; height:2.843ex;" alt="{\displaystyle F(x_{0},y_{0})=0}"> </noscript><span class="lazy-image-placeholder" style="width: 13.422ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78f23f124a1b5ff33397b39d81f7683a9862d01d" data-alt="{\displaystyle F(x_{0},y_{0})=0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  dobijamo opštu jednačinu konusne površi <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F({\frac {x}{z}}),{\frac {y}{z}})=0}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> F </mi> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> x </mi> <mi> z </mi> </mfrac> </mrow> <mo stretchy="false"> ) </mo> <mo> , </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> y </mi> <mi> z </mi> </mfrac> </mrow> <mo stretchy="false"> ) </mo> <mo> = </mo> <mn> 0 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle F({\frac {x}{z}}),{\frac {y}{z}})=0} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2590d703cb5494af06dd998760a88da69df746de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.907ex; height:4.843ex;" alt="{\displaystyle F({\frac {x}{z}}),{\frac {y}{z}})=0}"> </noscript><span class="lazy-image-placeholder" style="width: 13.907ex;height: 4.843ex;vertical-align: -1.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2590d703cb5494af06dd998760a88da69df746de" data-alt="{\displaystyle F({\frac {x}{z}}),{\frac {y}{z}})=0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Jednačina konusne površi čije izvodnice prolaze kroz koordinantni početak i kroz tačke krive <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x,z)=0}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> F </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> , </mo> <mi> z </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mn> 0 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle F(x,z)=0} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4a543b259a8b6b3a31a52394b27052ced5ae3b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.263ex; height:2.843ex;" alt="{\displaystyle F(x,z)=0}"> </noscript><span class="lazy-image-placeholder" style="width: 11.263ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4a543b259a8b6b3a31a52394b27052ced5ae3b8" data-alt="{\displaystyle F(x,z)=0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  i <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=1}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> y </mi> <mo> = </mo> <mn> 1 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle y=1} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10f53b404b1fdd041a589f1f2425e45a2edba110" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.416ex; height:2.509ex;" alt="{\displaystyle y=1}"> </noscript><span class="lazy-image-placeholder" style="width: 5.416ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10f53b404b1fdd041a589f1f2425e45a2edba110" data-alt="{\displaystyle y=1}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  je </dd> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F({\frac {x}{y}}),{\frac {z}{y}})=0}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> F </mi> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> x </mi> <mi> y </mi> </mfrac> </mrow> <mo stretchy="false"> ) </mo> <mo> , </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> z </mi> <mi> y </mi> </mfrac> </mrow> <mo stretchy="false"> ) </mo> <mo> = </mo> <mn> 0 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle F({\frac {x}{y}}),{\frac {z}{y}})=0} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb2466002eaa6b300a69d431b92d792233913d96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.907ex; height:5.176ex;" alt="{\displaystyle F({\frac {x}{y}}),{\frac {z}{y}})=0}"> </noscript><span class="lazy-image-placeholder" style="width: 13.907ex;height: 5.176ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb2466002eaa6b300a69d431b92d792233913d96" data-alt="{\displaystyle F({\frac {x}{y}}),{\frac {z}{y}})=0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(10)"> <h2 id="Valjkaste_površi">Valjkaste površi</h2> <a role="button" href="https://sh-m-wikipedia-org.translate.goog/w/index.php?title=Povr%C5%A1&action=edit&section=10&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Uredi odjeljak Valjkaste površi" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> uredi </a> </div> <section class="mf-section-10 collapsible-block" id="mf-section-10"> <ul class="gallery mw-gallery-traditional"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"> <a href="https://sh-m-wikipedia-org.translate.goog/wiki/Datoteka:Cylindrical_Coordinates.svg?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"> <noscript> <img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Cylindrical_Coordinates.svg/120px-Cylindrical_Coordinates.svg.png" decoding="async" width="120" height="120" class="mw-file-element" data-file-width="748" data-file-height="745"> </noscript><span class="lazy-image-placeholder" style="width: 120px;height: 120px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Cylindrical_Coordinates.svg/120px-Cylindrical_Coordinates.svg.png" data-width="120" data-height="120" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Cylindrical_Coordinates.svg/180px-Cylindrical_Coordinates.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Cylindrical_Coordinates.svg/240px-Cylindrical_Coordinates.svg.png 2x" data-class="mw-file-element"> </a> </div> <div class="gallerytext"></div></li> </ul> <ul> <li>Izvodnica je paralelna sa osom <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle OZ}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> O </mi> <mi> Z </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle OZ} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ef1fb6520fafba764040ed2928a06cdd5bf481f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.454ex; height:2.176ex;" alt="{\displaystyle OZ}"> </noscript><span class="lazy-image-placeholder" style="width: 3.454ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ef1fb6520fafba764040ed2928a06cdd5bf481f" data-alt="{\displaystyle OZ}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  i prolazi kroz krivu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x,y)=0}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> F </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> , </mo> <mi> y </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mn> 0 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle F(x,y)=0} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c910dd64922e23f212bc46e970af0540811c48f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.33ex; height:2.843ex;" alt="{\displaystyle F(x,y)=0}"> </noscript><span class="lazy-image-placeholder" style="width: 11.33ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c910dd64922e23f212bc46e970af0540811c48f2" data-alt="{\displaystyle F(x,y)=0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=0}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> z </mi> <mo> = </mo> <mn> 0 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle z=0} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b92bfc06485cc90286474b14a516a68d8bfdd7b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.349ex; height:2.176ex;" alt="{\displaystyle z=0}"> </noscript><span class="lazy-image-placeholder" style="width: 5.349ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b92bfc06485cc90286474b14a516a68d8bfdd7b3" data-alt="{\displaystyle z=0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </li> </ul> Opšta jednačina površi data je sa <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x,y)=0}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> F </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> , </mo> <mi> y </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mn> 0 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle F(x,y)=0} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c910dd64922e23f212bc46e970af0540811c48f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.33ex; height:2.843ex;" alt="{\displaystyle F(x,y)=0}"> </noscript><span class="lazy-image-placeholder" style="width: 11.33ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c910dd64922e23f212bc46e970af0540811c48f2" data-alt="{\displaystyle F(x,y)=0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> (nedostaje z) <ul> <li>Izvodnica je paralelna sa osom <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle OX}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> O </mi> <mi> X </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle OX} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67505ccf95552d7c42405b0988119f438861ed07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.753ex; height:2.176ex;" alt="{\displaystyle OX}"> </noscript><span class="lazy-image-placeholder" style="width: 3.753ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67505ccf95552d7c42405b0988119f438861ed07" data-alt="{\displaystyle OX}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  i prolazi kroz krivu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(y,z)=0}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> F </mi> <mo stretchy="false"> ( </mo> <mi> y </mi> <mo> , </mo> <mi> z </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mn> 0 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle F(y,z)=0} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2eb30ff0996c08d6928827f3b88cf737b0e2a7bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.089ex; height:2.843ex;" alt="{\displaystyle F(y,z)=0}"> </noscript><span class="lazy-image-placeholder" style="width: 11.089ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2eb30ff0996c08d6928827f3b88cf737b0e2a7bf" data-alt="{\displaystyle F(y,z)=0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=0}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> x </mi> <mo> = </mo> <mn> 0 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle x=0} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=0}"> </noscript><span class="lazy-image-placeholder" style="width: 5.591ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc" data-alt="{\displaystyle x=0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </li> </ul> Opšta jednačina površi data je sa <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(y,z)=0}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> F </mi> <mo stretchy="false"> ( </mo> <mi> y </mi> <mo> , </mo> <mi> z </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mn> 0 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle F(y,z)=0} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2eb30ff0996c08d6928827f3b88cf737b0e2a7bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.089ex; height:2.843ex;" alt="{\displaystyle F(y,z)=0}"> </noscript><span class="lazy-image-placeholder" style="width: 11.089ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2eb30ff0996c08d6928827f3b88cf737b0e2a7bf" data-alt="{\displaystyle F(y,z)=0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  (nedostaje x) <ul> <li>Izvodnica je paralelna sa osom <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle OY}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> O </mi> <mi> Y </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle OY} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b00a476cd78e7dcc2c9aa97d5cc2dded3b9e31df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.547ex; height:2.176ex;" alt="{\displaystyle OY}"> </noscript><span class="lazy-image-placeholder" style="width: 3.547ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b00a476cd78e7dcc2c9aa97d5cc2dded3b9e31df" data-alt="{\displaystyle OY}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  i prolazi kroz krivu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x,z)=0}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> F </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> , </mo> <mi> z </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mn> 0 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle F(x,z)=0} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4a543b259a8b6b3a31a52394b27052ced5ae3b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.263ex; height:2.843ex;" alt="{\displaystyle F(x,z)=0}"> </noscript><span class="lazy-image-placeholder" style="width: 11.263ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4a543b259a8b6b3a31a52394b27052ced5ae3b8" data-alt="{\displaystyle F(x,z)=0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=0}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> y </mi> <mo> = </mo> <mn> 0 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle y=0} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/094f824655138f6b11d96a0da32e7f0716ba6959" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.416ex; height:2.509ex;" alt="{\displaystyle y=0}"> </noscript><span class="lazy-image-placeholder" style="width: 5.416ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/094f824655138f6b11d96a0da32e7f0716ba6959" data-alt="{\displaystyle y=0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </li> </ul> Opšta jednačina površi data je sa <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x,z)=0}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> F </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> , </mo> <mi> z </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mn> 0 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle F(x,z)=0} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4a543b259a8b6b3a31a52394b27052ced5ae3b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.263ex; height:2.843ex;" alt="{\displaystyle F(x,z)=0}"> </noscript><span class="lazy-image-placeholder" style="width: 11.263ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4a543b259a8b6b3a31a52394b27052ced5ae3b8" data-alt="{\displaystyle F(x,z)=0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  (nedostaje y) </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(11)"> <h2 id="Rotacione_površi">Rotacione površi</h2> <a role="button" href="https://sh-m-wikipedia-org.translate.goog/w/index.php?title=Povr%C5%A1&action=edit&section=11&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Uredi odjeljak Rotacione površi" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> uredi </a> </div> <section class="mf-section-11 collapsible-block" id="mf-section-11"> <ul class="gallery mw-gallery-traditional"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"> <a href="https://sh-m-wikipedia-org.translate.goog/wiki/Datoteka:Sphere_wireframe_10deg_6r.svg?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"> <noscript> <img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Sphere_wireframe_10deg_6r.svg/120px-Sphere_wireframe_10deg_6r.svg.png" decoding="async" width="120" height="120" class="mw-file-element" data-file-width="800" data-file-height="800"> </noscript><span class="lazy-image-placeholder" style="width: 120px;height: 120px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Sphere_wireframe_10deg_6r.svg/120px-Sphere_wireframe_10deg_6r.svg.png" data-width="120" data-height="120" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Sphere_wireframe_10deg_6r.svg/180px-Sphere_wireframe_10deg_6r.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Sphere_wireframe_10deg_6r.svg/240px-Sphere_wireframe_10deg_6r.svg.png 2x" data-class="mw-file-element"> </a> </div> <div class="gallerytext"></div></li> </ul> Jednačina <a href="https://sh-m-wikipedia-org.translate.goog/w/index.php?title=Rotacione_povr%C5%A1i&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Rotacione površi (stranica ne postoji)">rotacione površi</a> koja nastaje rotacijom krive <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=f(y)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> z </mi> <mo> = </mo> <mi> f </mi> <mo stretchy="false"> ( </mo> <mi> y </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle z=f(y)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adb9c017b827a30228c8fb0349f8ac153e5236ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.43ex; height:2.843ex;" alt="{\displaystyle z=f(y)}"> </noscript><span class="lazy-image-placeholder" style="width: 8.43ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adb9c017b827a30228c8fb0349f8ac153e5236ca" data-alt="{\displaystyle z=f(y)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  oko ose <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle OZ}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> O </mi> <mi> Z </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle OZ} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ef1fb6520fafba764040ed2928a06cdd5bf481f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.454ex; height:2.176ex;" alt="{\displaystyle OZ}"> </noscript><span class="lazy-image-placeholder" style="width: 3.454ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ef1fb6520fafba764040ed2928a06cdd5bf481f" data-alt="{\displaystyle OZ}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Neka je <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho ={\sqrt {x^{2}+y^{2}}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> ρ </mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <msup> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \rho ={\sqrt {x^{2}+y^{2}}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6639197208572a96358af21debdbdaf96ae306a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:14.063ex; height:4.843ex;" alt="{\displaystyle \rho ={\sqrt {x^{2}+y^{2}}}}"> </noscript><span class="lazy-image-placeholder" style="width: 14.063ex;height: 4.843ex;vertical-align: -1.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6639197208572a96358af21debdbdaf96ae306a4" data-alt="{\displaystyle \rho ={\sqrt {x^{2}+y^{2}}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  udaljenost proizvoljne tačke <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(x,y,z)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> T </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> , </mo> <mi> y </mi> <mo> , </mo> <mi> z </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle T(x,y,z)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6897ec7f1a2b10374084a85c60f8f87fded5d140" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.087ex; height:2.843ex;" alt="{\displaystyle T(x,y,z)}"> </noscript><span class="lazy-image-placeholder" style="width: 9.087ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6897ec7f1a2b10374084a85c60f8f87fded5d140" data-alt="{\displaystyle T(x,y,z)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  rotacione povrsi od ose <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle OZ}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> O </mi> <mi> Z </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle OZ} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ef1fb6520fafba764040ed2928a06cdd5bf481f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.454ex; height:2.176ex;" alt="{\displaystyle OZ}"> </noscript><span class="lazy-image-placeholder" style="width: 3.454ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ef1fb6520fafba764040ed2928a06cdd5bf481f" data-alt="{\displaystyle OZ}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Tada je jednačina rotacione površi kojoj je osa <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle OZ}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> O </mi> <mi> Z </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle OZ} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ef1fb6520fafba764040ed2928a06cdd5bf481f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.454ex; height:2.176ex;" alt="{\displaystyle OZ}"> </noscript><span class="lazy-image-placeholder" style="width: 3.454ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ef1fb6520fafba764040ed2928a06cdd5bf481f" data-alt="{\displaystyle OZ}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  osa rotacije data sa <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=f(\rho )=f({\sqrt {x^{2}+y^{2}}})}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> z </mi> <mo> = </mo> <mi> f </mi> <mo stretchy="false"> ( </mo> <mi> ρ </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mi> f </mi> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <msup> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle z=f(\rho )=f({\sqrt {x^{2}+y^{2}}})} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dde3192f294bde2479eb291178e0c41233b9ac95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:24.426ex; height:4.843ex;" alt="{\displaystyle z=f(\rho )=f({\sqrt {x^{2}+y^{2}}})}"> </noscript><span class="lazy-image-placeholder" style="width: 24.426ex;height: 4.843ex;vertical-align: -1.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dde3192f294bde2479eb291178e0c41233b9ac95" data-alt="{\displaystyle z=f(\rho )=f({\sqrt {x^{2}+y^{2}}})}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  uopšteno sa <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(z,x^{2}+y^{2})=0}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> F </mi> <mo stretchy="false"> ( </mo> <mi> z </mi> <mo> , </mo> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <msup> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo stretchy="false"> ) </mo> <mo> = </mo> <mn> 0 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle F(z,x^{2}+y^{2})=0} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68e0046520204b07d7935f155828b6af76dac428" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.372ex; height:3.176ex;" alt="{\displaystyle F(z,x^{2}+y^{2})=0}"> </noscript><span class="lazy-image-placeholder" style="width: 17.372ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68e0046520204b07d7935f155828b6af76dac428" data-alt="{\displaystyle F(z,x^{2}+y^{2})=0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Jednačina rotacione površi koja nastaje rotacijom krive <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=f(z)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> x </mi> <mo> = </mo> <mi> f </mi> <mo stretchy="false"> ( </mo> <mi> z </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle x=f(z)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fed2b9fa0e42bd5570a5ee642891846975d906a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.604ex; height:2.843ex;" alt="{\displaystyle x=f(z)}"> </noscript><span class="lazy-image-placeholder" style="width: 8.604ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fed2b9fa0e42bd5570a5ee642891846975d906a" data-alt="{\displaystyle x=f(z)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  ili <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(y)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> f </mi> <mo stretchy="false"> ( </mo> <mi> y </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle f(y)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aaa9215c6afa4892692ba05ae4c44f23600ea79d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.243ex; height:2.843ex;" alt="{\displaystyle f(y)}"> </noscript><span class="lazy-image-placeholder" style="width: 4.243ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aaa9215c6afa4892692ba05ae4c44f23600ea79d" data-alt="{\displaystyle f(y)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  oko ose <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle OX}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> O </mi> <mi> X </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle OX} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67505ccf95552d7c42405b0988119f438861ed07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.753ex; height:2.176ex;" alt="{\displaystyle OX}"> </noscript><span class="lazy-image-placeholder" style="width: 3.753ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67505ccf95552d7c42405b0988119f438861ed07" data-alt="{\displaystyle OX}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  data je sa <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=f(\rho )=f({\sqrt {y^{2}+z^{2}}})}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> z </mi> <mo> = </mo> <mi> f </mi> <mo stretchy="false"> ( </mo> <mi> ρ </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mi> f </mi> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <msup> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle z=f(\rho )=f({\sqrt {y^{2}+z^{2}}})} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/590826204ad7ac6380f1134dfb1d4aa7689ee48b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:24.186ex; height:4.843ex;" alt="{\displaystyle z=f(\rho )=f({\sqrt {y^{2}+z^{2}}})}"> </noscript><span class="lazy-image-placeholder" style="width: 24.186ex;height: 4.843ex;vertical-align: -1.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/590826204ad7ac6380f1134dfb1d4aa7689ee48b" data-alt="{\displaystyle z=f(\rho )=f({\sqrt {y^{2}+z^{2}}})}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> uopšteno sa <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x,y^{2}+z^{2})=0}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> F </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> , </mo> <msup> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <msup> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo stretchy="false"> ) </mo> <mo> = </mo> <mn> 0 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle F(x,y^{2}+z^{2})=0} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a144d7de15c4a93fab9ca27874c62e56e4663fc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.374ex; height:3.176ex;" alt="{\displaystyle F(x,y^{2}+z^{2})=0}"> </noscript><span class="lazy-image-placeholder" style="width: 17.374ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a144d7de15c4a93fab9ca27874c62e56e4663fc3" data-alt="{\displaystyle F(x,y^{2}+z^{2})=0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Jednačina rotacione površi koja nastaje rotacijom krive <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=f(xz)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> x </mi> <mo> = </mo> <mi> f </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mi> z </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle x=f(xz)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f80e5ebf4a246fd9de8ca57bfd02fba585b0c438" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.934ex; height:2.843ex;" alt="{\displaystyle x=f(xz)}"> </noscript><span class="lazy-image-placeholder" style="width: 9.934ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f80e5ebf4a246fd9de8ca57bfd02fba585b0c438" data-alt="{\displaystyle x=f(xz)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  ili <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(z)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> f </mi> <mo stretchy="false"> ( </mo> <mi> z </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle f(z)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8dd568d570b390c337c0a911f0a1c5c214e8240" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.176ex; height:2.843ex;" alt="{\displaystyle f(z)}"> </noscript><span class="lazy-image-placeholder" style="width: 4.176ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8dd568d570b390c337c0a911f0a1c5c214e8240" data-alt="{\displaystyle f(z)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  oko ose <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle OX}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> O </mi> <mi> X </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle OX} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67505ccf95552d7c42405b0988119f438861ed07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.753ex; height:2.176ex;" alt="{\displaystyle OX}"> </noscript><span class="lazy-image-placeholder" style="width: 3.753ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67505ccf95552d7c42405b0988119f438861ed07" data-alt="{\displaystyle OX}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  je data sa <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=f(\rho )=f({\sqrt {x^{2}+z^{2}}})}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> z </mi> <mo> = </mo> <mi> f </mi> <mo stretchy="false"> ( </mo> <mi> ρ </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mi> f </mi> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <msup> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle z=f(\rho )=f({\sqrt {x^{2}+z^{2}}})} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9f225bf6307ca3d2c3ac5408f385ebe4fdf661d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.355ex; height:3.509ex;" alt="{\displaystyle z=f(\rho )=f({\sqrt {x^{2}+z^{2}}})}"> </noscript><span class="lazy-image-placeholder" style="width: 24.355ex;height: 3.509ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9f225bf6307ca3d2c3ac5408f385ebe4fdf661d" data-alt="{\displaystyle z=f(\rho )=f({\sqrt {x^{2}+z^{2}}})}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> uopšteno sa <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(y,x^{2}+z^{2})=0}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> F </mi> <mo stretchy="false"> ( </mo> <mi> y </mi> <mo> , </mo> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <msup> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo stretchy="false"> ) </mo> <mo> = </mo> <mn> 0 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle F(y,x^{2}+z^{2})=0} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/106ed181426e9f49c53892522313dfaa04544175" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.369ex; height:3.176ex;" alt="{\displaystyle F(y,x^{2}+z^{2})=0}"> </noscript><span class="lazy-image-placeholder" style="width: 17.369ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/106ed181426e9f49c53892522313dfaa04544175" data-alt="{\displaystyle F(y,x^{2}+z^{2})=0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(12)"> <h2 id="Izvori">Izvori</h2> <a role="button" href="https://sh-m-wikipedia-org.translate.goog/w/index.php?title=Povr%C5%A1&action=edit&section=12&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Uredi odjeljak Izvori" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> uredi </a> </div> <section class="mf-section-12 collapsible-block" id="mf-section-12"> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://lavica.fesb.hr/mat2/predavanja/node54.html">Plohe drugog reda</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://www.grad.hr/sgorjanc/Links/natkrivanje.htm">NATKRIVANJE PARABOLIČKIM KONOIDOM</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://www.mathos.unios.hr/fvv/plohe.pdf">Valjkaste (cilindrične) plohe</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://web.archive.org/web/20180713015126/http://www.mathos.unios.hr/fvv/plohe.pdf">Arhivirano</a> 2018-07-13 na <a href="https://sh-m-wikipedia-org.translate.goog/wiki/Wayback_Machine?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Wayback Machine">Wayback Machine-u</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://www.grad.hr/sgorjanc/Links/sanjaimario.pdf">Gaussova i srednja zakrivljenost ploha</a> <div class="infobox sisterproject" style="width: 300px"> <div style="float: left;"> <figure class="mw-halign-none" typeof="mw:File"> <a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://commons.wikimedia.org/wiki/Category:Surfaces" title="commons:Category:Surfaces"> <noscript> <img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/15px-Commons-logo.svg.png" decoding="async" width="15" height="20" class="mw-file-element" data-file-width="1024" data-file-height="1376"> </noscript><span class="lazy-image-placeholder" style="width: 15px;height: 20px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/15px-Commons-logo.svg.png" data-width="15" data-height="20" 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href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ar.wikipedia.org/wiki/%25D8%25B3%25D8%25B7%25D8%25AD" title="سطح — Arapski" lang="ar" hreflang="ar" data-title="سطح" data-language-autonym="العربية" data-language-local-name="Arapski" class="interlanguage-link-target">العربية</a></li> <li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ast.wikipedia.org/wiki/Superficie" title="Superficie — Asturijski" lang="ast" hreflang="ast" data-title="Superficie" data-language-autonym="Asturianu" data-language-local-name="Asturijski" class="interlanguage-link-target">Asturianu</a></li> <li class="interlanguage-link interwiki-az mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://az.wikipedia.org/wiki/S%25C9%2599th" title="Səth — Azerbejdžanski" lang="az" hreflang="az" data-title="Səth" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbejdžanski" class="interlanguage-link-target">Azərbaycanca</a></li> <li class="interlanguage-link interwiki-be mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://be.wikipedia.org/wiki/%25D0%259F%25D0%25B0%25D0%25B2%25D0%25B5%25D1%2580%25D1%2585%25D0%25BD%25D1%258F" title="Паверхня — Beloruski" lang="be" hreflang="be" data-title="Паверхня" data-language-autonym="Беларуская" data-language-local-name="Beloruski" class="interlanguage-link-target">Беларуская</a></li> <li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://bg.wikipedia.org/wiki/%25D0%259F%25D0%25BE%25D0%25B2%25D1%258A%25D1%2580%25D1%2585%25D0%25BD%25D0%25BE%25D1%2581%25D1%2582" title="Повърхност — Bugarski" lang="bg" hreflang="bg" data-title="Повърхност" data-language-autonym="Български" data-language-local-name="Bugarski" class="interlanguage-link-target">Български</a></li> <li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://bn.wikipedia.org/wiki/%25E0%25A6%25A4%25E0%25A6%25B2_(%25E0%25A6%259F%25E0%25A6%25AA%25E0%25A7%258B%25E0%25A6%25B2%25E0%25A6%259C%25E0%25A6%25BF)" title="তল (টপোলজি) — Bengalski" lang="bn" hreflang="bn" data-title="তল (টপোলজি)" data-language-autonym="বাংলা" data-language-local-name="Bengalski" class="interlanguage-link-target">বাংলা</a></li> <li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://bs.wikipedia.org/wiki/Povr%25C5%25A1" title="Površ — Bosanski" lang="bs" hreflang="bs" data-title="Površ" data-language-autonym="Bosanski" data-language-local-name="Bosanski" class="interlanguage-link-target">Bosanski</a></li> <li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ca.wikipedia.org/wiki/Superf%25C3%25ADcie_(matem%25C3%25A0tiques)" title="Superfície (matemàtiques) — Katalonski" lang="ca" hreflang="ca" data-title="Superfície (matemàtiques)" data-language-autonym="Català" data-language-local-name="Katalonski" class="interlanguage-link-target">Català</a></li> <li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ckb.wikipedia.org/wiki/%25DA%2595%25D9%2588%25D9%2588" title="ڕوو — centralnokurdski" lang="ckb" hreflang="ckb" data-title="ڕوو" data-language-autonym="کوردی" data-language-local-name="centralnokurdski" class="interlanguage-link-target">کوردی</a></li> <li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://cs.wikipedia.org/wiki/Plocha" title="Plocha — Češki" lang="cs" hreflang="cs" data-title="Plocha" data-language-autonym="Čeština" data-language-local-name="Češki" class="interlanguage-link-target">Čeština</a></li> <li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://cv.wikipedia.org/wiki/%25C3%2587%25D0%25B8%25D0%25B9" title="Çий — Čuvaški" lang="cv" hreflang="cv" data-title="Çий" data-language-autonym="Чӑвашла" data-language-local-name="Čuvaški" class="interlanguage-link-target">Чӑвашла</a></li> <li class="interlanguage-link interwiki-en mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://en.wikipedia.org/wiki/Surface_(topology)" title="Surface (topology) — Engleski" lang="en" hreflang="en" data-title="Surface (topology)" data-language-autonym="English" data-language-local-name="Engleski" class="interlanguage-link-target">English</a></li> <li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://eo.wikipedia.org/wiki/Surfaco" title="Surfaco — Esperanto" lang="eo" hreflang="eo" data-title="Surfaco" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li> <li class="interlanguage-link interwiki-es mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://es.wikipedia.org/wiki/Superficie_(topolog%25C3%25ADa)" title="Superficie (topología) — Španski" lang="es" hreflang="es" data-title="Superficie (topología)" data-language-autonym="Español" data-language-local-name="Španski" class="interlanguage-link-target">Español</a></li> <li class="interlanguage-link interwiki-et mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://et.wikipedia.org/wiki/Pind" title="Pind — Estonski" lang="et" hreflang="et" data-title="Pind" data-language-autonym="Eesti" data-language-local-name="Estonski" class="interlanguage-link-target">Eesti</a></li> <li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://eu.wikipedia.org/wiki/Gainazal" title="Gainazal — Baskijski" lang="eu" hreflang="eu" data-title="Gainazal" data-language-autonym="Euskara" data-language-local-name="Baskijski" class="interlanguage-link-target">Euskara</a></li> <li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://fa.wikipedia.org/wiki/%25D8%25B1%25D9%2588%25DB%258C%25D9%2587_(%25D8%25AA%25D9%2588%25D9%25BE%25D9%2588%25D9%2584%25D9%2588%25DA%2598%25DB%258C)" title="رویه (توپولوژی) — Persijski" lang="fa" hreflang="fa" data-title="رویه (توپولوژی)" data-language-autonym="فارسی" data-language-local-name="Persijski" class="interlanguage-link-target">فارسی</a></li> <li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://fi.wikipedia.org/wiki/Pinta_(geometria)" title="Pinta (geometria) — Finski" lang="fi" hreflang="fi" data-title="Pinta (geometria)" data-language-autonym="Suomi" data-language-local-name="Finski" class="interlanguage-link-target">Suomi</a></li> <li class="interlanguage-link interwiki-fr badge-Q70893996 mw-list-item" title=""><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://fr.wikipedia.org/wiki/Surface_(g%25C3%25A9om%25C3%25A9trie)" title="Surface (géométrie) — Francuski" lang="fr" hreflang="fr" data-title="Surface (géométrie)" data-language-autonym="Français" data-language-local-name="Francuski" class="interlanguage-link-target">Français</a></li> <li class="interlanguage-link interwiki-fur mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://fur.wikipedia.org/wiki/Superficie" title="Superficie — Friulijski" lang="fur" hreflang="fur" data-title="Superficie" data-language-autonym="Furlan" data-language-local-name="Friulijski" class="interlanguage-link-target">Furlan</a></li> <li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ga.wikipedia.org/wiki/Dromchla_(toipeola%25C3%25ADocht)" title="Dromchla (toipeolaíocht) — Irski" lang="ga" hreflang="ga" data-title="Dromchla (toipeolaíocht)" data-language-autonym="Gaeilge" data-language-local-name="Irski" class="interlanguage-link-target">Gaeilge</a></li> <li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://gl.wikipedia.org/wiki/Superficie" title="Superficie — Galski" lang="gl" hreflang="gl" data-title="Superficie" data-language-autonym="Galego" data-language-local-name="Galski" class="interlanguage-link-target">Galego</a></li> <li class="interlanguage-link interwiki-he mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://he.wikipedia.org/wiki/%25D7%259E%25D7%25A9%25D7%2598%25D7%2597_(%25D7%2598%25D7%2595%25D7%25A4%25D7%2595%25D7%259C%25D7%2595%25D7%2592%25D7%2599%25D7%2594)" title="משטח (טופולוגיה) — Hebrejski" lang="he" hreflang="he" data-title="משטח (טופולוגיה)" data-language-autonym="עברית" data-language-local-name="Hebrejski" class="interlanguage-link-target">עברית</a></li> <li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://hi.wikipedia.org/wiki/%25E0%25A4%25AA%25E0%25A5%2583%25E0%25A4%25B7%25E0%25A5%258D%25E0%25A4%259F" title="पृष्ट — Hindi" lang="hi" hreflang="hi" data-title="पृष्ट" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li> <li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://hr.wikipedia.org/wiki/Ploha_(geometrija)" title="Ploha (geometrija) — Hrvatski" lang="hr" hreflang="hr" data-title="Ploha (geometrija)" data-language-autonym="Hrvatski" data-language-local-name="Hrvatski" class="interlanguage-link-target">Hrvatski</a></li> <li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://hu.wikipedia.org/wiki/Felsz%25C3%25ADn" title="Felszín — Mađarski" lang="hu" hreflang="hu" data-title="Felszín" data-language-autonym="Magyar" data-language-local-name="Mađarski" class="interlanguage-link-target">Magyar</a></li> <li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://hy.wikipedia.org/wiki/%25D5%2584%25D5%25A1%25D5%25AF%25D5%25A5%25D6%2580%25D6%2587%25D5%25B8%25D6%2582%25D5%25B5%25D5%25A9" title="Մակերևույթ — Jermenski" lang="hy" hreflang="hy" data-title="Մակերևույթ" data-language-autonym="Հայերեն" data-language-local-name="Jermenski" class="interlanguage-link-target">Հայերեն</a></li> <li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ia.wikipedia.org/wiki/Superficie" title="Superficie — Interlingva" lang="ia" hreflang="ia" data-title="Superficie" data-language-autonym="Interlingua" data-language-local-name="Interlingva" class="interlanguage-link-target">Interlingua</a></li> <li class="interlanguage-link interwiki-id mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://id.wikipedia.org/wiki/Permukaan_(topologi)" title="Permukaan (topologi) — Indonezijski" lang="id" hreflang="id" data-title="Permukaan (topologi)" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonezijski" class="interlanguage-link-target">Bahasa Indonesia</a></li> <li class="interlanguage-link interwiki-inh mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://inh.wikipedia.org/wiki/%25D0%25A2%25D3%2580%25D0%25B5%25D1%2585%25D0%25B5" title="ТӀехе — Ingušetski" lang="inh" hreflang="inh" data-title="ТӀехе" data-language-autonym="ГӀалгӀай" data-language-local-name="Ingušetski" class="interlanguage-link-target">ГӀалгӀай</a></li> <li class="interlanguage-link interwiki-io mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://io.wikipedia.org/wiki/Surfaco" title="Surfaco — Ido" lang="io" hreflang="io" data-title="Surfaco" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target">Ido</a></li> <li class="interlanguage-link interwiki-is mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://is.wikipedia.org/wiki/Yfirbor%25C3%25B0" title="Yfirborð — Islandski" lang="is" hreflang="is" data-title="Yfirborð" data-language-autonym="Íslenska" data-language-local-name="Islandski" class="interlanguage-link-target">Íslenska</a></li> <li class="interlanguage-link interwiki-it mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://it.wikipedia.org/wiki/Superficie" title="Superficie — Italijanski" lang="it" hreflang="it" data-title="Superficie" data-language-autonym="Italiano" data-language-local-name="Italijanski" class="interlanguage-link-target">Italiano</a></li> <li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ja.wikipedia.org/wiki/%25E6%259B%25B2%25E9%259D%25A2" title="曲面 — Japanski" lang="ja" hreflang="ja" data-title="曲面" data-language-autonym="日本語" data-language-local-name="Japanski" class="interlanguage-link-target">日本語</a></li> <li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://kk.wikipedia.org/wiki/%25D0%2591%25D0%25B5%25D1%2582_(%25D0%25B3%25D0%25B5%25D0%25BE%25D0%25BC%25D0%25B5%25D1%2582%25D1%2580%25D0%25B8%25D1%258F)" title="Бет (геометрия) — Kozački" lang="kk" hreflang="kk" data-title="Бет (геометрия)" data-language-autonym="Қазақша" data-language-local-name="Kozački" class="interlanguage-link-target">Қазақша</a></li> <li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ko.wikipedia.org/wiki/%25EA%25B3%25A1%25EB%25A9%25B4" title="곡면 — Korejski" lang="ko" hreflang="ko" data-title="곡면" data-language-autonym="한국어" data-language-local-name="Korejski" class="interlanguage-link-target">한국어</a></li> <li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ky.wikipedia.org/wiki/%25D0%2591%25D0%25B5%25D1%2582_(%25D0%2593%25D0%25B5%25D0%25BE%25D0%25BC%25D0%25B5%25D1%2582%25D1%2580%25D0%25B8%25D1%258F)" title="Бет (Геометрия) — Kirgiski" lang="ky" hreflang="ky" data-title="Бет (Геометрия)" data-language-autonym="Кыргызча" data-language-local-name="Kirgiski" class="interlanguage-link-target">Кыргызча</a></li> <li class="interlanguage-link interwiki-lij mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://lij.wikipedia.org/wiki/Superfi%25C3%25A7ie_(matematica)" title="Superfiçie (matematica) — ligurski" lang="lij" hreflang="lij" data-title="Superfiçie (matematica)" data-language-autonym="Ligure" data-language-local-name="ligurski" class="interlanguage-link-target">Ligure</a></li> <li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://lt.wikipedia.org/wiki/Pavir%25C5%25A1ius" title="Paviršius — Litvanski" lang="lt" hreflang="lt" data-title="Paviršius" data-language-autonym="Lietuvių" data-language-local-name="Litvanski" class="interlanguage-link-target">Lietuvių</a></li> <li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://lv.wikipedia.org/wiki/Virsma" title="Virsma — Letonski" lang="lv" hreflang="lv" data-title="Virsma" data-language-autonym="Latviešu" data-language-local-name="Letonski" class="interlanguage-link-target">Latviešu</a></li> <li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://mr.wikipedia.org/wiki/%25E0%25A4%2586%25E0%25A4%25A1" title="आड — Marati" lang="mr" hreflang="mr" data-title="आड" data-language-autonym="मराठी" data-language-local-name="Marati" class="interlanguage-link-target">मराठी</a></li> <li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://nl.wikipedia.org/wiki/Oppervlak_(topologie)" title="Oppervlak (topologie) — Holandski" lang="nl" hreflang="nl" data-title="Oppervlak (topologie)" data-language-autonym="Nederlands" data-language-local-name="Holandski" class="interlanguage-link-target">Nederlands</a></li> <li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://nn.wikipedia.org/wiki/Flate" title="Flate — Norveški njorsk" lang="nn" hreflang="nn" data-title="Flate" data-language-autonym="Norsk nynorsk" data-language-local-name="Norveški njorsk" class="interlanguage-link-target">Norsk nynorsk</a></li> <li class="interlanguage-link interwiki-no mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://no.wikipedia.org/wiki/Flate" title="Flate — Norveški bokmål" lang="nb" hreflang="nb" data-title="Flate" data-language-autonym="Norsk bokmål" data-language-local-name="Norveški bokmål" class="interlanguage-link-target">Norsk bokmål</a></li> <li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://oc.wikipedia.org/wiki/Superf%25C3%25ADcia_(matematicas)" title="Superfícia (matematicas) — Provansalski" lang="oc" hreflang="oc" data-title="Superfícia (matematicas)" data-language-autonym="Occitan" data-language-local-name="Provansalski" class="interlanguage-link-target">Occitan</a></li> <li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://pl.wikipedia.org/wiki/Powierzchnia" title="Powierzchnia — Poljski" lang="pl" hreflang="pl" data-title="Powierzchnia" data-language-autonym="Polski" data-language-local-name="Poljski" class="interlanguage-link-target">Polski</a></li> <li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://pms.wikipedia.org/wiki/Surfassa" title="Surfassa — Piedmontese" lang="pms" hreflang="pms" data-title="Surfassa" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target">Piemontèis</a></li> <li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://pt.wikipedia.org/wiki/Superf%25C3%25ADcie" title="Superfície — Portugalski" lang="pt" hreflang="pt" data-title="Superfície" data-language-autonym="Português" data-language-local-name="Portugalski" class="interlanguage-link-target">Português</a></li> <li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ro.wikipedia.org/wiki/Suprafa%25C8%259B%25C4%2583" title="Suprafață — Rumunski" lang="ro" hreflang="ro" data-title="Suprafață" data-language-autonym="Română" data-language-local-name="Rumunski" class="interlanguage-link-target">Română</a></li> <li class="interlanguage-link interwiki-rsk mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://rsk.wikipedia.org/wiki/%25D0%259F%25D0%25BE%25D0%25B2%25D0%25B5%25D1%2580%25D1%2585%25D0%25BD%25D0%25BE%25D1%2581%25D1%2586" title="Поверхносц — Pannonian Rusyn" lang="rsk" hreflang="rsk" data-title="Поверхносц" data-language-autonym="Руски" data-language-local-name="Pannonian Rusyn" class="interlanguage-link-target">Руски</a></li> <li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ru.wikipedia.org/wiki/%25D0%259F%25D0%25BE%25D0%25B2%25D0%25B5%25D1%2580%25D1%2585%25D0%25BD%25D0%25BE%25D1%2581%25D1%2582%25D1%258C" title="Поверхность — Ruski" lang="ru" hreflang="ru" data-title="Поверхность" data-language-autonym="Русский" data-language-local-name="Ruski" class="interlanguage-link-target">Русский</a></li> <li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://simple.wikipedia.org/wiki/Surface" title="Surface — Simple English" lang="en-simple" hreflang="en-simple" data-title="Surface" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li> <li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sk.wikipedia.org/wiki/Povrch" title="Povrch — Slovački" lang="sk" hreflang="sk" data-title="Povrch" data-language-autonym="Slovenčina" data-language-local-name="Slovački" class="interlanguage-link-target">Slovenčina</a></li> <li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sl.wikipedia.org/wiki/Ploskev" title="Ploskev — Slovenački" lang="sl" hreflang="sl" data-title="Ploskev" data-language-autonym="Slovenščina" data-language-local-name="Slovenački" class="interlanguage-link-target">Slovenščina</a></li> <li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sn.wikipedia.org/wiki/Chiso_(Chiumbwa)" title="Chiso (Chiumbwa) — Šona" lang="sn" hreflang="sn" data-title="Chiso (Chiumbwa)" data-language-autonym="ChiShona" data-language-local-name="Šona" class="interlanguage-link-target">ChiShona</a></li> <li class="interlanguage-link interwiki-so mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://so.wikipedia.org/wiki/Oogo_(dhul)" title="Oogo (dhul) — Somalski" lang="so" hreflang="so" data-title="Oogo (dhul)" data-language-autonym="Soomaaliga" data-language-local-name="Somalski" class="interlanguage-link-target">Soomaaliga</a></li> <li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sr.wikipedia.org/wiki/%25D0%259F%25D0%25BE%25D0%25B2%25D1%2580%25D1%2588" title="Површ — Srpski" lang="sr" hreflang="sr" data-title="Површ" data-language-autonym="Српски / srpski" data-language-local-name="Srpski" class="interlanguage-link-target">Српски / srpski</a></li> <li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sv.wikipedia.org/wiki/Yta" title="Yta — Švedski" lang="sv" hreflang="sv" data-title="Yta" data-language-autonym="Svenska" data-language-local-name="Švedski" class="interlanguage-link-target">Svenska</a></li> <li class="interlanguage-link interwiki-te mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://te.wikipedia.org/wiki/%25E0%25B0%2589%25E0%25B0%25AA%25E0%25B0%25B0%25E0%25B0%25BF%25E0%25B0%25A4%25E0%25B0%25B2%25E0%25B0%2582" title="ఉపరితలం — Telugu" lang="te" hreflang="te" data-title="ఉపరితలం" data-language-autonym="తెలుగు" data-language-local-name="Telugu" class="interlanguage-link-target">తెలుగు</a></li> <li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://tr.wikipedia.org/wiki/Y%25C3%25BCzey" title="Yüzey — Turski" lang="tr" hreflang="tr" data-title="Yüzey" data-language-autonym="Türkçe" data-language-local-name="Turski" class="interlanguage-link-target">Türkçe</a></li> <li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://uk.wikipedia.org/wiki/%25D0%259F%25D0%25BE%25D0%25B2%25D0%25B5%25D1%2580%25D1%2585%25D0%25BD%25D1%258F" title="Поверхня — Ukrajinski" lang="uk" hreflang="uk" data-title="Поверхня" data-language-autonym="Українська" data-language-local-name="Ukrajinski" class="interlanguage-link-target">Українська</a></li> <li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ur.wikipedia.org/wiki/%25D8%25B3%25D8%25B7%25D8%25AD" title="سطح — Urdu" lang="ur" hreflang="ur" data-title="سطح" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target">اردو</a></li> <li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://uz.wikipedia.org/wiki/Sirt" title="Sirt — Uzbečki" lang="uz" hreflang="uz" data-title="Sirt" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbečki" class="interlanguage-link-target">Oʻzbekcha / ўзбекча</a></li> <li class="interlanguage-link interwiki-vec mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://vec.wikipedia.org/wiki/Superficie" title="Superficie — venecijanski" lang="vec" hreflang="vec" data-title="Superficie" data-language-autonym="Vèneto" data-language-local-name="venecijanski" class="interlanguage-link-target">Vèneto</a></li> <li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://vi.wikipedia.org/wiki/M%25E1%25BA%25B7t_(t%25C3%25B4_p%25C3%25B4)" title="Mặt (tô pô) — Vijetnamski" lang="vi" hreflang="vi" data-title="Mặt (tô pô)" data-language-autonym="Tiếng Việt" data-language-local-name="Vijetnamski" class="interlanguage-link-target">Tiếng Việt</a></li> <li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://wuu.wikipedia.org/wiki/%25E6%259B%25B2%25E9%259D%25A2" title="曲面 — Wu kineski" lang="wuu" hreflang="wuu" data-title="曲面" 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Površ – Wikipedija/Википедија