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class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"> Most of the terms listed in Wikipedia glossaries are already defined and explained within Wikipedia itself. However, glossaries like this one are useful for looking up, comparing and reviewing large numbers of terms together. You can help enhance this page by adding new terms or writing definitions for existing ones. This glossary of calculus is a list of definitions about <a href="/wiki/Calculus" title="Calculus">calculus</a>, its sub-disciplines, and related fields. <div class="noprint" style="text-align:center;"><div role="navigation" id="toc" class="toc plainlinks" aria-labelledby="tocheading" style="margin-left:auto;margin-right:auto; text-align:left;"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul 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dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style><div class="hlist"> <div id="toctitle" class="toctitle" style="text-align:center;display:inline-block;">Contents: </div> <div style="margin:auto; display:inline-block;"> <ul><li><a href="#A">A</a></li> <li><a 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href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><td class="sidebar-pretitle">Part of a series of articles about</td></tr><tr><th class="sidebar-title-with-pretitle" style="padding-bottom:0.25em;"><a href="/wiki/Calculus" title="Calculus">Calculus</a></th></tr><tr><td class="sidebar-image"><big><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17d063dc86a53a2efb1fe86f4a5d47d498652766" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.228ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}"></big></td></tr><tr><td class="sidebar-above" style="padding:0.15em 0.25em 0.3em;font-weight:normal;"> <ul><li><a href="/wiki/Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus">Fundamental theorem</a></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Limit_of_a_function" title="Limit of a function">Limits</a></li> <li><a href="/wiki/Continuous_function" title="Continuous function">Continuity</a></li></ul> </div><div class="hlist"> <ul><li><a href="/wiki/Rolle%27s_theorem" title="Rolle's theorem">Rolle's theorem</a></li> <li><a href="/wiki/Mean_value_theorem" title="Mean value theorem">Mean value theorem</a></li> <li><a href="/wiki/Inverse_function_theorem" title="Inverse function theorem">Inverse function theorem</a></li></ul> </div></td></tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base);display:block;margin-top:0.65em;"><a href="/wiki/Differential_calculus" title="Differential calculus">Differential</a></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><th class="sidebar-heading"> Definitions</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Derivative" title="Derivative">Derivative</a> (<a href="/wiki/Generalizations_of_the_derivative" title="Generalizations of the derivative">generalizations</a>)</li> <li><a href="/wiki/Differential_(mathematics)" title="Differential (mathematics)">Differential</a> <ul><li><a href="/wiki/Differential_(infinitesimal)" class="mw-redirect" title="Differential (infinitesimal)">infinitesimal</a></li> <li><a href="/wiki/Differential_of_a_function" title="Differential of a function">of a function</a></li> <li><a href="/wiki/Differential_of_a_function#Differentials_in_several_variables" title="Differential of a function">total</a></li></ul></li></ul></td> </tr><tr><th class="sidebar-heading"> Concepts</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Notation_for_differentiation" title="Notation for differentiation">Differentiation notation</a></li> <li><a href="/wiki/Second_derivative" title="Second derivative">Second derivative</a></li> <li><a href="/wiki/Implicit_function" title="Implicit function">Implicit differentiation</a></li> <li><a href="/wiki/Logarithmic_differentiation" title="Logarithmic differentiation">Logarithmic differentiation</a></li> <li><a href="/wiki/Related_rates" title="Related rates">Related rates</a></li> <li><a href="/wiki/Taylor%27s_theorem" title="Taylor's theorem">Taylor's theorem</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Differentiation_rules" title="Differentiation rules">Rules and identities</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Sum_rule_in_differentiation" class="mw-redirect" title="Sum rule in differentiation">Sum</a></li> <li><a href="/wiki/Product_rule" title="Product rule">Product</a></li> <li><a href="/wiki/Chain_rule" title="Chain rule">Chain</a></li> <li><a href="/wiki/Power_rule" title="Power rule">Power</a></li> <li><a href="/wiki/Quotient_rule" title="Quotient rule">Quotient</a></li> <li><a href="/wiki/L%27H%C3%B4pital%27s_rule" title="L'Hôpital's rule">L'Hôpital's rule</a></li> <li><a href="/wiki/Inverse_function_rule" title="Inverse function rule">Inverse</a></li> <li><a href="/wiki/General_Leibniz_rule" title="General Leibniz rule">General Leibniz</a></li> <li><a href="/wiki/Fa%C3%A0_di_Bruno%27s_formula" title="Faà di Bruno's formula">Faà di Bruno's formula</a></li> <li><a href="/wiki/Reynolds_transport_theorem" title="Reynolds transport theorem">Reynolds</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="/wiki/Integral" title="Integral">Integral</a></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Lists_of_integrals" title="Lists of integrals">Lists of integrals</a></li> <li><a href="/wiki/Integral_transform" title="Integral transform">Integral transform</a></li> <li><a href="/wiki/Leibniz_integral_rule" title="Leibniz integral rule">Leibniz integral rule</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Definitions</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Antiderivative" title="Antiderivative">Antiderivative</a></li> <li><a href="/wiki/Integral" title="Integral">Integral</a> (<a href="/wiki/Improper_integral" title="Improper integral">improper</a>)</li> <li><a href="/wiki/Riemann_integral" title="Riemann integral">Riemann integral</a></li> <li><a href="/wiki/Lebesgue_integration" class="mw-redirect" title="Lebesgue integration">Lebesgue integration</a></li> <li><a href="/wiki/Contour_integration" title="Contour integration">Contour integration</a></li> <li><a href="/wiki/Integral_of_inverse_functions" title="Integral of inverse functions">Integral of inverse functions</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Integration by</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Integration_by_parts" title="Integration by parts">Parts</a></li> <li><a href="/wiki/Disc_integration" title="Disc integration">Discs</a></li> <li><a href="/wiki/Shell_integration" title="Shell integration">Cylindrical shells</a></li> <li><a href="/wiki/Integration_by_substitution" title="Integration by substitution">Substitution</a> (<a href="/wiki/Trigonometric_substitution" title="Trigonometric substitution">trigonometric</a>, <a href="/wiki/Tangent_half-angle_substitution" title="Tangent half-angle substitution">tangent half-angle</a>, <a href="/wiki/Euler_substitution" title="Euler substitution">Euler</a>)</li> <li><a href="/wiki/Integration_using_Euler%27s_formula" title="Integration using Euler's formula">Euler's formula</a></li> <li><a href="/wiki/Partial_fractions_in_integration" class="mw-redirect" title="Partial fractions in integration">Partial fractions</a> (<a href="/wiki/Heaviside_cover-up_method" title="Heaviside cover-up method">Heaviside's method</a>)</li> <li><a href="/wiki/Order_of_integration_(calculus)" title="Order of integration (calculus)">Changing order</a></li> <li><a href="/wiki/Integration_by_reduction_formulae" title="Integration by reduction formulae">Reduction formulae</a></li> <li><a href="/wiki/Leibniz_integral_rule#Evaluating_definite_integrals" title="Leibniz integral rule">Differentiating under the integral sign</a></li> <li><a href="/wiki/Risch_algorithm" title="Risch algorithm">Risch algorithm</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="/wiki/Series_(mathematics)" title="Series (mathematics)">Series</a></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Geometric_series" title="Geometric series">Geometric</a> (<a href="/wiki/Arithmetico%E2%80%93geometric_sequence" class="mw-redirect" title="Arithmetico–geometric sequence">arithmetico-geometric</a>)</li> <li><a href="/wiki/Harmonic_series_(mathematics)" title="Harmonic series (mathematics)">Harmonic</a></li> <li><a href="/wiki/Alternating_series" title="Alternating series">Alternating</a></li> <li><a href="/wiki/Power_series" title="Power series">Power</a></li> <li><a href="/wiki/Binomial_series" title="Binomial series">Binomial</a></li> <li><a href="/wiki/Taylor_series" title="Taylor series">Taylor</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Convergence_tests" title="Convergence tests">Convergence tests</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Term_test" class="mw-redirect" title="Term test">Summand limit (term test)</a></li> <li><a href="/wiki/Ratio_test" title="Ratio test">Ratio</a></li> <li><a href="/wiki/Root_test" title="Root test">Root</a></li> <li><a href="/wiki/Integral_test_for_convergence" title="Integral test for convergence">Integral</a></li> <li><a href="/wiki/Direct_comparison_test" title="Direct comparison test">Direct comparison</a></li> <li> <a href="/wiki/Limit_comparison_test" title="Limit comparison test">Limit comparison</a></li> <li><a href="/wiki/Alternating_series_test" title="Alternating series test">Alternating series</a></li> <li><a href="/wiki/Cauchy_condensation_test" title="Cauchy condensation test">Cauchy condensation</a></li> <li><a href="/wiki/Dirichlet%27s_test" title="Dirichlet's test">Dirichlet</a></li> <li><a href="/wiki/Abel%27s_test" title="Abel's test">Abel</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="/wiki/Vector_calculus" title="Vector calculus">Vector</a></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Gradient" title="Gradient">Gradient</a></li> <li><a href="/wiki/Divergence" title="Divergence">Divergence</a></li> <li><a href="/wiki/Curl_(mathematics)" title="Curl (mathematics)">Curl</a></li> <li><a href="/wiki/Laplace_operator" title="Laplace operator">Laplacian</a></li> <li><a href="/wiki/Directional_derivative" title="Directional derivative">Directional derivative</a></li> <li><a href="/wiki/Vector_calculus_identities" title="Vector calculus identities">Identities</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Theorems</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Gradient_theorem" title="Gradient theorem">Gradient</a></li> <li><a href="/wiki/Green%27s_theorem" title="Green's theorem">Green's</a></li> <li><a href="/wiki/Stokes%27_theorem" title="Stokes' theorem">Stokes'</a></li> <li><a href="/wiki/Divergence_theorem" title="Divergence theorem">Divergence</a></li> <li><a href="/wiki/Generalized_Stokes_theorem" title="Generalized Stokes theorem">generalized Stokes</a></li> <li><a href="/wiki/Helmholtz_decomposition" title="Helmholtz decomposition">Helmholtz decomposition</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="/wiki/Multivariable_calculus" title="Multivariable calculus">Multivariable</a></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><th class="sidebar-heading"> Formalisms</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Matrix_calculus" title="Matrix calculus">Matrix</a></li> <li><a href="/wiki/Tensor_calculus" class="mw-redirect" title="Tensor calculus">Tensor</a></li> <li><a href="/wiki/Exterior_derivative" title="Exterior derivative">Exterior</a></li> <li><a href="/wiki/Geometric_calculus" title="Geometric calculus">Geometric</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Definitions</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Partial_derivative" title="Partial derivative">Partial derivative</a></li> <li><a href="/wiki/Multiple_integral" title="Multiple integral">Multiple integral</a></li> <li><a href="/wiki/Line_integral" title="Line integral">Line integral</a></li> <li><a href="/wiki/Surface_integral" title="Surface integral">Surface integral</a></li> <li><a href="/wiki/Volume_integral" title="Volume integral">Volume integral</a></li> <li><a href="/wiki/Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian</a></li> <li><a href="/wiki/Hessian_matrix" title="Hessian matrix">Hessian</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Advanced</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Calculus_on_Euclidean_space" title="Calculus on Euclidean space">Calculus on Euclidean space</a></li> <li><a href="/wiki/Generalized_function" title="Generalized function">Generalized functions</a></li> <li><a href="/wiki/Limit_of_distributions" title="Limit of distributions">Limit of distributions</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Specialized</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Fractional_calculus" title="Fractional calculus">Fractional</a></li> <li><a href="/wiki/Malliavin_calculus" title="Malliavin calculus">Malliavin</a></li> <li><a href="/wiki/Stochastic_calculus" title="Stochastic calculus">Stochastic</a></li> <li><a href="/wiki/Calculus_of_variations" title="Calculus of variations">Variations</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Miscellanea</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Precalculus" title="Precalculus">Precalculus</a></li> <li><a href="/wiki/History_of_calculus" title="History of calculus">History</a></li> <li><a class="mw-selflink selflink">Glossary</a></li> <li><a href="/wiki/List_of_calculus_topics" title="List of calculus topics">List of topics</a></li> <li><a href="/wiki/Integration_Bee" title="Integration Bee">Integration Bee</a></li> <li><a href="/wiki/Mathematical_analysis" title="Mathematical analysis">Mathematical analysis</a></li> <li><a href="/wiki/Nonstandard_analysis" title="Nonstandard analysis">Nonstandard analysis</a></li></ul></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output 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title="Template talk:Calculus"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Calculus" title="Special:EditPage/Template:Calculus"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="A">A</h2>[<a href="/w/index.php?title=Glossary_of_calculus&action=edit&section=1" title="Edit section: A">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1228772891">.mw-parser-output .glossary dt{margin-top:0.4em}.mw-parser-output .glossary dt+dt{margin-top:-0.2em}.mw-parser-output .glossary .templatequote{margin-top:0;margin-bottom:-0.5em}</style> <dl class="glossary"> <dt id="abel's_test"><dfn><a href="/wiki/Abel%27s_test" title="Abel's test">Abel's test</a></dfn></dt> <dd>A method of testing for the <a href="/wiki/Convergent_series" title="Convergent series">convergence</a> of an <a href="/wiki/Series_(mathematics)" title="Series (mathematics)">infinite series</a>.</dd> <dt id="absolute_convergence"><dfn><a href="/wiki/Absolute_convergence" title="Absolute convergence">absolute convergence</a></dfn></dt> <dd>An <a href="/wiki/Series_(mathematics)" title="Series (mathematics)">infinite series</a> of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the <a href="/wiki/Absolute_value" title="Absolute value">absolute values</a> of the summands is finite. More precisely, a real or complex series <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \sum _{n=0}^{\infty }a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \sum _{n=0}^{\infty }a_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e82e9cba73339069c69edf3d9e3553754ea73080" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.608ex; height:3.176ex;" alt="{\displaystyle \textstyle \sum _{n=0}^{\infty }a_{n}}"> is said to converge absolutely if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \sum _{n=0}^{\infty }\left|a_{n}\right|=L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow> <mo>|</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>|</mo> </mrow> <mo>=</mo> <mi>L</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \sum _{n=0}^{\infty }\left|a_{n}\right|=L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6b0869fe4d41a583b5abefcc70e358004978183" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.583ex; height:3.176ex;" alt="{\displaystyle \textstyle \sum _{n=0}^{\infty }\left|a_{n}\right|=L}"> for some real number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mi>L</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cda6bb9fb712344826da560f9cee7f13da48cfb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle \textstyle L}">. Similarly, an <a href="/wiki/Improper_integral" title="Improper integral">improper integral</a> of a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \int _{0}^{\infty }f(x)\,dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \int _{0}^{\infty }f(x)\,dx}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91df6d76bfaef17c8351fd26bb3dae444a117b37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.127ex; height:3.176ex;" alt="{\displaystyle \textstyle \int _{0}^{\infty }f(x)\,dx}">, is said to converge absolutely if the integral of the absolute value of the integrand is finite—that is, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \int _{0}^{\infty }\left|f(x)\right|dx=L.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow> <mo>|</mo> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mi>L</mi> <mo>.</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \int _{0}^{\infty }\left|f(x)\right|dx=L.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc2311c86069ed781d51780751952f16bf8c97ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.749ex; height:3.176ex;" alt="{\displaystyle \textstyle \int _{0}^{\infty }\left|f(x)\right|dx=L.}"></dd> <dt id="absolute_maximum"><dfn><a href="/wiki/Maxima_and_minima" class="mw-redirect" title="Maxima and minima">absolute maximum</a></dfn></dt> <dd>The highest value a function attains.</dd> <dt id="absolute_minimum"><dfn><a href="/wiki/Maxima_and_minima" class="mw-redirect" title="Maxima and minima">absolute minimum</a></dfn></dt> <dd>The lowest value a function attains.</dd> <dt id="absolute_value"><dfn><a href="/wiki/Absolute_value" title="Absolute value">absolute value</a></dfn></dt> <dd>The absolute value or modulus |x| of a <a href="/wiki/Real_number" title="Real number">real number</a> x is the <a href="/wiki/Non-negative" class="mw-redirect" title="Non-negative">non-negative</a> value of x without regard to its <a href="/wiki/Sign_(mathematics)" title="Sign (mathematics)">sign</a>. Namely, |x| = x for a <a href="/wiki/Positive_number" class="mw-redirect" title="Positive number">positive</a> x, |x| = −x for a <a href="/wiki/Negative_number" title="Negative number">negative</a> x (in which case −x is positive), and |0| = 0. For example, the absolute value of 3 is 3, and the absolute value of −3 is also 3. The absolute value of a number may be thought of as its <a href="/wiki/Distance" title="Distance">distance</a> from zero.</dd> <dt id="alternating_series"><dfn><a href="/wiki/Alternating_series" title="Alternating series">alternating series</a></dfn></dt> <dd>An <a href="/wiki/Infinite_series" class="mw-redirect" title="Infinite series">infinite series</a> whose terms alternate between positive and negative.</dd> <dt id="alternating_series_test"><dfn><a href="/wiki/Alternating_series_test" title="Alternating series test">alternating series test</a></dfn></dt> <dd>Is the method used to prove that an <a href="/wiki/Alternating_series" title="Alternating series">alternating series</a> with terms that decrease in absolute value is a <a href="/wiki/Convergent_series" title="Convergent series">convergent series</a>. The test was used by <a href="/wiki/Gottfried_Leibniz" class="mw-redirect" title="Gottfried Leibniz">Gottfried Leibniz</a> and is sometimes known as Leibniz's test, Leibniz's rule, or the Leibniz criterion.</dd> <dt id="annulus"><dfn><a href="/wiki/Annulus_(mathematics)" title="Annulus (mathematics)">annulus</a></dfn></dt> <dd>A ring-shaped object, a region bounded by two <a href="/wiki/Concentric_circles" class="mw-redirect" title="Concentric circles">concentric circles</a>.</dd> <dt id="antiderivative"><dfn><a href="/wiki/Antiderivative" title="Antiderivative">antiderivative</a></dfn></dt> <dd>An antiderivative, primitive function, primitive integral or indefinite integral<a href="#cite_note-1">[Note 1]</a> of a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> f is a differentiable function F whose <a href="/wiki/Derivative" title="Derivative">derivative</a> is equal to the original function f. This can be stated symbolically as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F'=f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>F</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F'=f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c10131b70a7b0b4c39667f5386b14c449a5217e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.877ex; height:2.843ex;" alt="{\displaystyle F'=f}">.<a href="#cite_note-2">[1]</a><a href="#cite_note-3">[2]</a> The process of solving for antiderivatives is called antidifferentiation (or indefinite integration) and its opposite operation is called differentiation, which is the process of finding a derivative.</dd> <dt id="arcsin"><dfn><a href="/wiki/Inverse_trigonometric_functions" title="Inverse trigonometric functions">arcsin</a></dfn></dt> <dd></dd> <dt id="area_under_a_curve"><dfn><a href="/wiki/Area_under_a_curve" class="mw-redirect" title="Area under a curve">area under a curve</a></dfn></dt> <dd></dd> <dt id="asymptote"><dfn><a href="/wiki/Asymptote" title="Asymptote">asymptote</a></dfn></dt> <dd>In <a href="/wiki/Analytic_geometry" title="Analytic geometry">analytic geometry</a>, an asymptote of a <a href="/wiki/Curve" title="Curve">curve</a> is a line such that the distance between the curve and the line approaches zero as one or both of the x or y coordinates <a href="/wiki/Limit_of_a_function#Limits_at_infinity" title="Limit of a function">tends to infinity</a>. Some sources include the requirement that the curve may not cross the line infinitely often, but this is unusual for modern authors.<a href="#cite_note-4">[3]</a> In <a href="/wiki/Projective_geometry" title="Projective geometry">projective geometry</a> and related contexts, an asymptote of a curve is a line which is <a href="/wiki/Tangent" title="Tangent">tangent</a> to the curve at a <a href="/wiki/Point_at_infinity" title="Point at infinity">point at infinity</a>.<a href="#cite_note-5">[4]</a><a href="#cite_note-6">[5]</a></dd> <dt id="automatic_differentiation"><dfn><a href="/wiki/Automatic_differentiation" title="Automatic differentiation">automatic differentiation</a></dfn></dt> <dd>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> and <a href="/wiki/Computer_algebra" title="Computer algebra">computer algebra</a>, automatic differentiation (AD), also called algorithmic differentiation or computational differentiation,<a href="#cite_note-7">[6]</a><a href="#cite_note-baydin2018automatic-8">[7]</a> is a set of techniques to numerically evaluate the <a href="/wiki/Derivative" title="Derivative">derivative</a> of a function specified by a computer program. AD exploits the fact that every computer program, no matter how complicated, executes a sequence of elementary arithmetic operations (addition, subtraction, multiplication, division, etc.) and elementary functions (exp, log, sin, cos, etc.). By applying the <a href="/wiki/Chain_rule" title="Chain rule">chain rule</a> repeatedly to these operations, derivatives of arbitrary order can be computed automatically, accurately to working precision, and using at most a small constant factor more arithmetic operations than the original program.</dd> <dt id="average_rate_of_change"><dfn><a href="/wiki/Rate_(mathematics)#rate_of_change" title="Rate (mathematics)">average rate of change</a></dfn></dt> <dd></dd> </dl> <div class="mw-heading mw-heading2"><h2 id="B">B</h2>[<a href="/w/index.php?title=Glossary_of_calculus&action=edit&section=2" title="Edit section: B">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1228772891"> <dl class="glossary"> <dt id="binomial_coefficient"><dfn><a href="/wiki/Binomial_coefficient" title="Binomial coefficient">binomial coefficient</a></dfn></dt> <dd>Any of the positive <a href="/wiki/Integer" title="Integer">integers</a> that occurs as a <a href="/wiki/Coefficient" title="Coefficient">coefficient</a> in the <a href="/wiki/Binomial_theorem" title="Binomial theorem">binomial theorem</a> is a binomial coefficient. Commonly, a binomial coefficient is indexed by a pair of integers n ≥ k ≥ 0 and is written <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tbinom {n}{k}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tbinom {n}{k}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39c7822635a8fa426d00ca72733ea1bd6fe90b01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.763ex; height:3.176ex;" alt="{\displaystyle {\tbinom {n}{k}}.}"> It is the <a href="/wiki/Coefficient" title="Coefficient">coefficient</a> of the xk term in the <a href="/wiki/Polynomial_expansion" title="Polynomial expansion">polynomial expansion</a> of the <a href="/wiki/Binomial_(polynomial)" title="Binomial (polynomial)">binomial</a> <a href="/wiki/Exponentiation" title="Exponentiation">power</a> (1 + x)n, and it is given by the formula <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\binom {n}{k}}={\frac {n!}{k!(n-k)!}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>!</mo> </mrow> <mrow> <mi>k</mi> <mo>!</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\binom {n}{k}}={\frac {n!}{k!(n-k)!}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2457a7ef3c77831e34e06a1fe17a80b84a03181" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:19.158ex; height:6.343ex;" alt="{\displaystyle {\binom {n}{k}}={\frac {n!}{k!(n-k)!}}.}"></dd></dl></dd></dl> <dt id="binomial_theorem'''_(or_'''binomial_expansion)"><dfn><a href="/wiki/Binomial_theorem" title="Binomial theorem">binomial theorem</a> (or <a href="/wiki/Binomial_expansion" class="mw-redirect" title="Binomial expansion">binomial expansion</a>)</dfn></dt> <dd> Describes the algebraic expansion of <a href="/wiki/Exponentiation" title="Exponentiation">powers</a> of a <a href="/wiki/Binomial_(polynomial)" title="Binomial (polynomial)">binomial</a>.</dd> <dt id="bounded_function"><dfn><a href="/wiki/Bounded_function" title="Bounded function">bounded function</a></dfn></dt> <dd>A <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> f defined on some <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> X with <a href="/wiki/Real_number" title="Real number">real</a> or <a href="/wiki/Complex_number" title="Complex number">complex</a> values is called bounded, if the set of its values is <a href="/wiki/Bounded_set" title="Bounded set">bounded</a>. In other words, <a href="/wiki/There_exists" class="mw-redirect" title="There exists">there exists</a> a real number M such that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |f(x)|\leq M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |f(x)|\leq M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfc8ec4d3c57aa126bee5459febb9832b99cbcc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.252ex; height:2.843ex;" alt="{\displaystyle |f(x)|\leq M}"></dd></dl> <a href="/wiki/For_all" class="mw-redirect" title="For all">for all</a> x in X. A function that is not bounded is said to be unbounded. Sometimes, if f(x) ≤ A for all x in X, then the function is said to be bounded above by A. On the other hand, if f(x) ≥ B for all x in X, then the function is said to be bounded below by B.</dd> <dt id="bounded_sequence"><dfn><a href="/wiki/Bounded_function#bounded_sequence" title="Bounded function">bounded sequence</a></dfn></dt> <dd> .</dd> <div class="mw-heading mw-heading2"><h2 id="C">C</h2>[<a href="/w/index.php?title=Glossary_of_calculus&action=edit&section=3" title="Edit section: C">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1228772891"> <dl class="glossary"> <dt id="calculus"><dfn><a href="/wiki/Calculus" title="Calculus">calculus</a></dfn></dt> <dd>(From <a href="/wiki/Latin" title="Latin">Latin</a> calculus, literally 'small pebble', used for counting and calculations, as on an <a href="/wiki/Abacus" title="Abacus">abacus</a>)<a href="#cite_note-oxdic-9">[8]</a> is the <a href="/wiki/Mathematics" title="Mathematics">mathematical</a> study of continuous change, in the same way that <a href="/wiki/Geometry" title="Geometry">geometry</a> is the study of shape and <a href="/wiki/Algebra" title="Algebra">algebra</a> is the study of generalizations of <a href="/wiki/Arithmetic_operations" class="mw-redirect" title="Arithmetic operations">arithmetic operations</a>.</dd> <dt id="cavalieri's_principle"><dfn><a href="/wiki/Cavalieri%27s_principle" title="Cavalieri's principle">Cavalieri's principle</a></dfn></dt> <dd> Cavalieri's principle, a modern implementation of the method of indivisibles, named after <a href="/wiki/Bonaventura_Cavalieri" title="Bonaventura Cavalieri">Bonaventura Cavalieri</a>, is as follows:<a href="#cite_note-10">[9]</a> <ul><li>2-dimensional case: Suppose two regions in a plane are included between two parallel lines in that plane. If every line parallel to these two lines intersects both regions in line segments of equal length, then the two regions have equal areas.</li> <li>3-dimensional case: Suppose two regions in three-space (solids) are included between two parallel planes. If every plane parallel to these two planes intersects both regions in <a href="/wiki/Cross_section_(geometry)" title="Cross section (geometry)">cross-sections</a> of equal area, then the two regions have equal volumes.</li></ul></dd> <dt id="chain_rule"><dfn><a href="/wiki/Chain_rule" title="Chain rule">chain rule</a></dfn></dt> <dd>The chain rule is a <a href="/wiki/Formula" title="Formula">formula</a> for computing the <a href="/wiki/Derivative" title="Derivative">derivative</a> of the <a href="/wiki/Function_composition" title="Function composition">composition</a> of two or more <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">functions</a>. That is, if f and g are functions, then the chain rule expresses the derivative of their composition f ∘ g (the function which maps x to f(g(x)) ) in terms of the derivatives of f and g and the <a href="/wiki/Pointwise_product" class="mw-redirect" title="Pointwise product">product of functions</a> as follows: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f\circ g)'=(f'\circ g)\cdot g'.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∘</mo> <mi>g</mi> <msup> <mo stretchy="false">)</mo> <mo>′</mo> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo>∘</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f\circ g)'=(f'\circ g)\cdot g'.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb55cd5448d4bed6da3b79283d92eec2ab9bb95d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.436ex; height:3.009ex;" alt="{\displaystyle (f\circ g)'=(f'\circ g)\cdot g'.}"></dd></dl> This may equivalently be expressed in terms of the variable. Let F = f ∘ g, or equivalently, F(x) = f(g(x)) for all x. Then one can also write <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F'(x)=f'(g(x))g'(x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>F</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F'(x)=f'(g(x))g'(x).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63b54293b9ca115caa553c556cc7faa6a022888a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.395ex; height:3.009ex;" alt="{\displaystyle F'(x)=f'(g(x))g'(x).}"></dd></dl> The chain rule may be written in <a href="/wiki/Leibniz%27s_notation" title="Leibniz's notation">Leibniz's notation</a> in the following way. If a variable z depends on the variable y, which itself depends on the variable x, so that y and z are therefore <a href="/wiki/Dependent_variable" class="mw-redirect" title="Dependent variable">dependent variables</a>, then z, via the intermediate variable of y, depends on x as well. The chain rule then states, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dz}{dx}}={\frac {dz}{dy}}\cdot {\frac {dy}{dx}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>z</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>z</mi> </mrow> <mrow> <mi>d</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dz}{dx}}={\frac {dz}{dy}}\cdot {\frac {dy}{dx}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1dfc1ad15251bd965f3737d0b8fa8b67ef3cccb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.395ex; height:5.843ex;" alt="{\displaystyle {\frac {dz}{dx}}={\frac {dz}{dy}}\cdot {\frac {dy}{dx}}.}"></dd></dl> The two versions of the chain rule are related; if <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><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)}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=g(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=g(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26c08f8fd3471dad5e2c45c2f753ffd7c9aba4ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.509ex; height:2.843ex;" alt="{\displaystyle y=g(x)}">, then <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dz}{dx}}={\frac {dz}{dy}}\cdot {\frac {dy}{dx}}=f'(y)g'(x)=f'(g(x))g'(x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>z</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>z</mi> </mrow> <mrow> <mi>d</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dz}{dx}}={\frac {dz}{dy}}\cdot {\frac {dy}{dx}}=f'(y)g'(x)=f'(g(x))g'(x).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/075ea30eb4897a37e57bd9ac7bd6a097a1f42167" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:44.516ex; height:5.843ex;" alt="{\displaystyle {\frac {dz}{dx}}={\frac {dz}{dy}}\cdot {\frac {dy}{dx}}=f'(y)g'(x)=f'(g(x))g'(x).}"></dd></dl> In <a href="/wiki/Integral" title="Integral">integration</a>, the counterpart to the chain rule is the <a href="/wiki/Substitution_rule" class="mw-redirect" title="Substitution rule">substitution rule</a>.</dd> <dt id="change_of_variables"><dfn><a href="/wiki/Change_of_variables" title="Change of variables">change of variables</a></dfn></dt> <dd> Is a basic technique used to simplify problems in which the original <a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variables</a> are replaced with <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">functions</a> of other variables. The intent is that when expressed in new variables, the problem may become simpler, or equivalent to a better understood problem.</dd> <dt id="cofunction"><dfn><a href="/wiki/Cofunction" title="Cofunction">cofunction</a></dfn></dt> <dd>A <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> f is cofunction of a function g if f(A) = g(B) whenever A and B are <a href="/wiki/Complementary_angles" class="mw-redirect" title="Complementary angles">complementary angles</a>.<a href="#cite_note-Hall_1909-11">[10]</a> This definition typically applies to <a href="/wiki/Trigonometric_functions" title="Trigonometric functions">trigonometric functions</a>.<a href="#cite_note-Aufmann_Nation_2014-12">[11]</a><a href="#cite_note-Bales_2012-13">[12]</a> The prefix "co-" can be found already in <a href="/wiki/Edmund_Gunter" title="Edmund Gunter">Edmund Gunter</a>'s Canon triangulorum (1620).<a href="#cite_note-Gunter_1620-14">[13]</a><a href="#cite_note-Roegel_2010-15">[14]</a></dd> <dt id="concave_function"><dfn><a href="/wiki/Concave_function" title="Concave function">concave function</a></dfn></dt> <dd> Is the <a href="/wiki/Additive_inverse" title="Additive inverse">negative</a> of a <a href="/wiki/Convex_function" title="Convex function">convex function</a>. A concave function is also <a href="/wiki/Synonym" title="Synonym">synonymously</a> called concave downwards, concave down, convex upwards, convex cap or upper convex.</dd> <dt id="constant_of_integration"><dfn><a href="/wiki/Constant_of_integration" title="Constant of integration">constant of integration</a></dfn></dt> <dd>The <a href="/wiki/Indefinite_integral" class="mw-redirect" title="Indefinite integral">indefinite integral</a> of a given function (i.e., the <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> of all <a href="/wiki/Antiderivative" title="Antiderivative">antiderivatives</a> of the function) on a <a href="/wiki/Connected_set" class="mw-redirect" title="Connected set">connected domain</a> is only defined <a href="/wiki/Up_to" title="Up to">up to</a> an additive constant, the constant of integration.<a href="#cite_note-16">[15]</a><a href="#cite_note-17">[16]</a> This constant expresses an ambiguity inherent in the construction of antiderivatives. If a function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"> is defined on an <a href="/wiki/Interval_(mathematics)" title="Interval (mathematics)">interval</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71a82805d469cdfa7856c11d6ee756acd1dc7174" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.88ex; height:2.843ex;" alt="{\displaystyle F(x)}"> is an antiderivative of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}">, then the set of all antiderivatives of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"> is given by the functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x)+C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x)+C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84bfd7719983cec6643eb997d3aa006ad1c3bf26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.486ex; height:2.843ex;" alt="{\displaystyle F(x)+C}">, where C is an arbitrary constant (meaning that any value for C makes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x)+C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x)+C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84bfd7719983cec6643eb997d3aa006ad1c3bf26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.486ex; height:2.843ex;" alt="{\displaystyle F(x)+C}"> a valid antiderivative). The constant of integration is sometimes omitted in <a href="/wiki/Lists_of_integrals" title="Lists of integrals">lists of integrals</a> for simplicity.</dd> <dt id="continuous_function"><dfn><a href="/wiki/Continuous_function" title="Continuous function">continuous function</a></dfn></dt> <dd>Is a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> for which sufficiently small changes in the input result in arbitrarily small changes in the output. Otherwise, a function is said to be a discontinuous function. A continuous function with a continuous <a href="/wiki/Inverse_function" title="Inverse function">inverse function</a> is called a <a href="/wiki/Homeomorphism" title="Homeomorphism">homeomorphism</a>.</dd> <dt id="continuously_differentiable"><dfn><a href="/wiki/Differentiable_function#Differentiability_classes" title="Differentiable function">continuously differentiable</a></dfn></dt> <dd>A function f is said to be continuously differentiable if the derivative f′(x) exists and is itself a continuous function.</dd> <dt id="contour_integration"><dfn><a href="/wiki/Methods_of_contour_integration" class="mw-redirect" title="Methods of contour integration">contour integration</a></dfn></dt> <dd>In the mathematical field of <a href="/wiki/Complex_analysis" title="Complex analysis">complex analysis</a>, contour integration is a method of evaluating certain <a href="/wiki/Integral" title="Integral">integrals</a> along paths in the complex plane.<a href="#cite_note-Stalker-18">[17]</a><a href="#cite_note-Bak-19">[18]</a><a href="#cite_note-Krantz-20">[19]</a></dd> <dt id="convergence_tests"><dfn><a href="/wiki/Convergence_test" class="mw-redirect" title="Convergence test">convergence tests</a></dfn></dt> <dd>Are methods of testing for the <a href="/wiki/Convergent_series" title="Convergent series">convergence</a>, <a href="/wiki/Conditional_convergence" title="Conditional convergence">conditional convergence</a>, <a href="/wiki/Absolute_convergence" title="Absolute convergence">absolute convergence</a>, <a href="/wiki/Interval_of_convergence" class="mw-redirect" title="Interval of convergence">interval of convergence</a> or divergence of an <a href="/wiki/Series_(mathematics)" title="Series (mathematics)">infinite series</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=1}^{\infty }a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=1}^{\infty }a_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbf33b91e1eb05d0530e73e355823f3c07821381" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:6.19ex; height:6.843ex;" alt="{\displaystyle \sum _{n=1}^{\infty }a_{n}}">.</dd> <dt id="convergent_series"><dfn><a href="/wiki/Convergent_series" title="Convergent series">convergent series</a></dfn></dt> <dd>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a <a href="/wiki/Series_(mathematics)" title="Series (mathematics)">series</a> is the <a href="/wiki/Summation" title="Summation">sum</a> of the terms of an <a href="/wiki/Infinite_sequence" class="mw-redirect" title="Infinite sequence">infinite sequence</a> of numbers. Given an infinite sequence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(a_{1},\ a_{2},\ a_{3},\dots \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(a_{1},\ a_{2},\ a_{3},\dots \right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da1a7105fe28981ce02d106be4ab12e093be5f59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.648ex; height:2.843ex;" alt="{\displaystyle \left(a_{1},\ a_{2},\ a_{3},\dots \right)}">, the nth <a href="/wiki/Partial_sum" class="mw-redirect" title="Partial sum">partial sum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f049ac28d4ac8097b625f9d71c1f22b2ebd1bc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.643ex; height:2.509ex;" alt="{\displaystyle S_{n}}"> is the sum of the first n terms of the sequence, that is, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{n}=\sum _{k=1}^{n}a_{k}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{n}=\sum _{k=1}^{n}a_{k}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e782ba72c493a2b78b1c14eec13fc58140febeff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:12.449ex; height:6.843ex;" alt="{\displaystyle S_{n}=\sum _{k=1}^{n}a_{k}.}"></dd></dl> A series is convergent if the sequence of its partial sums <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{S_{1},\ S_{2},\ S_{3},\dots \right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{S_{1},\ S_{2},\ S_{3},\dots \right\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89250263c344e97cc83f572e346064e5fbed19f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.749ex; height:2.843ex;" alt="{\displaystyle \left\{S_{1},\ S_{2},\ S_{3},\dots \right\}}"> tends to a <a href="/wiki/Limit_of_a_sequence" title="Limit of a sequence">limit</a>; that means that the partial sums become closer and closer to a given number when the number of their terms increases. More precisely, a series converges, if there exists a number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f066e981e530bacc07efc6a10fa82deee985929e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.97ex; height:2.176ex;" alt="{\displaystyle \ell }"> such that for any arbitrarily small positive number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }">, there is a (sufficiently large) <a href="/wiki/Integer" title="Integer">integer</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"> such that for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq \ N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mtext> </mtext> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq \ N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7963de0d41caf7e03362087d13e04db9eac57b52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.137ex; height:2.343ex;" alt="{\displaystyle n\geq \ N}">, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|S_{n}-\ell \right\vert \leq \ \varepsilon .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mi>ℓ</mi> </mrow> <mo>|</mo> </mrow> <mo>≤</mo> <mtext> </mtext> <mi>ε</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|S_{n}-\ell \right\vert \leq \ \varepsilon .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a11ae7f0b6f3f6bd68713334ff444ea123f8fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.157ex; height:2.843ex;" alt="{\displaystyle \left|S_{n}-\ell \right\vert \leq \ \varepsilon .}"></dd></dl> If the series is convergent, the number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f066e981e530bacc07efc6a10fa82deee985929e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.97ex; height:2.176ex;" alt="{\displaystyle \ell }"> (necessarily unique) is called the sum of the series. Any series that is not convergent is said to be <a href="/wiki/Divergent_series" title="Divergent series">divergent</a>.</dd> <dt id="convex_function"><dfn><a href="/wiki/Convex_function" title="Convex function">convex function</a></dfn></dt> <dd>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a <a href="/wiki/Real-valued_function" title="Real-valued function">real-valued function</a> defined on an <a href="/wiki/Interval_(mathematics)#Multi-dimensional_intervals" title="Interval (mathematics)">n-dimensional interval</a> is called convex (or convex downward or concave upward) if the <a href="/wiki/Line_segment" title="Line segment">line segment</a> between any two points on the <a href="/wiki/Graph_of_a_function" title="Graph of a function">graph of the function</a> lies above or on the graph, in a <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a> (or more generally a <a href="/wiki/Vector_space" title="Vector space">vector space</a>) of at least two dimensions. Equivalently, a function is convex if its <a href="/wiki/Epigraph_(mathematics)" title="Epigraph (mathematics)">epigraph</a> (the set of points on or above the graph of the function) is a <a href="/wiki/Convex_set" title="Convex set">convex set</a>. For a twice differentiable function of a single variable, if the second derivative is always greater than or equal to zero for its entire domain then the function is convex.<a href="#cite_note-21">[20]</a> Well-known examples of convex functions include the <a href="/wiki/Quadratic_function" title="Quadratic function">quadratic function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf0bf28fd28f45d07e1ceb909ce333c18c558c93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.384ex; height:2.676ex;" alt="{\displaystyle x^{2}}"> and the <a href="/wiki/Exponential_function" title="Exponential function">exponential function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{x}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/841c0d168e64191c45a45e54c7e447defd17ec6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.256ex; height:2.343ex;" alt="{\displaystyle e^{x}}">.</dd> <dt id="cramer's_rule"><dfn><a href="/wiki/Cramer%27s_rule" title="Cramer's rule">Cramer's rule</a></dfn></dt> <dd> In <a href="/wiki/Linear_algebra" title="Linear algebra">linear algebra</a>, Cramer's rule is an explicit formula for the solution of a <a href="/wiki/System_of_linear_equations" title="System of linear equations">system of linear equations</a> with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the <a href="/wiki/Determinant" title="Determinant">determinants</a> of the (square) coefficient <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> and of matrices obtained from it by replacing one column by the column vector of right-hand-sides of the equations. It is named after <a href="/wiki/Gabriel_Cramer" title="Gabriel Cramer">Gabriel Cramer</a> (1704–1752), who published the rule for an arbitrary number of unknowns in 1750,<a href="#cite_note-22">[21]</a><a href="#cite_note-23">[22]</a> although <a href="/wiki/Colin_Maclaurin" title="Colin Maclaurin">Colin Maclaurin</a> also published special cases of the rule in 1748<a href="#cite_note-24">[23]</a> (and possibly knew of it as early as 1729).<a href="#cite_note-25">[24]</a><a href="#cite_note-26">[25]</a><a href="#cite_note-27">[26]</a></dd> <dt id="critical_point"><dfn><a href="/wiki/Critical_point_(mathematics)" title="Critical point (mathematics)">critical point</a></dfn></dt> <dd>A critical point or stationary point of a <a href="/wiki/Differentiable_function" title="Differentiable function">differentiable function</a> of a <a href="/wiki/Function_of_a_real_variable" title="Function of a real variable">real</a> or <a href="/wiki/Complex_variable" class="mw-redirect" title="Complex variable">complex variable</a> is any value in its <a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a> where its <a href="/wiki/Derivative" title="Derivative">derivative</a> is 0.<a href="#cite_note-28">[27]</a><a href="#cite_note-29">[28]</a></dd> <dt id="curve"><dfn><a href="/wiki/Curve" title="Curve">curve</a></dfn></dt> <dd>A curve (also called a curved line in older texts) is, generally speaking, an object similar to a <a href="/wiki/Line_(geometry)" title="Line (geometry)">line</a> but that need not be <a href="/wiki/Linearity" title="Linearity">straight</a>.</dd> <dt id="curve_sketching"><dfn><a href="/wiki/Curve_sketching" title="Curve sketching">curve sketching</a></dfn></dt> <dd>In <a href="/wiki/Geometry" title="Geometry">geometry</a>, curve sketching (or curve tracing) includes techniques that can be used to produce a rough idea of overall shape of a <a href="/wiki/Plane_curve" title="Plane curve">plane curve</a> given its equation without computing the large numbers of points required for a detailed plot. It is an application of the theory of curves to find their main features. Here input is an equation. In <a href="/wiki/Digital_geometry" title="Digital geometry">digital geometry</a> it is a method of drawing a curve pixel by pixel. Here input is an array (digital image).</dd> </dl> <div class="mw-heading mw-heading2"><h2 id="D">D</h2>[<a href="/w/index.php?title=Glossary_of_calculus&action=edit&section=4" title="Edit section: D">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1228772891"> <dl class="glossary"> <dt id="damped_sine_wave"><dfn><a href="/wiki/Damped_sine_wave" class="mw-redirect" title="Damped sine wave">damped sine wave</a></dfn></dt> <dd>Is a <a href="/wiki/Sine_wave" title="Sine wave">sinusoidal function</a> whose amplitude approaches zero as time increases.<a href="#cite_note-30">[29]</a></dd> <dt id="degree_of_a_polynomial"><dfn><a href="/wiki/Degree_of_a_polynomial" title="Degree of a polynomial">degree of a polynomial</a></dfn></dt> <dd>Is the highest degree of its <a href="/wiki/Monomial" title="Monomial">monomials</a> (individual terms) with non-zero coefficients. The <a href="/wiki/Degree_of_a_monomial" class="mw-redirect" title="Degree of a monomial">degree of a term</a> is the sum of the exponents of the <a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variables</a> that appear in it, and thus is a non-negative integer.</dd> <dt id="derivative"><dfn><a href="/wiki/Derivative" title="Derivative">derivative</a></dfn></dt> <dd>The derivative of a <a href="/wiki/Function_of_a_real_variable" title="Function of a real variable">function of a real variable</a> measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of <a href="/wiki/Calculus" title="Calculus">calculus</a>. For example, the derivative of the position of a moving object with respect to <a href="/wiki/Time" title="Time">time</a> is the object's <a href="/wiki/Velocity" title="Velocity">velocity</a>: this measures how quickly the position of the object changes when time advances.</dd> <dt id="derivative_test"><dfn><a href="/wiki/Derivative_test" title="Derivative test">derivative test</a></dfn></dt> <dd>A derivative test uses the <a href="/wiki/Derivative" title="Derivative">derivatives</a> of a function to locate the <a href="/wiki/Stationary_point" title="Stationary point">critical points</a> of a function and determine whether each point is a <a href="/wiki/Local_maximum" class="mw-redirect" title="Local maximum">local maximum</a>, a <a href="/wiki/Local_minimum" class="mw-redirect" title="Local minimum">local minimum</a>, or a <a href="/wiki/Saddle_point" title="Saddle point">saddle point</a>. Derivative tests can also give information about the <a href="/wiki/Concave_function" title="Concave function">concavity</a> of a function.</dd> <dt id="differentiable_function"><dfn><a href="/wiki/Differentiable_function" title="Differentiable function">differentiable function</a></dfn></dt> <dd>A differentiable function of one <a href="/wiki/Real_number" title="Real number">real</a> variable is a function whose <a href="/wiki/Derivative" title="Derivative">derivative</a> exists at each point in its <a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a>. As a result, the <a href="/wiki/Graph_of_a_function" title="Graph of a function">graph</a> of a differentiable function must have a (non-<a href="/wiki/Vertical_tangent" title="Vertical tangent">vertical</a>) <a href="/wiki/Tangent_line" class="mw-redirect" title="Tangent line">tangent line</a> at each point in its domain, be relatively smooth, and cannot contain any breaks, bends, or <a href="/wiki/Cusp_(singularity)" title="Cusp (singularity)">cusps</a>.</dd> <dt id="differential_(infinitesimal)"><dfn><a href="/wiki/Differential_(infinitesimal)" class="mw-redirect" title="Differential (infinitesimal)">differential (infinitesimal)</a></dfn></dt> <dd>The term differential is used in <a href="/wiki/Calculus" title="Calculus">calculus</a> to refer to an <a href="/wiki/Infinitesimal" title="Infinitesimal">infinitesimal</a> (infinitely small) change in some <a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">varying quantity</a>. For example, if x is a <a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variable</a>, then a change in the value of x is often denoted Δx (pronounced <a href="/wiki/Delta_(Greek)" class="mw-redirect" title="Delta (Greek)">delta</a> x). The differential dx represents an infinitely small change in the variable x. The idea of an infinitely small or infinitely slow change is extremely useful intuitively, and there are a number of ways to make the notion mathematically precise. Using calculus, it is possible to relate the infinitely small changes of various variables to each other mathematically using <a href="/wiki/Derivative" title="Derivative">derivatives</a>. If y is a function of x, then the differential dy of y is related to dx by the formula <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dy={\frac {dy}{dx}}\,dx,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dy={\frac {dy}{dx}}\,dx,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8690f9df1066aadc3c2999e8633a040038558df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.431ex; height:5.509ex;" alt="{\displaystyle dy={\frac {dy}{dx}}\,dx,}"></dd></dl> where dy/dx denotes the <a href="/wiki/Derivative" title="Derivative">derivative</a> of y with respect to x. This formula summarizes the intuitive idea that the derivative of y with respect to x is the limit of the ratio of differences Δy/Δx as Δx becomes infinitesimal.</dd> <dt id="differential_calculus"><dfn><a href="/wiki/Differential_calculus" title="Differential calculus">differential calculus</a></dfn></dt> <dd>Is a subfield of calculus<a href="#cite_note-31">[30]</a> concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being <a href="/wiki/Integral_calculus" class="mw-redirect" title="Integral calculus">integral calculus</a>, the study of the area beneath a curve.<a href="#cite_note-32">[31]</a></dd> <dt id="differential_equation"><dfn><a href="/wiki/Differential_equation" title="Differential equation">differential equation</a></dfn></dt> <dd>Is a <a href="/wiki/Mathematics" title="Mathematics">mathematical</a> <a href="/wiki/Equation" title="Equation">equation</a> that relates some <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> with its <a href="/wiki/Derivative" title="Derivative">derivatives</a>. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, and the equation defines a relationship between the two.</dd> <dt id="differential_operator"><dfn><a href="/wiki/Differential_operator" title="Differential operator">differential operator</a></dfn></dt> <dd>.</dd> <dt id="differential_of_a_function"><dfn><a href="/wiki/Differential_of_a_function" title="Differential of a function">differential of a function</a></dfn></dt> <dd>In <a href="/wiki/Calculus" title="Calculus">calculus</a>, the differential represents the <a href="/wiki/Principal_part#Calculus" title="Principal part">principal part</a> of the change in a function y = f(x) with respect to changes in the independent variable. The differential dy is defined by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dy=f'(x)\,dx,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>y</mi> <mo>=</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dy=f'(x)\,dx,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f133dae980e28d3d9865a0fef74cf9632c26d3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.194ex; height:3.009ex;" alt="{\displaystyle dy=f'(x)\,dx,}"></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f'(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f'(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0cd7d7c75340e779d82658e19d1720ce84ab127" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.144ex; height:3.009ex;" alt="{\displaystyle f'(x)}"> is the <a href="/wiki/Derivative" title="Derivative">derivative</a> of f with respect to x, and dx is an additional real <a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variable</a> (so that dy is a function of x and dx). The notation is such that the equation <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dy={\frac {dy}{dx}}\,dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dy={\frac {dy}{dx}}\,dx}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4af882e478e7345731264e1984ccdbc067af8df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.784ex; height:5.509ex;" alt="{\displaystyle dy={\frac {dy}{dx}}\,dx}"></dd></dl> holds, where the derivative is represented in the <a href="/wiki/Leibniz_notation" class="mw-redirect" title="Leibniz notation">Leibniz notation</a> dy/dx, and this is consistent with regarding the derivative as the quotient of the differentials. One also writes <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle df(x)=f'(x)\,dx.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle df(x)=f'(x)\,dx.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1036d9e3de8d36ab7126d94f2e0603458b3307b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.456ex; height:3.009ex;" alt="{\displaystyle df(x)=f'(x)\,dx.}"></dd></dl> The precise meaning of the variables dy and dx depends on the context of the application and the required level of mathematical rigor. The domain of these variables may take on a particular geometrical significance if the differential is regarded as a particular <a href="/wiki/Differential_form" title="Differential form">differential form</a>, or analytical significance if the differential is regarded as a <a href="/wiki/Linear_approximation" title="Linear approximation">linear approximation</a> to the increment of a function. Traditionally, the variables dx and dy are considered to be very small (<a href="/wiki/Infinitesimal" title="Infinitesimal">infinitesimal</a>), and this interpretation is made rigorous in <a href="/wiki/Non-standard_analysis" class="mw-redirect" title="Non-standard analysis">non-standard analysis</a>.</dd> <dt id="differentiation_rules"><dfn><a href="/wiki/Differentiation_rules" title="Differentiation rules">differentiation rules</a></dfn></dt> <dd>.</dd> <dt id="direct_comparison_test"><dfn><a href="/wiki/Direct_comparison_test" title="Direct comparison test">direct comparison test</a></dfn></dt> <dd>A convergence test in which an infinite series or an improper integral is compared to one with known convergence properties.</dd> <dt id="dirichlet's_test"><dfn><a href="/wiki/Dirichlet%27s_test" title="Dirichlet's test">Dirichlet's test</a></dfn></dt> <dd>Is a method of testing for the <a href="/wiki/Convergent_series" title="Convergent series">convergence</a> of a <a href="/wiki/Series_(mathematics)" title="Series (mathematics)">series</a>. It is named after its author <a href="/wiki/Peter_Gustav_Lejeune_Dirichlet" title="Peter Gustav Lejeune Dirichlet">Peter Gustav Lejeune Dirichlet</a>, and was published posthumously in the <a href="/wiki/Journal_de_Math%C3%A9matiques_Pures_et_Appliqu%C3%A9es" title="Journal de Mathématiques Pures et Appliquées">Journal de Mathématiques Pures et Appliquées</a> in 1862.<a href="#cite_note-33">[32]</a> The test states that if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{a_{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{a_{n}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/339d461b497fb8f8670bb29308fe09f0e7bfd34a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.773ex; height:2.843ex;" alt="{\displaystyle \{a_{n}\}}"> is a <a href="/wiki/Sequence" title="Sequence">sequence</a> of <a href="/wiki/Real_number" title="Real number">real numbers</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{b_{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{b_{n}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cad2485b9672375982ec521a53ee5a4104001a15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.541ex; height:2.843ex;" alt="{\displaystyle \{b_{n}\}}"> a sequence of <a href="/wiki/Complex_number" title="Complex number">complex numbers</a> satisfying <dl><dd><ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n+1}\leq a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>≤</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n+1}\leq a_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02713e1514bdb8004c9c47d119dafbb2d0499179" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.096ex; height:2.343ex;" alt="{\displaystyle a_{n+1}\leq a_{n}}"></li></ul></dd></dl> <dl><dd><ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\rightarrow \infty }a_{n}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\rightarrow \infty }a_{n}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d7b1e35359928f755f4b2e11910157bf977816d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.369ex; height:3.676ex;" alt="{\displaystyle \lim _{n\rightarrow \infty }a_{n}=0}"></li></ul></dd></dl> <dl><dd><ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|\sum _{n=1}^{N}b_{n}\right|\leq M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>|</mo> </mrow> <mo>≤</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|\sum _{n=1}^{N}b_{n}\right|\leq M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17ad87aaef6dc6c7f3a52c1dfcd5b7c7fbddbe2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:12.792ex; height:7.509ex;" alt="{\displaystyle \left|\sum _{n=1}^{N}b_{n}\right|\leq M}"> for every positive integer N</li></ul></dd></dl> where M is some constant, then the series <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=1}^{\infty }a_{n}b_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=1}^{\infty }a_{n}b_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1da4118b971026424454fb80f2458ef9b4cac33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:8.406ex; height:6.843ex;" alt="{\displaystyle \sum _{n=1}^{\infty }a_{n}b_{n}}"></dd></dl> converges.</dd> <dt id="disc_integration"><dfn><a href="/wiki/Disc_integration" title="Disc integration">disc integration</a></dfn></dt> <dd>Also known in <a href="/wiki/Integral_calculus" class="mw-redirect" title="Integral calculus">integral calculus</a> as the disc method, is a means of calculating the <a href="/wiki/Volume" title="Volume">volume</a> of a <a href="/wiki/Solid_of_revolution" title="Solid of revolution">solid of revolution</a> of a solid-state material when <a href="/wiki/Integral" title="Integral">integrating</a> along an axis "parallel" to the <a href="/wiki/Axis_of_revolution" class="mw-redirect" title="Axis of revolution">axis of revolution</a>.</dd> <dt id="divergent_series"><dfn><a href="/wiki/Divergent_series" title="Divergent series">divergent series</a></dfn></dt> <dd>Is an <a href="/wiki/Infinite_series" class="mw-redirect" title="Infinite series">infinite series</a> that is not <a href="/wiki/Convergent_series" title="Convergent series">convergent</a>, meaning that the infinite <a href="/wiki/Sequence" title="Sequence">sequence</a> of the <a href="/wiki/Partial_sum" class="mw-redirect" title="Partial sum">partial sums</a> of the series does not have a finite <a href="/wiki/Limit_of_a_sequence" title="Limit of a sequence">limit</a>.</dd> <dt id="discontinuity"><dfn><a href="/wiki/Discontinuity_(mathematics)" class="mw-redirect" title="Discontinuity (mathematics)">discontinuity</a></dfn></dt> <dd><a href="/wiki/Continuous_function" title="Continuous function">Continuous functions</a> are of utmost importance in <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, functions and applications. However, not all <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">functions</a> are continuous. If a function is not continuous at a point in its <a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a>, one says that it has a discontinuity there. The set of all points of discontinuity of a function may be a <a href="/wiki/Discrete_set" class="mw-redirect" title="Discrete set">discrete set</a>, a <a href="/wiki/Dense_set" title="Dense set">dense set</a>, or even the entire domain of the function.</dd> <dt id="dot_product"><dfn><a href="/wiki/Dot_product" title="Dot product">dot product</a></dfn></dt> <dd>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, the dot product or scalar product<a href="#cite_note-34">[note 1]</a> is an <a href="/wiki/Algebraic_operation" title="Algebraic operation">algebraic operation</a> that takes two equal-length sequences of numbers (usually <a href="/wiki/Coordinate_vector" title="Coordinate vector">coordinate vectors</a>) and returns a single number. In <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>, the dot product of the <a href="/wiki/Cartesian_coordinates" class="mw-redirect" title="Cartesian coordinates">Cartesian coordinates</a> of two <a href="/wiki/Vector_(mathematics_and_physics)" title="Vector (mathematics and physics)">vectors</a> is widely used and often called "the" inner product (or rarely projection product) of Euclidean space even though it is not the only inner product that can be defined on Euclidean space; see also <a href="/wiki/Inner_product_space" title="Inner product space">inner product space</a>.</dd> <dt id="double_integral"><dfn><a href="/wiki/Double_integral" class="mw-redirect" title="Double integral">double integral</a></dfn></dt> <dd>The multiple integral is a <a href="/wiki/Definite_integral" class="mw-redirect" title="Definite integral">definite integral</a> of a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> of more than one real <a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variable</a>, for example, f(x, y) or f(x, y, z). Integrals of a function of two variables over a region in R2 are called <a href="/wiki/Double_integrals" class="mw-redirect" title="Double integrals">double integrals</a>, and integrals of a function of three variables over a region of R3 are called <a href="/wiki/Triple_integrals" class="mw-redirect" title="Triple integrals">triple integrals</a>.<a href="#cite_note-Stewart-35">[33]</a></dd> </dl> <div class="mw-heading mw-heading2"><h2 id="E">E</h2>[<a href="/w/index.php?title=Glossary_of_calculus&action=edit&section=5" title="Edit section: E">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1228772891"> <dl class="glossary"> <dt id="e_(mathematical_constant)"><dfn><a href="/wiki/E_(mathematical_constant)" title="E (mathematical constant)">e (mathematical constant)</a></dfn></dt> <dd>The number e is a <a href="/wiki/Mathematical_constant" title="Mathematical constant">mathematical constant</a> that is the base of the <a href="/wiki/Natural_logarithm" title="Natural logarithm">natural logarithm</a>: the unique number whose natural logarithm is equal to one. It is approximately equal to 2.71828,<a href="#cite_note-36">[34]</a> and is the <a href="/wiki/Limit_of_a_sequence" title="Limit of a sequence">limit</a> of (1 + 1/n)n as n approaches <a href="/wiki/Infinity" title="Infinity">infinity</a>, an expression that arises in the study of <a href="/wiki/Compound_interest" title="Compound interest">compound interest</a>. It can also be calculated as the sum of the infinite <a href="/wiki/Series_(mathematics)" title="Series (mathematics)">series</a><a href="#cite_note-37">[35]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e=\displaystyle \sum \limits _{n=0}^{\infty }{\dfrac {1}{n!}}={\frac {1}{1}}+{\frac {1}{1}}+{\frac {1}{1\cdot 2}}+{\frac {1}{1\cdot 2\cdot 3}}+\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>=</mo> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo movablelimits="false">∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>1</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>1</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>⋅</mo> <mn>2</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>⋅</mo> <mn>2</mn> <mo>⋅</mo> <mn>3</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e=\displaystyle \sum \limits _{n=0}^{\infty }{\dfrac {1}{n!}}={\frac {1}{1}}+{\frac {1}{1}}+{\frac {1}{1\cdot 2}}+{\frac {1}{1\cdot 2\cdot 3}}+\cdots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/534a55992d112b62c1163de47b1dd9268f20153a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:44.504ex; height:6.843ex;" alt="{\displaystyle e=\displaystyle \sum \limits _{n=0}^{\infty }{\dfrac {1}{n!}}={\frac {1}{1}}+{\frac {1}{1}}+{\frac {1}{1\cdot 2}}+{\frac {1}{1\cdot 2\cdot 3}}+\cdots }"></dd></dl></dd></dl> <dt id="elliptic_integral"><dfn><a href="/wiki/Elliptic_integral" title="Elliptic integral">elliptic integral</a></dfn></dt> <dd>In <a href="/wiki/Integral_calculus" class="mw-redirect" title="Integral calculus">integral calculus</a>, elliptic integrals originally arose in connection with the problem of giving the <a href="/wiki/Arc_length" title="Arc length">arc length</a> of an <a href="/wiki/Ellipse" title="Ellipse">ellipse</a>. They were first studied by <a href="/wiki/Giulio_Carlo_de%27_Toschi_di_Fagnano" title="Giulio Carlo de' Toschi di Fagnano">Giulio Fagnano</a> and <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> (<abbr title="circa">c.</abbr> 1750). Modern mathematics defines an "elliptic integral" as any <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> f which can be expressed in the form <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=\int _{c}^{x}R\left(t,{\sqrt {P(t)}}\right)\,dt,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <mi>R</mi> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>P</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=\int _{c}^{x}R\left(t,{\sqrt {P(t)}}\right)\,dt,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/066d7c985940c1aa0b4458a9ce2abafe1c3c6fcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:28.535ex; height:5.843ex;" alt="{\displaystyle f(x)=\int _{c}^{x}R\left(t,{\sqrt {P(t)}}\right)\,dt,}"></dd></dl> where R is a <a href="/wiki/Rational_function" title="Rational function">rational function</a> of its two arguments, P is a <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> of degree 3 or 4 with no repeated roots, and c is a constant..</dd> <dt id="essential_discontinuity"><dfn><a href="/wiki/Classification_of_discontinuities#Essential_discontinuity" title="Classification of discontinuities">essential discontinuity</a></dfn></dt> <dd>For an essential discontinuity, only one of the two one-sided limits needs not exist or be infinite. Consider the function <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)={\begin{cases}\sin {\frac {5}{x-1}}&{\mbox{ for }}x<1\\0&{\mbox{ for }}x=1\\{\frac {1}{x-1}}&{\mbox{ for }}x>1\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> for </mtext> </mstyle> </mrow> <mi>x</mi> <mo><</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> for </mtext> </mstyle> </mrow> <mi>x</mi> <mo>=</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> for </mtext> </mstyle> </mrow> <mi>x</mi> <mo>></mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)={\begin{cases}\sin {\frac {5}{x-1}}&{\mbox{ for }}x<1\\0&{\mbox{ for }}x=1\\{\frac {1}{x-1}}&{\mbox{ for }}x>1\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b55cbb95006aa9cec84fe74b94c6882e0570ee5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:29.313ex; height:10.176ex;" alt="{\displaystyle f(x)={\begin{cases}\sin {\frac {5}{x-1}}&{\mbox{ for }}x<1\\0&{\mbox{ for }}x=1\\{\frac {1}{x-1}}&{\mbox{ for }}x>1\end{cases}}}"></dd></dl> Then, the point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle x_{0}\;=\;1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mspace width="thickmathspace" /> <mo>=</mo> <mspace width="thickmathspace" /> <mn>1</mn> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle x_{0}\;=\;1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0734fed27aaad5439e2e56e7d2a7d20ebb2a9c71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.163ex; height:2.009ex;" alt="{\displaystyle \scriptstyle x_{0}\;=\;1}"> is an essential discontinuity. In this case, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle L^{-}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle L^{-}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d01d94927d7928caa2d568c0e8381ed1171346" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.321ex; height:2.009ex;" alt="{\displaystyle \scriptstyle L^{-}}"> doesn't exist and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle L^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle L^{+}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fa48b6e6c408920e1b58a0fccabb7a6542e5754" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.321ex; height:2.009ex;" alt="{\displaystyle \scriptstyle L^{+}}"> is infinite – thus satisfying twice the conditions of essential discontinuity. So x0 is an essential discontinuity, infinite discontinuity, or discontinuity of the second kind. (This is distinct from the term <a href="/wiki/Essential_singularity" title="Essential singularity">essential singularity</a> which is often used when studying <a href="/wiki/Complex_analysis" title="Complex analysis">functions of complex variables</a>.</dd> <dt id="euler_method"><dfn><a href="/wiki/Euler_method" title="Euler method">Euler method</a></dfn></dt> <dd>Euler's method is a numerical method to solve first order first degree differential equation with a given initial value. It is the most basic <a href="/wiki/Explicit_and_implicit_methods" title="Explicit and implicit methods">explicit method</a> for <a href="/wiki/Numerical_ordinary_differential_equations" class="mw-redirect" title="Numerical ordinary differential equations">numerical integration of ordinary differential equations</a> and is the simplest <a href="/wiki/Runge%E2%80%93Kutta_method" class="mw-redirect" title="Runge–Kutta method">Runge–Kutta method</a>. The Euler method is named after <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a>, who treated it in his book <a href="/wiki/Institutionum_calculi_integralis" class="mw-redirect" title="Institutionum calculi integralis">Institutionum calculi integralis</a> (published 1768–1870).<a href="#cite_note-38">[36]</a></dd> <dt id="exponential_function"><dfn><a href="/wiki/Exponential_function" title="Exponential function">exponential function</a></dfn></dt> <dd>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, an exponential function is a function of the form <style data-mw-deduplicate="TemplateStyles:r996643573">.mw-parser-output .block-indent{padding-left:3em;padding-right:0;overflow:hidden}</style><div class="block-indent"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=ab^{x},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=ab^{x},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8f7e0207f9410fd16506b8cab68b31b78c4d219" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.563ex; height:2.843ex;" alt="{\displaystyle f(x)=ab^{x},}"></div> where b is a positive real number, and in which the argument x occurs as an exponent. For real numbers c and d, a function of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=ab^{cx+d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>x</mi> <mo>+</mo> <mi>d</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=ab^{cx+d}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ee482ef884c8151184d4bc75c2188e650c7eb96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.766ex; height:3.176ex;" alt="{\displaystyle f(x)=ab^{cx+d}}"> is also an exponential function, as it can be rewritten as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ab^{cx+d}=\left(ab^{d}\right)\left(b^{c}\right)^{x}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>x</mi> <mo>+</mo> <mi>d</mi> </mrow> </msup> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ab^{cx+d}=\left(ab^{d}\right)\left(b^{c}\right)^{x}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07197937aa9e72da60331fdccca784b145755967" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.755ex; height:3.343ex;" alt="{\displaystyle ab^{cx+d}=\left(ab^{d}\right)\left(b^{c}\right)^{x}.}"></dd></dl></dd> <dt id="extreme_value_theorem"><dfn><a href="/wiki/Extreme_value_theorem" title="Extreme value theorem">extreme value theorem</a></dfn></dt> <dd>States that if a real-valued <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> f is <a href="/wiki/Continuous_function" title="Continuous function">continuous</a> on the <a href="/wiki/Bounded_interval#Classification_of_intervals" class="mw-redirect" title="Bounded interval">closed</a> interval [a,b], then f must attain a <a href="/wiki/Maximum" class="mw-redirect" title="Maximum">maximum</a> and a <a href="/wiki/Minimum" class="mw-redirect" title="Minimum">minimum</a>, each at least once. That is, there exist numbers c and d in [a,b] such that: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(c)\geq f(x)\geq f(d)\quad {\text{for all }}x\in [a,b].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>for all </mtext> </mrow> <mi>x</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(c)\geq f(x)\geq f(d)\quad {\text{for all }}x\in [a,b].}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ccc399772dc99e3a3970432b84382c0ffda9560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.111ex; height:2.843ex;" alt="{\displaystyle f(c)\geq f(x)\geq f(d)\quad {\text{for all }}x\in [a,b].}"></dd></dl> A related theorem is the boundedness theorem which states that a continuous function f in the closed interval [a,b] is <a href="/wiki/Bounded_function" title="Bounded function">bounded</a> on that interval. That is, there exist real numbers m and M such that: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m<f(x)<M\quad {\text{for all }}x\in [a,b].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo><</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo><</mo> <mi>M</mi> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>for all </mtext> </mrow> <mi>x</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m<f(x)<M\quad {\text{for all }}x\in [a,b].}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bea84dcd500745a81c211a93cc3b650db8ea98d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.195ex; height:2.843ex;" alt="{\displaystyle m<f(x)<M\quad {\text{for all }}x\in [a,b].}"></dd></dl> The extreme value theorem enriches the boundedness theorem by saying that not only is the function bounded, but it also attains its least upper bound as its maximum and its greatest lower bound as its minimum.</dd> <dt id="extremum"><dfn><a href="/wiki/Maxima_and_minima" class="mw-redirect" title="Maxima and minima">extremum</a></dfn></dt> <dd>In <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">mathematical analysis</a>, the maxima and minima (the respective plurals of maximum and minimum) of a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a>, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given range (the local or relative extrema) or on the entire <a href="/wiki/Domain_of_a_function" title="Domain of a function">domain of a function</a> (the global or absolute extrema).<a href="#cite_note-39">[37]</a><a href="#cite_note-40">[38]</a><a href="#cite_note-41">[39]</a> <a href="/wiki/Pierre_de_Fermat" title="Pierre de Fermat">Pierre de Fermat</a> was one of the first mathematicians to propose a general technique, <a href="/wiki/Adequality" title="Adequality">adequality</a>, for finding the maxima and minima of functions. As defined in <a href="/wiki/Set_theory" title="Set theory">set theory</a>, the maximum and minimum of a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> are the <a href="/wiki/Greatest_and_least_elements" class="mw-redirect" title="Greatest and least elements">greatest and least elements</a> in the set, respectively. Unbounded infinite sets, such as the set of <a href="/wiki/Real_number" title="Real number">real numbers</a>, have no minimum or maximum.</dd> <div class="mw-heading mw-heading2"><h2 id="F">F</h2>[<a href="/w/index.php?title=Glossary_of_calculus&action=edit&section=6" title="Edit section: F">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1228772891"> <dl class="glossary"> <dt id="faà_di_bruno's_formula"><dfn><a href="/wiki/Fa%C3%A0_di_Bruno%27s_formula" title="Faà di Bruno's formula">Faà di Bruno's formula</a></dfn></dt> <dd>Is an identity in <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> generalizing the <a href="/wiki/Chain_rule" title="Chain rule">chain rule</a> to higher derivatives, named after <a href="/wiki/Francesco_Fa%C3%A0_di_Bruno" title="Francesco Faà di Bruno">Francesco Faà di Bruno</a> (<a href="#CITEREFFaà_di_Bruno1855">1855</a>, <a href="#CITEREFFaà_di_Bruno1857">1857</a>), though he was not the first to state or prove the formula. In 1800, more than 50 years before Faà di Bruno, the French mathematician <a href="/wiki/Louis_Fran%C3%A7ois_Antoine_Arbogast" title="Louis François Antoine Arbogast">Louis François Antoine Arbogast</a> stated the formula in a calculus textbook,<a href="#cite_note-42">[40]</a> considered the first published reference on the subject.<a href="#cite_note-43">[41]</a> Perhaps the most well-known form of Faà di Bruno's formula says that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {d^{n} \over dx^{n}}f(g(x))=\sum {\frac {n!}{m_{1}!\,1!^{m_{1}}\,m_{2}!\,2!^{m_{2}}\,\cdots \,m_{n}!\,n!^{m_{n}}}}\cdot f^{(m_{1}+\cdots +m_{n})}(g(x))\cdot \prod _{j=1}^{n}\left(g^{(j)}(x)\right)^{m_{j}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>!</mo> </mrow> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>!</mo> <mspace width="thinmathspace" /> <mn>1</mn> <msup> <mo>!</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> <mspace width="thinmathspace" /> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>!</mo> <mspace width="thinmathspace" /> <mn>2</mn> <msup> <mo>!</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msup> <mspace width="thinmathspace" /> <mo>⋯</mo> <mspace width="thinmathspace" /> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>!</mo> <mspace width="thinmathspace" /> <mi>n</mi> <msup> <mo>!</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>⋅</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>⋅</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>j</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {d^{n} \over dx^{n}}f(g(x))=\sum {\frac {n!}{m_{1}!\,1!^{m_{1}}\,m_{2}!\,2!^{m_{2}}\,\cdots \,m_{n}!\,n!^{m_{n}}}}\cdot f^{(m_{1}+\cdots +m_{n})}(g(x))\cdot \prod _{j=1}^{n}\left(g^{(j)}(x)\right)^{m_{j}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bef53bf2898c47370c394592ed95a17ced35a04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:87.175ex; height:7.176ex;" alt="{\displaystyle {d^{n} \over dx^{n}}f(g(x))=\sum {\frac {n!}{m_{1}!\,1!^{m_{1}}\,m_{2}!\,2!^{m_{2}}\,\cdots \,m_{n}!\,n!^{m_{n}}}}\cdot f^{(m_{1}+\cdots +m_{n})}(g(x))\cdot \prod _{j=1}^{n}\left(g^{(j)}(x)\right)^{m_{j}},}"></dd></dl> where the sum is over all n-<a href="/wiki/Tuple" title="Tuple">tuples</a> of nonnegative integers (m1, …, mn) satisfying the constraint <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\cdot m_{1}+2\cdot m_{2}+3\cdot m_{3}+\cdots +n\cdot m_{n}=n.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>⋅</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>2</mn> <mo>⋅</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mn>3</mn> <mo>⋅</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mi>n</mi> <mo>⋅</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>n</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\cdot m_{1}+2\cdot m_{2}+3\cdot m_{3}+\cdots +n\cdot m_{n}=n.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2386ea9ee11ffa7f538a1d45e3595cd66c38e885" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:43.366ex; height:2.509ex;" alt="{\displaystyle 1\cdot m_{1}+2\cdot m_{2}+3\cdot m_{3}+\cdots +n\cdot m_{n}=n.}"></dd></dl> Sometimes, to give it a memorable pattern, it is written in a way in which the coefficients that have the combinatorial interpretation discussed below are less explicit: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {d^{n} \over dx^{n}}f(g(x))=\sum {\frac {n!}{m_{1}!\,m_{2}!\,\cdots \,m_{n}!}}\cdot f^{(m_{1}+\cdots +m_{n})}(g(x))\cdot \prod _{j=1}^{n}\left({\frac {g^{(j)}(x)}{j!}}\right)^{m_{j}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>!</mo> </mrow> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>!</mo> <mspace width="thinmathspace" /> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>!</mo> <mspace width="thinmathspace" /> <mo>⋯</mo> <mspace width="thinmathspace" /> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>⋅</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>⋅</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>j</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>j</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {d^{n} \over dx^{n}}f(g(x))=\sum {\frac {n!}{m_{1}!\,m_{2}!\,\cdots \,m_{n}!}}\cdot f^{(m_{1}+\cdots +m_{n})}(g(x))\cdot \prod _{j=1}^{n}\left({\frac {g^{(j)}(x)}{j!}}\right)^{m_{j}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/115a3532fcb90d6c323e14a5a4fdd8f5eef065af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:74.443ex; height:7.676ex;" alt="{\displaystyle {d^{n} \over dx^{n}}f(g(x))=\sum {\frac {n!}{m_{1}!\,m_{2}!\,\cdots \,m_{n}!}}\cdot f^{(m_{1}+\cdots +m_{n})}(g(x))\cdot \prod _{j=1}^{n}\left({\frac {g^{(j)}(x)}{j!}}\right)^{m_{j}}.}"></dd></dl> Combining the terms with the same value of m1 + m2 + ... + mn = k and noticing that m j has to be zero for j > n − k + 1 leads to a somewhat simpler formula expressed in terms of <a href="/wiki/Bell_polynomial" class="mw-redirect" title="Bell polynomial">Bell polynomials</a> Bn,k(x1,...,xn−k+1): <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {d^{n} \over dx^{n}}f(g(x))=\sum _{k=1}^{n}f^{(k)}(g(x))\cdot B_{n,k}\left(g'(x),g''(x),\dots ,g^{(n-k+1)}(x)\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>⋅</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msup> <mi>g</mi> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {d^{n} \over dx^{n}}f(g(x))=\sum _{k=1}^{n}f^{(k)}(g(x))\cdot B_{n,k}\left(g'(x),g''(x),\dots ,g^{(n-k+1)}(x)\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5e9078bc352969d6e2aed7629d56dbbdfca0209" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:66.247ex; height:6.843ex;" alt="{\displaystyle {d^{n} \over dx^{n}}f(g(x))=\sum _{k=1}^{n}f^{(k)}(g(x))\cdot B_{n,k}\left(g'(x),g''(x),\dots ,g^{(n-k+1)}(x)\right).}"></dd></dl></dd></dl> <dt id="first-degree_polynomial"><dfn><a href="/wiki/Degree_of_a_polynomial" title="Degree of a polynomial">first-degree polynomial</a></dfn></dt> <dd></dd> <dt id="first_derivative_test"><dfn><a href="/wiki/Derivative_test#First-derivative_test" title="Derivative test">first derivative test</a></dfn></dt> <dd>The first derivative test examines a function's <a href="/wiki/Monotonic_function" title="Monotonic function">monotonic</a> properties (where the function is increasing or decreasing) focusing on a particular point in its domain. If the function "switches" from increasing to decreasing at the point, then the function will achieve a highest value at that point. Similarly, if the function "switches" from decreasing to increasing at the point, then it will achieve a least value at that point. If the function fails to "switch", and remains increasing or remains decreasing, then no highest or least value is achieved.</dd> <dt id="fractional_calculus"><dfn><a href="/wiki/Fractional_calculus" title="Fractional calculus">Fractional calculus</a></dfn></dt> <dd>Is a branch of <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">mathematical analysis</a> that studies the several different possibilities of defining <a href="/wiki/Real_number" title="Real number">real number</a> powers or <a href="/wiki/Complex_number" title="Complex number">complex number</a> powers of the <a href="/wiki/Derivative" title="Derivative">differentiation operator</a> D <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Df(x)={\dfrac {d}{dx}}f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mstyle> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Df(x)={\dfrac {d}{dx}}f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/512629e0b77e80cae8d8ae9bb7494d9be2e5b8cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.24ex; height:5.509ex;" alt="{\displaystyle Df(x)={\dfrac {d}{dx}}f(x)}">,</dd></dl> and of the integration operator J <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Jf(x)=\int _{0}^{x}\!\!\!\!f(s){ds}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mi>f</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>s</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Jf(x)=\int _{0}^{x}\!\!\!\!f(s){ds}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b0c95e0ce73c7f2a4b46f7361c7d4bfed960fad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.947ex; height:5.843ex;" alt="{\displaystyle Jf(x)=\int _{0}^{x}\!\!\!\!f(s){ds}}">,<a href="#cite_note-44">[Note 2]</a></dd></dl> and developing a <a href="/wiki/Calculus" title="Calculus">calculus</a> for such operators generalizing the classical one. In this context, the term powers refers to iterative application of a linear operator to a function, in some analogy to <a href="/wiki/Function_composition" title="Function composition">function composition</a> acting on a variable, i.e. f ∘2(x) = f ∘ f (x) = f ( f (x) ).</dd> <dt id="frustum"><dfn><a href="/wiki/Frustum" title="Frustum">frustum</a></dfn></dt> <dd>In <a href="/wiki/Geometry" title="Geometry">geometry</a>, a frustum (plural: frusta or frustums) is the portion of a <a href="/wiki/Polyhedron" title="Polyhedron">solid</a> (normally a <a href="/wiki/Cone_(geometry)" class="mw-redirect" title="Cone (geometry)">cone</a> or <a href="/wiki/Pyramid_(geometry)" title="Pyramid (geometry)">pyramid</a>) that lies between one or two <a href="/wiki/Parallel_planes" class="mw-redirect" title="Parallel planes">parallel planes</a> cutting it. A right frustum is a parallel <a href="/wiki/Truncation_(geometry)" title="Truncation (geometry)">truncation</a> of a <a href="/wiki/Right_pyramid" class="mw-redirect" title="Right pyramid">right pyramid</a> or right cone.<a href="#cite_note-45">[42]</a></dd> <dt id="function"><dfn><a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a></dfn></dt> <dd>Is a process or a relation that associates each element x of a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> X, the <a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a> of the function, to a single element y of another set Y (possibly the same set), the codomain of the function. If the function is called f, this relation is denoted y = f (x) (read f of x), the element x is the <a href="/wiki/Argument_of_a_function" title="Argument of a function">argument</a> or input of the function, and y is the value of the function, the output, or the image of x by f.<a href="#cite_note-MacLane-46">[43]</a> The symbol that is used for representing the input is the <a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variable</a> of the function (one often says that f is a function of the variable x).</dd> <dt id="function_composition"><dfn><a href="/wiki/Function_composition" title="Function composition">function composition</a></dfn></dt> <dd>Is an operation that takes two <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">functions</a> f and g and produces a function h such that h(x) = g(f(x)). In this operation, the function g is <a href="/wiki/Function_application" title="Function application">applied</a> to the result of applying the function f to x. That is, the functions f : X → Y and g : Y → Z are composed to yield a function that maps x in X to g(f(x)) in Z.</dd> <dt id="fundamental_theorem_of_calculus"><dfn><a href="/wiki/Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus">fundamental theorem of calculus</a></dfn></dt> <dd>The fundamental theorem of calculus is a <a href="/wiki/Theorem" title="Theorem">theorem</a> that links the concept of <a href="/wiki/Derivative" title="Derivative">differentiating</a> a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> with the concept of <a href="/wiki/Integral" title="Integral">integrating</a> a function. The first part of the theorem, sometimes called the first fundamental theorem of calculus, states that one of the <a href="/wiki/Antiderivative" title="Antiderivative">antiderivatives</a> (also called indefinite integral), say F, of some function f may be obtained as the integral of f with a variable bound of integration. This implies the existence of <a href="/wiki/Antiderivative" title="Antiderivative">antiderivatives</a> for <a href="/wiki/Continuous_function" title="Continuous function">continuous functions</a>.<a href="#cite_note-47">[44]</a> Conversely, the second part of the theorem, sometimes called the second fundamental theorem of calculus, states that the integral of a function f over some interval can be computed by using any one, say F, of its infinitely many <a href="/wiki/Antiderivative" title="Antiderivative">antiderivatives</a>. This part of the theorem has key practical applications, because explicitly finding the antiderivative of a function by <a href="/wiki/Symbolic_integration" title="Symbolic integration">symbolic integration</a> avoids <a href="/wiki/Numerical_integration" title="Numerical integration">numerical integration</a> to compute integrals. This provides generally a better numerical accuracy.</dd> <div class="mw-heading mw-heading2"><h2 id="G">G</h2>[<a href="/w/index.php?title=Glossary_of_calculus&action=edit&section=7" title="Edit section: G">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1228772891"> <dl class="glossary"> <dt id="general_leibniz_rule"><dfn><a href="/wiki/General_Leibniz_rule" title="General Leibniz rule">general Leibniz rule</a></dfn></dt> <dd>The general Leibniz rule,<a href="#cite_note-48">[45]</a> named after <a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a>, generalizes the <a href="/wiki/Product_rule" title="Product rule">product rule</a> (which is also known as "Leibniz's rule"). It states that if <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><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}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"> are <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">-times <a href="/wiki/Differentiable_function" title="Differentiable function">differentiable functions</a>, then the product <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle fg}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle fg}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06bac4638bb56f14688118ce88c188c7a021eb29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.395ex; height:2.509ex;" alt="{\displaystyle fg}"> is also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">-times differentiable and its <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">th derivative is given by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (fg)^{(n)}=\sum _{k=0}^{n}{n \choose k}f^{(n-k)}g^{(k)},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mi>g</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (fg)^{(n)}=\sum _{k=0}^{n}{n \choose k}f^{(n-k)}g^{(k)},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b5fbf529aa458b37f32e4cf1839132d83af06e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:28.445ex; height:7.009ex;" alt="{\displaystyle (fg)^{(n)}=\sum _{k=0}^{n}{n \choose k}f^{(n-k)}g^{(k)},}"></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {n \choose k}={n! \over k!(n-k)!}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>!</mo> </mrow> <mrow> <mi>k</mi> <mo>!</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {n \choose k}={n! \over k!(n-k)!}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ed1bdd61047e98b30df11a23956723badc802bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:18.511ex; height:6.343ex;" alt="{\displaystyle {n \choose k}={n! \over k!(n-k)!}}"> is the <a href="/wiki/Binomial_coefficient" title="Binomial coefficient">binomial coefficient</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{(0)}\equiv f.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo>≡</mo> <mi>f</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{(0)}\equiv f.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2330050abc564f048e15651f273829cbc00cfee9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.678ex; height:3.176ex;" alt="{\displaystyle f^{(0)}\equiv f.}"> This can be proved by using the product rule and <a href="/wiki/Mathematical_induction" title="Mathematical induction">mathematical induction</a>.</dd> <dt id="global_maximum"><dfn><a href="/wiki/Maxima_and_minima" class="mw-redirect" title="Maxima and minima">global maximum</a></dfn></dt> <dd>In <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">mathematical analysis</a>, the maxima and minima (the respective plurals of maximum and minimum) of a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a>, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given range (the local or relative extrema) or on the entire <a href="/wiki/Domain_of_a_function" title="Domain of a function">domain of a function</a> (the global or absolute extrema).<a href="#cite_note-49">[46]</a><a href="#cite_note-50">[47]</a><a href="#cite_note-51">[48]</a> <a href="/wiki/Pierre_de_Fermat" title="Pierre de Fermat">Pierre de Fermat</a> was one of the first mathematicians to propose a general technique, <a href="/wiki/Adequality" title="Adequality">adequality</a>, for finding the maxima and minima of functions. As defined in <a href="/wiki/Set_theory" title="Set theory">set theory</a>, the maximum and minimum of a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> are the <a href="/wiki/Greatest_and_least_elements" class="mw-redirect" title="Greatest and least elements">greatest and least elements</a> in the set, respectively. Unbounded infinite sets, such as the set of <a href="/wiki/Real_number" title="Real number">real numbers</a>, have no minimum or maximum.</dd> <dt id="global_minimum"><dfn><a href="/wiki/Maxima_and_minima" class="mw-redirect" title="Maxima and minima">global minimum</a></dfn></dt> <dd>In <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">mathematical analysis</a>, the maxima and minima (the respective plurals of maximum and minimum) of a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a>, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given range (the local or relative extrema) or on the entire <a href="/wiki/Domain_of_a_function" title="Domain of a function">domain of a function</a> (the global or absolute extrema).<a href="#cite_note-52">[49]</a><a href="#cite_note-53">[50]</a><a href="#cite_note-54">[51]</a> <a href="/wiki/Pierre_de_Fermat" title="Pierre de Fermat">Pierre de Fermat</a> was one of the first mathematicians to propose a general technique, <a href="/wiki/Adequality" title="Adequality">adequality</a>, for finding the maxima and minima of functions. As defined in <a href="/wiki/Set_theory" title="Set theory">set theory</a>, the maximum and minimum of a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> are the <a href="/wiki/Greatest_and_least_elements" class="mw-redirect" title="Greatest and least elements">greatest and least elements</a> in the set, respectively. Unbounded infinite sets, such as the set of <a href="/wiki/Real_number" title="Real number">real numbers</a>, have no minimum or maximum.</dd> <dt id="golden_spiral"><dfn><a href="/wiki/Golden_spiral" title="Golden spiral">golden spiral</a></dfn></dt> <dd>In <a href="/wiki/Geometry" title="Geometry">geometry</a>, a golden spiral is a <a href="/wiki/Logarithmic_spiral" title="Logarithmic spiral">logarithmic spiral</a> whose growth factor is <a href="/wiki/Phi" title="Phi">φ</a>, the <a href="/wiki/Golden_ratio" title="Golden ratio">golden ratio</a>.<a href="#cite_note-55">[52]</a> That is, a golden spiral gets wider (or further from its origin) by a factor of φ for every quarter turn it makes.</dd> <dt id="gradient"><dfn><a href="/wiki/Gradient" title="Gradient">gradient</a></dfn></dt> <dd>Is a multi-variable generalization of the <a href="/wiki/Derivative" title="Derivative">derivative</a>. While a derivative can be defined on functions of a single variable, for <a href="/wiki/Function_of_several_variables" class="mw-redirect" title="Function of several variables">functions of several variables</a>, the gradient takes its place. The gradient is a <a href="/wiki/Vector-valued_function" title="Vector-valued function">vector-valued function</a>, as opposed to a derivative, which is <a href="/wiki/Scalar-valued_function" class="mw-redirect" title="Scalar-valued function">scalar-valued</a>.</dd> </dl> <div class="mw-heading mw-heading2"><h2 id="H">H</h2>[<a href="/w/index.php?title=Glossary_of_calculus&action=edit&section=8" title="Edit section: H">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1228772891"> <dl class="glossary"> <dt id="harmonic_progression"><dfn><a href="/wiki/Harmonic_progression_(mathematics)" title="Harmonic progression (mathematics)">harmonic progression</a></dfn></dt> <dd>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a harmonic progression (or harmonic sequence) is a progression formed by taking the reciprocals of an <a href="/wiki/Arithmetic_progression" title="Arithmetic progression">arithmetic progression</a>. It is a <a href="/wiki/Sequence" title="Sequence">sequence</a> of the form <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{a}},\ {\frac {1}{a+d}}\ ,{\frac {1}{a+2d}}\ ,{\frac {1}{a+3d}}\ ,\cdots ,{\frac {1}{a+kd}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>a</mi> </mfrac> </mrow> <mo>,</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>a</mi> <mo>+</mo> <mi>d</mi> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>a</mi> <mo>+</mo> <mn>2</mn> <mi>d</mi> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>a</mi> <mo>+</mo> <mn>3</mn> <mi>d</mi> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>a</mi> <mo>+</mo> <mi>k</mi> <mi>d</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{a}},\ {\frac {1}{a+d}}\ ,{\frac {1}{a+2d}}\ ,{\frac {1}{a+3d}}\ ,\cdots ,{\frac {1}{a+kd}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb3d0a8448a2b942e3b1886a90ce71ea096c4a7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:41.34ex; height:5.509ex;" alt="{\displaystyle {\frac {1}{a}},\ {\frac {1}{a+d}}\ ,{\frac {1}{a+2d}}\ ,{\frac {1}{a+3d}}\ ,\cdots ,{\frac {1}{a+kd}},}"></dd></dl> where −a/d is not a <a href="/wiki/Natural_number" title="Natural number">natural number</a> and k is a natural number. Equivalently, a sequence is a harmonic progression when each term is the <a href="/wiki/Harmonic_mean" title="Harmonic mean">harmonic mean</a> of the neighboring terms. It is not possible for a harmonic progression (other than the trivial case where a = 1 and k = 0) to sum to an <a href="/wiki/Integer" title="Integer">integer</a>. The reason is that, necessarily, at least one denominator of the progression will be divisible by a <a href="/wiki/Prime_number" title="Prime number">prime number</a> that does not divide any other denominator.<a href="#cite_note-56">[53]</a></dd> <dt id="higher_derivative"><dfn><a href="/wiki/Derivative#Higher_derivatives" title="Derivative">higher derivative</a></dfn></dt> <dd>Let f be a differentiable function, and let f ′ be its derivative. The derivative of f ′ (if it has one) is written f ′′ and is called the <a href="/wiki/Second_derivative" title="Second derivative">second derivative</a> of f. Similarly, the derivative of the second derivative, if it exists, is written f ′′′ and is called the <a href="/wiki/Third_derivative" title="Third derivative">third derivative</a> of f. Continuing this process, one can define, if it exists, the nth derivative as the derivative of the (n-1)th derivative. These repeated derivatives are called higher-order derivatives. The nth derivative is also called the derivative of order n.</dd> <dt id="homogeneous_linear_differential_equation"><dfn><a href="/wiki/Homogeneous_differential_equation" title="Homogeneous differential equation">homogeneous linear differential equation</a></dfn></dt> <dd>A <a href="/wiki/Differential_equation" title="Differential equation">differential equation</a> can be homogeneous in either of two respects. A <a href="/wiki/First_order_differential_equation" class="mw-redirect" title="First order differential equation">first order differential equation</a> is said to be homogeneous if it may be written <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x,y)dy=g(x,y)dx,}"> <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> <mi>d</mi> <mi>y</mi> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x,y)dy=g(x,y)dx,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4b0be82f9ea4a314c2c3adad396acb548c7adb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.714ex; height:2.843ex;" alt="{\displaystyle f(x,y)dy=g(x,y)dx,}"></dd></dl> where f and g are <a href="/wiki/Homogeneous_function" title="Homogeneous function">homogeneous functions</a> of the same degree of x and y. In this case, the change of variable y = ux leads to an equation of the form <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dx}{x}}=h(u)du,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mi>x</mi> </mfrac> </mrow> <mo>=</mo> <mi>h</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>u</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dx}{x}}=h(u)du,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/655ac39b47bf14555c0efeb383326d24ac558657" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.151ex; height:5.343ex;" alt="{\displaystyle {\frac {dx}{x}}=h(u)du,}"></dd></dl> which is easy to solve by <a href="/wiki/Integral" title="Integral">integration</a> of the two members. Otherwise, a differential equation is homogeneous if it is a homogeneous function of the unknown function and its derivatives. In the case of <a href="/wiki/Linear_differential_equation" title="Linear differential equation">linear differential equations</a>, this means that there are no constant terms. The solutions of any linear <a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">ordinary differential equation</a> of any order may be deduced by integration from the solution of the homogeneous equation obtained by removing the constant term.</dd> <dt id="hyperbolic_function"><dfn><a href="/wiki/Hyperbolic_function" class="mw-redirect" title="Hyperbolic function">hyperbolic function</a></dfn></dt> <dd>Hyperbolic functions are analogs of the ordinary <a href="/wiki/Trigonometric_function" class="mw-redirect" title="Trigonometric function">trigonometric</a>, or <a href="/wiki/Unit_circle" title="Unit circle">circular</a>, functions.</dd> </dl> <div class="mw-heading mw-heading2"><h2 id="I">I</h2>[<a href="/w/index.php?title=Glossary_of_calculus&action=edit&section=9" title="Edit section: I">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1228772891"> <dl class="glossary"> <dt id="identity_function"><dfn><a href="/wiki/Identity_function" title="Identity function">identity function</a></dfn></dt> <dd>Also called an identity relation or identity map or identity transformation, is a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> that always returns the same value that was used as its argument. In <a href="/wiki/Equation" title="Equation">equations</a>, the function is given by f(x) = x.</dd> <dt id="imaginary_number"><dfn><a href="/wiki/Imaginary_number" title="Imaginary number">imaginary number</a></dfn></dt> <dd>Is a <a href="/wiki/Complex_number" title="Complex number">complex number</a> that can be written as a <a href="/wiki/Real_number" title="Real number">real number</a> multiplied by the <a href="/wiki/Imaginary_unit" title="Imaginary unit">imaginary unit</a> i,<a href="#cite_note-57">[note 2]</a> which is defined by its property i2 = −1.<a href="#cite_note-58">[54]</a> The <a href="/wiki/Square_(algebra)" title="Square (algebra)">square</a> of an imaginary number bi is −b2. For example, 5i is an imaginary number, and its square is −25. Zero is considered to be both real and imaginary.<a href="#cite_note-59">[55]</a></dd> <dt id="implicit_function"><dfn><a href="/wiki/Implicit_function" title="Implicit function">implicit function</a></dfn></dt> <dd>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, an implicit equation is a <a href="/wiki/Relation_(mathematics)" title="Relation (mathematics)">relation</a> of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R(x_{1},\ldots ,x_{n})=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R(x_{1},\ldots ,x_{n})=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2da5e7cf8f171a3430f3a4a66679be85e22dcbc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.945ex; height:2.843ex;" alt="{\displaystyle R(x_{1},\ldots ,x_{n})=0}">, where <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/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"> is a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> of several variables (often a <a href="/wiki/Polynomial" title="Polynomial">polynomial</a>). For example, the implicit equation of the <a href="/wiki/Unit_circle" title="Unit circle">unit circle</a> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+y^{2}-1=0}"> <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> <mn>1</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+y^{2}-1=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ee594b8851d760d0e2d44aba714907aca657b8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.703ex; height:3.009ex;" alt="{\displaystyle x^{2}+y^{2}-1=0}">. An implicit function is a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> that is defined implicitly by an implicit equation, by associating one of the variables (the <a href="/wiki/Value_(mathematics)" title="Value (mathematics)">value</a>) with the others (the <a href="/wiki/Argument_of_a_function" title="Argument of a function">arguments</a>).<a href="#cite_note-Chiang-60">[56]</a>: 204–206  Thus, an implicit function for <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><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}"> in the context of the <a href="/wiki/Unit_circle" title="Unit circle">unit circle</a> is defined implicitly by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+f(x)^{2}-1=0}"> <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> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+f(x)^{2}-1=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6c01bd908dc53c2263a66c8b118de726665a1b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.96ex; height:3.176ex;" alt="{\displaystyle x^{2}+f(x)^{2}-1=0}">. This implicit equation defines <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><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}"> as a function of <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}"> only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1\leq x\leq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> <mo>≤</mo> <mi>x</mi> <mo>≤</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1\leq x\leq 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c07d30937eb8028698054698012f2d76caad7fe4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.66ex; height:2.343ex;" alt="{\displaystyle -1\leq x\leq 1}"> and one considers only non-negative (or non-positive) values for the values of the function. The <a href="/wiki/Implicit_function_theorem" title="Implicit function theorem">implicit function theorem</a> provides conditions under which some kinds of relations define an implicit function, namely relations defined as the <a href="/wiki/Indicator_function" title="Indicator function">indicator function</a> of the <a href="/wiki/Zero_set" class="mw-redirect" title="Zero set">zero set</a> of some <a href="/wiki/Continuously_differentiable" class="mw-redirect" title="Continuously differentiable">continuously differentiable</a> <a href="/wiki/Multivariable_calculus" title="Multivariable calculus">multivariate</a> function.</dd> <dt id="improper_fraction"><dfn><a href="/wiki/Improper_fraction" class="mw-redirect" title="Improper fraction">improper fraction</a></dfn></dt> <dd>Common fractions can be classified as either proper or improper. When the numerator and the denominator are both positive, the fraction is called proper if the numerator is less than the denominator, and improper otherwise.<a href="#cite_note-61">[57]</a><a href="#cite_note-62">[58]</a> In general, a common fraction is said to be a proper fraction if the <a href="/wiki/Absolute_value" title="Absolute value">absolute value</a> of the fraction is strictly less than one—that is, if the fraction is greater than −1 and less than 1.<a href="#cite_note-63">[59]</a><a href="#cite_note-64">[60]</a> It is said to be an improper fraction, or sometimes top-heavy fraction,<a href="#cite_note-65">[61]</a> if the absolute value of the fraction is greater than or equal to 1. Examples of proper fractions are 2/3, –3/4, and 4/9; examples of improper fractions are 9/4, –4/3, and 3/3.</dd> <dt id="improper_integral"><dfn><a href="/wiki/Improper_integral" title="Improper integral">improper integral</a></dfn></dt> <dd>In <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">mathematical analysis</a>, an improper integral is the <a href="/wiki/Limit_(mathematics)" title="Limit (mathematics)">limit</a> of a <a href="/wiki/Definite_integral" class="mw-redirect" title="Definite integral">definite integral</a> as an endpoint of the interval(s) of integration approaches either a specified <a href="/wiki/Real_number" title="Real number">real number</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c26c105004f30c27aa7c2a9c601550a4183b1f21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.676ex;" alt="{\displaystyle \infty }">, <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><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 }">, or in some instances as both endpoints approach limits. Such an integral is often written symbolically just like a standard definite integral, in some cases with infinity as a limit of integration. Specifically, an improper integral is a limit of the form: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{b\to \infty }\int _{a}^{b}f(x)\,dx,\qquad \lim _{a\to -\infty }\int _{a}^{b}f(x)\,dx,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>,</mo> <mspace width="2em" /> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo stretchy="false">→</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{b\to \infty }\int _{a}^{b}f(x)\,dx,\qquad \lim _{a\to -\infty }\int _{a}^{b}f(x)\,dx,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c6ee8f1ea309bed84da5b0455d98662ec4bd2e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:38.804ex; height:6.343ex;" alt="{\displaystyle \lim _{b\to \infty }\int _{a}^{b}f(x)\,dx,\qquad \lim _{a\to -\infty }\int _{a}^{b}f(x)\,dx,}"></dd></dl> or <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{c\to b^{-}}\int _{a}^{c}f(x)\,dx,\quad \lim _{c\to a^{+}}\int _{c}^{b}f(x)\,dx,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mo stretchy="false">→</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> </mrow> </munder> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>,</mo> <mspace width="1em" /> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mo stretchy="false">→</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mrow> </munder> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{c\to b^{-}}\int _{a}^{c}f(x)\,dx,\quad \lim _{c\to a^{+}}\int _{c}^{b}f(x)\,dx,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15ee4275fcc67f68715da0be89d3f7bf976e9be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:35.752ex; height:6.343ex;" alt="{\displaystyle \lim _{c\to b^{-}}\int _{a}^{c}f(x)\,dx,\quad \lim _{c\to a^{+}}\int _{c}^{b}f(x)\,dx,}"></dd></dl> in which one takes a limit in one or the other (or sometimes both) endpoints (<a href="#CITEREFApostol1967">Apostol 1967</a>, §10.23).</dd> <dt id="inflection_point"><dfn><a href="/wiki/Inflection_point" title="Inflection point">inflection point</a></dfn></dt> <dd>In <a href="/wiki/Differential_calculus" title="Differential calculus">differential calculus</a>, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a <a href="/wiki/Continuous_function" title="Continuous function">continuous</a> <a href="/wiki/Plane_curve" title="Plane curve">plane curve</a> at which the curve changes from being <a href="/wiki/Concave_function" title="Concave function">concave</a> (concave downward) to <a href="/wiki/Convex_function" title="Convex function">convex</a> (concave upward), or vice versa.</dd> <dt id="instantaneous_rate_of_change"><dfn><a href="/wiki/Derivative" title="Derivative">instantaneous rate of change</a></dfn></dt> <dd>The derivative of a function of a single variable at a chosen input value, when it exists, is the <a href="/wiki/Slope" title="Slope">slope</a> of the <a href="/wiki/Tangent" title="Tangent">tangent line</a> to the <a href="/wiki/Graph_of_a_function" title="Graph of a function">graph of the function</a> at that point. The tangent line is the best <a href="/wiki/Linear_approximation" title="Linear approximation">linear approximation</a> of the function near that input value. For this reason, the derivative is often described as the "instantaneous rate of change", the ratio of the instantaneous change in the dependent variable to that of the independent variable. .</dd> <dt id="instantaneous_velocity"><dfn><a href="/wiki/Velocity#Instantaneous_velocity" title="Velocity">instantaneous velocity</a></dfn></dt> <dd>If we consider v as velocity and x as the displacement (change in position) vector, then we can express the (instantaneous) velocity of a particle or object, at any particular time t, as the <a href="/wiki/Derivative" title="Derivative">derivative</a> of the position with respect to time: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {v}}=\lim _{{\Delta t}\to 0}{\frac {\Delta {\boldsymbol {x}}}{\Delta t}}={\frac {d{\boldsymbol {x}}}{d{\mathit {t}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mrow> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> </mrow> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">t</mi> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {v}}=\lim _{{\Delta t}\to 0}{\frac {\Delta {\boldsymbol {x}}}{\Delta t}}={\frac {d{\boldsymbol {x}}}{d{\mathit {t}}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98326656eb93e2b598177b2a80ff949240888f99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:20.864ex; height:5.676ex;" alt="{\displaystyle {\boldsymbol {v}}=\lim _{{\Delta t}\to 0}{\frac {\Delta {\boldsymbol {x}}}{\Delta t}}={\frac {d{\boldsymbol {x}}}{d{\mathit {t}}}}.}"></dd></dl> From this derivative equation, in the one-dimensional case it can be seen that the area under a velocity vs. time (v vs. t graph) is the displacement, x. In calculus terms, the <a href="/wiki/Integral" title="Integral">integral</a> of the velocity function v(t) is the displacement function x(t). In the figure, this corresponds to the yellow area under the curve labeled s (s being an alternative notation for displacement). <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {x}}=\int {\boldsymbol {v}}\ d{\mathit {t}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <mo>=</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> <mtext> </mtext> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">t</mi> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {x}}=\int {\boldsymbol {v}}\ d{\mathit {t}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfc65107366d82be9fb36c9d2c13a1bb91405838" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.745ex; height:5.676ex;" alt="{\displaystyle {\boldsymbol {x}}=\int {\boldsymbol {v}}\ d{\mathit {t}}.}"></dd></dl> Since the derivative of the position with respect to time gives the change in position (in <a href="/wiki/Metre" title="Metre">metres</a>) divided by the change in time (in <a href="/wiki/Second" title="Second">seconds</a>), velocity is measured in <a href="/wiki/Metre_per_second" title="Metre per second">metres per second</a> (m/s). Although the concept of an instantaneous velocity might at first seem counter-intuitive, it may be thought of as the velocity that the object would continue to travel at if it stopped accelerating at that moment. .</dd> <dt id="integral"><dfn><a href="/wiki/Integral" title="Integral">integral</a></dfn></dt> <dd>An integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining <a href="/wiki/Infinitesimal" title="Infinitesimal">infinitesimal</a> data. Integration is one of the two main operations of calculus, with its inverse operation, <a href="/wiki/Derivative" title="Derivative">differentiation</a>, being the other. .</dd> <dt id="integral_symbol"><dfn><a href="/wiki/Integral_symbol" title="Integral symbol">integral symbol</a></dfn></dt> <dd>The integral symbol: <dl><dd>∫ (<a href="/wiki/Unicode" title="Unicode">Unicode</a>), <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \displaystyle \int }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \displaystyle \int }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e594c389846abbae8442b2c57a4dd70533071047" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:2.194ex; height:5.676ex;" alt="{\displaystyle \displaystyle \int }"> (<a href="/wiki/LaTeX" title="LaTeX">LaTeX</a>)</dd></dl> is used to denote <a href="/wiki/Integral" title="Integral">integrals</a> and <a href="/wiki/Antiderivatives" class="mw-redirect" title="Antiderivatives">antiderivatives</a> in <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>. .</dd> <dt id="integrand"><dfn><a href="/wiki/Integrand" class="mw-redirect" title="Integrand">integrand</a></dfn></dt> <dd>The function to be integrated in an integral.</dd> <dt id="integration_by_parts"><dfn><a href="/wiki/Integration_by_parts" title="Integration by parts">integration by parts</a></dfn></dt> <dd>In calculus, and more generally in <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">mathematical analysis</a>, integration by parts or partial integration is a process that finds the <a href="/wiki/Integral_(mathematics)" class="mw-redirect" title="Integral (mathematics)">integral</a> of a <a href="/wiki/Product_(mathematics)" title="Product (mathematics)">product</a> of functions in terms of the integral of their derivative and antiderivative. It is frequently used to transform the antiderivative of a product of functions into an antiderivative for which a solution can be more easily found. The rule can be readily derived by integrating the <a href="/wiki/Product_rule" title="Product rule">product rule</a> of <a href="/wiki/Derivative" title="Derivative">differentiation</a>. If u = u(x) and du = u′(x) dx, while v = v(x) and dv = v′(x) dx, then integration by parts states that: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\int _{a}^{b}u(x)v'(x)\,dx&={\Big [}u(x)v(x){\Big ]}_{a}^{b}-\int _{a}^{b}u'(x)v(x)\,dx\\&=u(b)v(b)-u(a)v(a)-\int _{a}^{b}u'(x)v(x)\,dx\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>u</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>v</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">[</mo> </mrow> </mrow> <mi>u</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>v</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mo>−</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <msup> <mi>u</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>v</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>u</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mi>v</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>u</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mi>v</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <msup> <mi>u</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>v</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\int _{a}^{b}u(x)v'(x)\,dx&={\Big [}u(x)v(x){\Big ]}_{a}^{b}-\int _{a}^{b}u'(x)v(x)\,dx\\&=u(b)v(b)-u(a)v(a)-\int _{a}^{b}u'(x)v(x)\,dx\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5198e7dca933740bd3c1b7dbbfc484d39c18427d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:58.42ex; height:12.843ex;" alt="{\displaystyle {\begin{aligned}\int _{a}^{b}u(x)v'(x)\,dx&={\Big [}u(x)v(x){\Big ]}_{a}^{b}-\int _{a}^{b}u'(x)v(x)\,dx\\&=u(b)v(b)-u(a)v(a)-\int _{a}^{b}u'(x)v(x)\,dx\end{aligned}}}"></dd></dl> or more compactly: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int u\,dv=uv-\int v\,du.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mi>u</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>v</mi> <mo>=</mo> <mi>u</mi> <mi>v</mi> <mo>−</mo> <mo>∫</mo> <mi>v</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>u</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int u\,dv=uv-\int v\,du.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7543879c4840a6b9d2a9ae0846cd7b30360d59b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.325ex; height:5.676ex;" alt="{\displaystyle \int u\,dv=uv-\int v\,du.}"></dd></dl> Mathematician <a href="/wiki/Brook_Taylor" title="Brook Taylor">Brook Taylor</a> discovered integration by parts, first publishing the idea in <a href="/wiki/1715_in_science" title="1715 in science">1715</a>.<a href="#cite_note-Brook_Taylor_biography,_St._Andrews-66">[62]</a><a href="#cite_note-Brook_Taylor_biography,_Stetson-67">[63]</a> More general formulations of integration by parts exist for the <a href="/wiki/Riemann%E2%80%93Stieltjes_integral#Properties_and_relation_to_the_Riemann_integral" title="Riemann–Stieltjes integral">Riemann–Stieltjes</a> and <a href="/wiki/Lebesgue%E2%80%93Stieltjes_integral#Integration_by_parts" class="mw-redirect" title="Lebesgue–Stieltjes integral">Lebesgue–Stieltjes integrals</a>. The discrete analogue for sequences is called <a href="/wiki/Summation_by_parts" title="Summation by parts">summation by parts</a>. .</dd> <dt id="integration_by_substitution"><dfn><a href="/wiki/Integration_by_substitution" title="Integration by substitution">integration by substitution</a></dfn></dt> <dd>Also known as u-substitution, is a method for solving <a href="/wiki/Integral" title="Integral">integrals</a>. Using the <a href="/wiki/Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus">fundamental theorem of calculus</a> often requires finding an <a href="/wiki/Antiderivative" title="Antiderivative">antiderivative</a>. For this and other reasons, integration by substitution is an important tool in mathematics. It is the counterpart to the <a href="/wiki/Chain_rule" title="Chain rule">chain rule</a> for <a href="/wiki/Derivative" title="Derivative">differentiation</a>. .</dd> <dt id="intermediate_value_theorem"><dfn><a href="/wiki/Intermediate_value_theorem" title="Intermediate value theorem">intermediate value theorem</a></dfn></dt> <dd>In <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">mathematical analysis</a>, the intermediate value theorem states that if a <a href="/wiki/Continuous_function" title="Continuous function">continuous function</a>, f, with an <a href="/wiki/Interval_(mathematics)" title="Interval (mathematics)">interval</a>, [a, b], as its <a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a>, takes values f(a) and f(b) at each end of the interval, then it also takes any value between f(a) and f(b) at some point within the interval. This has two important <a href="/wiki/Corollary" title="Corollary">corollaries</a>: <ol><li>If a continuous function has values of opposite sign inside an interval, then it has a root in that interval (Bolzano's theorem).<a href="#cite_note-68">[64]</a></li> <li>The <a href="/wiki/Image_(mathematics)" title="Image (mathematics)">image</a> of a continuous function over an interval is itself an interval. .</li></ol></dd> <dt id="inverse_trigonometric_functions"><dfn><a href="/wiki/Inverse_trigonometric_functions" title="Inverse trigonometric functions">inverse trigonometric functions</a></dfn></dt> <dd>(Also called arcus functions,<a href="#cite_note-69">[65]</a><a href="#cite_note-70">[66]</a><a href="#cite_note-71">[67]</a><a href="#cite_note-72">[68]</a><a href="#cite_note-73">[69]</a> antitrigonometric functions<a href="#cite_note-74">[70]</a> or cyclometric functions<a href="#cite_note-75">[71]</a><a href="#cite_note-76">[72]</a><a href="#cite_note-77">[73]</a>) are the <a href="/wiki/Inverse_function" title="Inverse function">inverse functions</a> of the <a href="/wiki/Trigonometric_functions" title="Trigonometric functions">trigonometric functions</a> (with suitably restricted <a href="/wiki/Domain_of_a_function" title="Domain of a function">domains</a>). Specifically, they are the inverses of the <a href="/wiki/Sine" class="mw-redirect" title="Sine">sine</a>, <a href="/wiki/Cosine" class="mw-redirect" title="Cosine">cosine</a>, <a href="/wiki/Tangent_(trigonometry)" class="mw-redirect" title="Tangent (trigonometry)">tangent</a>, <a href="/wiki/Cotangent" class="mw-redirect" title="Cotangent">cotangent</a>, <a href="/wiki/Secant_(trigonometry)" class="mw-redirect" title="Secant (trigonometry)">secant</a>, and <a href="/wiki/Cosecant" class="mw-redirect" title="Cosecant">cosecant</a> functions, and are used to obtain an angle from any of the angle's trigonometric ratios.</dd> </dl> <div class="mw-heading mw-heading2"><h2 id="J">J</h2>[<a href="/w/index.php?title=Glossary_of_calculus&action=edit&section=10" title="Edit section: J">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1228772891"> <dl class="glossary"> <dt id="jump_discontinuity"><dfn><a href="/wiki/Classification_of_discontinuities#Jump_discontinuity" title="Classification of discontinuities">jump discontinuity</a></dfn></dt> <dd>Consider the function <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)={\begin{cases}x^{2}&{\mbox{ for }}x<1\\0&{\mbox{ for }}x=1\\2-(x-1)^{2}&{\mbox{ for }}x>1\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> for </mtext> </mstyle> </mrow> <mi>x</mi> <mo><</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> for </mtext> </mstyle> </mrow> <mi>x</mi> <mo>=</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <mo>−</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> for </mtext> </mstyle> </mrow> <mi>x</mi> <mo>></mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)={\begin{cases}x^{2}&{\mbox{ for }}x<1\\0&{\mbox{ for }}x=1\\2-(x-1)^{2}&{\mbox{ for }}x>1\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32713a375739d98e9d9258ed8eeb193419e0ec84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:34.393ex; height:8.843ex;" alt="{\displaystyle f(x)={\begin{cases}x^{2}&{\mbox{ for }}x<1\\0&{\mbox{ for }}x=1\\2-(x-1)^{2}&{\mbox{ for }}x>1\end{cases}}}"></dd></dl> Then, the point x0 = 1 is a jump discontinuity. In this case, a single limit does not exist because the one-sided limits, L− and L+, exist and are finite, but are not equal: since, L− ≠ L+, the limit L does not exist. Then, x0 is called a jump discontinuity, step discontinuity, or discontinuity of the first kind. For this type of discontinuity, the function f may have any value at x0.</dd> </dl> <div class="mw-heading mw-heading2"><h2 id="L">L</h2>[<a href="/w/index.php?title=Glossary_of_calculus&action=edit&section=11" title="Edit section: L">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1228772891"> <dl class="glossary"> <dt id="lebesgue_integration"><dfn><a href="/wiki/Lebesgue_integration" class="mw-redirect" title="Lebesgue integration">Lebesgue integration</a></dfn></dt> <dd>In mathematics, the <a href="/wiki/Integral" title="Integral">integral</a> of a non-negative <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> of a single variable can be regarded, in the simplest case, as the <a href="/wiki/Area" title="Area">area</a> between the <a href="/wiki/Graph_of_a_function" title="Graph of a function">graph</a> of that function and the x-axis. The Lebesgue integral extends the integral to a larger class of functions. It also extends the <a href="/wiki/Domain_of_a_function" title="Domain of a function">domains</a> on which these functions can be defined.</dd> <dt id="l'hôpital's_rule"><dfn><a href="/wiki/L%27H%C3%B4pital%27s_rule" title="L'Hôpital's rule">L'Hôpital's rule</a></dfn></dt> <dd>L'Hôpital's rule or L'Hospital's rule uses <a href="/wiki/Derivative" title="Derivative">derivatives</a> to help evaluate <a href="/wiki/Limit_of_a_function" title="Limit of a function">limits</a> involving <a href="/wiki/Indeterminate_form" title="Indeterminate form">indeterminate forms</a>. Application (or repeated application) of the rule often converts an indeterminate form to an expression that can be evaluated by substitution, allowing easier evaluation of the limit. The rule is named after the 17th-century <a href="/wiki/France" title="France">French</a> <a href="/wiki/Mathematician" title="Mathematician">mathematician</a> <a href="/wiki/Guillaume_de_l%27H%C3%B4pital" title="Guillaume de l'Hôpital">Guillaume de l'Hôpital</a>. Although the contribution of the rule is often attributed to L'Hôpital, the theorem was first introduced to L'Hôpital in 1694 by the Swiss mathematician <a href="/wiki/Johann_Bernoulli" title="Johann Bernoulli">Johann Bernoulli</a>. L'Hôpital's rule states that for functions <var style="padding-right: 1px;">f</var> and <var style="padding-right: 1px;">g</var> which are <a href="/wiki/Differentiable_function" title="Differentiable function">differentiable</a> on an open <a href="/wiki/Interval_(mathematics)" title="Interval (mathematics)">interval</a> <var style="padding-right: 1px;">I</var> except possibly at a point <var style="padding-right: 1px;">c</var> contained in <var style="padding-right: 1px;">I</var>, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{x\to c}f(x)=\lim _{x\to c}g(x)=0{\text{ or }}\pm \infty ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mi>c</mi> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mi>c</mi> </mrow> </munder> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> or </mtext> </mrow> <mo>±</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\to c}f(x)=\lim _{x\to c}g(x)=0{\text{ or }}\pm \infty ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d45752c4c4c55ffb31e1cc7b15e0729f73097d11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:32.443ex; height:3.843ex;" alt="{\displaystyle \lim _{x\to c}f(x)=\lim _{x\to c}g(x)=0{\text{ or }}\pm \infty ,}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g'(x)\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g'(x)\neq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f022fde7dae42ed7a4bf38125e329293ac4d5ce5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.203ex; height:3.009ex;" alt="{\displaystyle g'(x)\neq 0}"> for all <var style="padding-right: 1px;">x</var> in <var style="padding-right: 1px;">I</var> with <var style="padding-right: 1px;">x</var> ≠ <var style="padding-right: 1px;">c</var>, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{x\to c}{\frac {f'(x)}{g'(x)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mi>c</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\to c}{\frac {f'(x)}{g'(x)}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9186ac40d31fd1b92d69283a5a91702a30dbc034" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:9.663ex; height:6.509ex;" alt="{\displaystyle \lim _{x\to c}{\frac {f'(x)}{g'(x)}}}"> exists, then <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{x\to c}{\frac {f(x)}{g(x)}}=\lim _{x\to c}{\frac {f'(x)}{g'(x)}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mi>c</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mi>c</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\to c}{\frac {f(x)}{g(x)}}=\lim _{x\to c}{\frac {f'(x)}{g'(x)}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d7785840386a94cc10bf59bffa7fe39cc946c43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:22.344ex; height:6.509ex;" alt="{\displaystyle \lim _{x\to c}{\frac {f(x)}{g(x)}}=\lim _{x\to c}{\frac {f'(x)}{g'(x)}}.}"></dd></dl> The differentiation of the numerator and denominator often simplifies the quotient or converts it to a limit that can be evaluated directly.</dd> <dt id="limit_comparison_test"><dfn><a href="/wiki/Limit_comparison_test" title="Limit comparison test">limit comparison test</a></dfn></dt> <dd>The limit comparison test allows one to determine the convergence of one series based on the convergence of another.</dd> <dt id="limit_of_a_function"><dfn><a href="/wiki/Limit_of_a_function" title="Limit of a function">limit of a function</a></dfn></dt> <dd>.</dd> <dt id="limits_of_integration"><dfn><a href="/wiki/Limits_of_integration" title="Limits of integration">limits of integration</a></dfn></dt> <dd>.</dd> <dt id="linear_combination"><dfn><a href="/wiki/Linear_combination" title="Linear combination">linear combination</a></dfn></dt> <dd>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a linear combination is an <a href="/wiki/Expression_(mathematics)" title="Expression (mathematics)">expression</a> constructed from a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of x and y would be any expression of the form ax + by, where a and b are constants).<a href="#cite_note-78">[74]</a><a href="#cite_note-79">[75]</a><a href="#cite_note-80">[76]</a> The concept of linear combinations is central to <a href="/wiki/Linear_algebra" title="Linear algebra">linear algebra</a> and related fields of mathematics.</dd> <dt id="linear_equation"><dfn><a href="/wiki/Linear_equation" title="Linear equation">linear equation</a></dfn></dt> <dd>A linear equation is an equation relating two or more variables to each other in the form of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}x_{1}+\cdots +a_{n}x_{n}+b=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <mi>b</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}x_{1}+\cdots +a_{n}x_{n}+b=0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d95216163a96c7ec2c25539daf889dbe01c816ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:26.814ex; height:2.509ex;" alt="{\displaystyle a_{1}x_{1}+\cdots +a_{n}x_{n}+b=0,}"> with the highest power of each variable being 1.</dd> <dt id="linear_system"><dfn><a href="/wiki/Linear_system" title="Linear system">linear system</a></dfn></dt> <dd>.</dd> <dt id="list_of_integrals"><dfn><a href="/wiki/List_of_integrals" class="mw-redirect" title="List of integrals">list of integrals</a></dfn></dt> <dd>.</dd> <dt id="logarithm"><dfn><a href="/wiki/Logarithm" title="Logarithm">logarithm</a></dfn></dt> <dd>.</dd> <dt id="logarithmic_differentiation"><dfn><a href="/wiki/Logarithmic_differentiation" title="Logarithmic differentiation">logarithmic differentiation</a></dfn></dt> <dd>.</dd> <dt id="lower_bound"><dfn><a href="/wiki/Upper_and_lower_bounds#Bounds_of_functions" title="Upper and lower bounds">lower bound</a></dfn></dt> <dd>.</dd> </dl> <div class="mw-heading mw-heading2"><h2 id="M">M</h2>[<a href="/w/index.php?title=Glossary_of_calculus&action=edit&section=12" title="Edit section: M">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1228772891"> <dl class="glossary"> <dt id="mean_value_theorem"><dfn><a href="/wiki/Mean_value_theorem" title="Mean value theorem">mean value theorem</a></dfn></dt> <dd>.</dd> <dt id="monotonic_function"><dfn><a href="/wiki/Monotonic_function#Monotonicity_in_calculus_and_analysis" title="Monotonic function">monotonic function</a></dfn></dt> <dd>.</dd> <dt id="multiple_integral"><dfn><a href="/wiki/Multiple_integral" title="Multiple integral">multiple integral</a></dfn></dt> <dd>.</dd> <dt id="multiplicative_calculus"><dfn>Multiplicative calculus</dfn></dt> <dd>.</dd> <dt id="multivariable_calculus"><dfn><a href="/wiki/Multivariable_calculus" title="Multivariable calculus">multivariable calculus</a></dfn></dt> <dd>.</dd> </dl> <div class="mw-heading mw-heading2"><h2 id="N">N</h2>[<a href="/w/index.php?title=Glossary_of_calculus&action=edit&section=13" title="Edit section: N">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1228772891"> <dl class="glossary"> <dt id="natural_logarithm"><dfn><a href="/wiki/Natural_logarithm" title="Natural logarithm">natural logarithm</a></dfn></dt> <dd>The natural logarithm of a number is its <a href="/wiki/Logarithm" title="Logarithm">logarithm</a> to the <a href="/wiki/Base_(exponentiation)" title="Base (exponentiation)">base</a> of the <a href="/wiki/Mathematical_constant" title="Mathematical constant">mathematical constant</a> <a href="/wiki/E_(mathematical_constant)" title="E (mathematical constant)">e</a>, where e is an <a href="/wiki/Irrational_number" title="Irrational number">irrational</a> and <a href="/wiki/Transcendental_number" title="Transcendental number">transcendental</a> number approximately equal to 2.718281828459. The natural logarithm of x is generally written as ln x, loge x, or sometimes, if the base e is implicit, simply log x.<a href="#cite_note-81">[77]</a> <a href="/wiki/Parentheses" class="mw-redirect" title="Parentheses">Parentheses</a> are sometimes added for clarity, giving ln(x), loge(x) or log(x). This is done in particular when the argument to the logarithm is not a single symbol, to prevent ambiguity.</dd> <dt id="non-newtonian_calculus"><dfn><a href="/w/index.php?title=Non-Newtonian_calculus&action=edit&redlink=1" class="new" title="Non-Newtonian calculus (page does not exist)">non-Newtonian calculus</a></dfn></dt> <dd>.</dd> <dt id="nonstandard_calculus"><dfn><a href="/wiki/Nonstandard_calculus" title="Nonstandard calculus">nonstandard calculus</a></dfn></dt> <dd>.</dd> <dt id="notation_for_differentiation"><dfn><a href="/wiki/Notation_for_differentiation" title="Notation for differentiation">notation for differentiation</a></dfn></dt> <dd>.</dd> <dt id="numerical_integration"><dfn><a href="/wiki/Numerical_integration" title="Numerical integration">numerical integration</a></dfn></dt> <dd>.</dd> </dl> <div class="mw-heading mw-heading2"><h2 id="O">O</h2>[<a href="/w/index.php?title=Glossary_of_calculus&action=edit&section=14" title="Edit section: O">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1228772891"> <dl class="glossary"> <dt id="one-sided_limit"><dfn><a href="/wiki/One-sided_limit" title="One-sided limit">one-sided limit</a></dfn></dt> <dd>.</dd> <dt id="ordinary_differential_equation"><dfn><a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">ordinary differential equation</a></dfn></dt> <dd>.</dd> </dl> <div class="mw-heading mw-heading2"><h2 id="P">P</h2>[<a href="/w/index.php?title=Glossary_of_calculus&action=edit&section=15" title="Edit section: P">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1228772891"> <dl class="glossary"> <dt id="pappus's_centroid_theorem"><dfn><a href="/wiki/Pappus%27s_centroid_theorem" title="Pappus's centroid theorem">Pappus's centroid theorem</a></dfn></dt> <dd>(Also known as the Guldinus theorem, Pappus–Guldinus theorem or Pappus's theorem) is either of two related <a href="/wiki/Theorem" title="Theorem">theorems</a> dealing with the <a href="/wiki/Surface_area" title="Surface area">surface areas</a> and <a href="/wiki/Volume" title="Volume">volumes</a> of <a href="/wiki/Surface_of_revolution" title="Surface of revolution">surfaces</a> and <a href="/wiki/Solid_of_revolution" title="Solid of revolution">solids</a> of revolution.</dd> <dt id="parabola"><dfn><a href="/wiki/Parabola" title="Parabola">parabola</a></dfn></dt> <dd>Is a <a href="/wiki/Plane_curve" title="Plane curve">plane curve</a> that is <a href="/wiki/Reflection_symmetry" title="Reflection symmetry">mirror-symmetrical</a> and is approximately U-<a href="/wiki/Shape" title="Shape">shaped</a>. It fits several superficially different other <a href="/wiki/Mathematics" title="Mathematics">mathematical</a> descriptions, which can all be proved to define exactly the same curves.</dd> <dt id="paraboloid"><dfn><a href="/wiki/Paraboloid" title="Paraboloid">paraboloid</a></dfn></dt> <dd>.</dd> <dt id="partial_derivative"><dfn><a href="/wiki/Partial_derivative" title="Partial derivative">partial derivative</a></dfn></dt> <dd>.</dd> <dt id="partial_differential_equation"><dfn><a href="/wiki/Partial_differential_equation" title="Partial differential equation">partial differential equation</a></dfn></dt> <dd>.</dd> <dt id="partial_fraction_decomposition"><dfn><a href="/wiki/Partial_fraction_decomposition" title="Partial fraction decomposition">partial fraction decomposition</a></dfn></dt> <dd>.</dd> <dt id="particular_solution"><dfn><a href="/wiki/Particular_solution" class="mw-redirect" title="Particular solution">particular solution</a></dfn></dt> <dd>.</dd> <dt id="piecewise-defined_function"><dfn><a href="/wiki/Piecewise" class="mw-redirect" title="Piecewise">piecewise-defined function</a></dfn></dt> <dd>A function defined by multiple sub-functions that apply to certain intervals of the function's domain.</dd> <dt id="position_vector"><dfn><a href="/wiki/Position_(vector)" class="mw-redirect" title="Position (vector)">position vector</a></dfn></dt> <dd>.</dd> <dt id="power_rule"><dfn><a href="/wiki/Power_rule" title="Power rule">power rule</a></dfn></dt> <dd>.</dd> <dt id="product_integral"><dfn><a href="/wiki/Product_integral" title="Product integral">product integral</a></dfn></dt> <dd>.</dd> <dt id="product_rule"><dfn><a href="/wiki/Product_rule" title="Product rule">product rule</a></dfn></dt> <dd>.</dd> <dt id="proper_fraction"><dfn><a href="/wiki/Proper_fraction" class="mw-redirect" title="Proper fraction">proper fraction</a></dfn></dt> <dd>.</dd> <dt id="proper_rational_function"><dfn><a href="/wiki/Proper_rational_function" class="mw-redirect" title="Proper rational function">proper rational function</a></dfn></dt> <dd>.</dd> <dt id="pythagorean_theorem"><dfn><a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a></dfn></dt> <dd>.</dd> <dt id="pythagorean_trigonometric_identity"><dfn><a href="/wiki/Pythagorean_trigonometric_identity" title="Pythagorean trigonometric identity">Pythagorean trigonometric identity</a></dfn></dt> <dd>.</dd> </dl> <div class="mw-heading mw-heading2"><h2 id="Q">Q</h2>[<a href="/w/index.php?title=Glossary_of_calculus&action=edit&section=16" title="Edit section: Q">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1228772891"> <dl class="glossary"> <dt id="quadratic_function"><dfn><a href="/wiki/Quadratic_function" title="Quadratic function">quadratic function</a></dfn></dt> <dd>In <a href="/wiki/Algebra" title="Algebra">algebra</a>, a quadratic function, a quadratic polynomial, a polynomial of degree 2, or simply a quadratic, is a <a href="/wiki/Polynomial_function" class="mw-redirect" title="Polynomial function">polynomial function</a> with one or more variables in which the highest-degree term is of the second degree. For example, a quadratic function in three variables x, y, and z contains exclusively terms x2, y2, z2, xy, xz, yz, x, y, z, and a constant: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x,y,z)=ax^{2}+by^{2}+cz^{2}+dxy+exz+fyz+gx+hy+iz+j,}"> <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> <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>j</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x,y,z)=ax^{2}+by^{2}+cz^{2}+dxy+exz+fyz+gx+hy+iz+j,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27b2570145a686a65d4317f8e7e58c95213a7e39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:66.529ex; height:3.176ex;" alt="{\displaystyle f(x,y,z)=ax^{2}+by^{2}+cz^{2}+dxy+exz+fyz+gx+hy+iz+j,}"></dd></dl> with at least one of the <a href="/wiki/Coefficient" title="Coefficient">coefficients</a> a, b, c, d, e, or f of the second-degree terms being non-zero. A univariate (single-variable) quadratic function has the form<a href="#cite_note-wolfram-82">[78]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=ax^{2}+bx+c,\quad a\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>b</mi> <mi>x</mi> <mo>+</mo> <mi>c</mi> <mo>,</mo> <mspace width="1em" /> <mi>a</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=ax^{2}+bx+c,\quad a\neq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60f6deb5a9c814c726d2185ccaca925726f564d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.992ex; height:3.176ex;" alt="{\displaystyle f(x)=ax^{2}+bx+c,\quad a\neq 0}"></dd></dl> in the single variable x. The <a href="/wiki/Graph_of_a_function" title="Graph of a function">graph</a> of a univariate quadratic function is a <a href="/wiki/Parabola" title="Parabola">parabola</a> whose axis of symmetry is parallel to the y-axis, as shown at right. If the quadratic function is set equal to zero, then the result is a <a href="/wiki/Quadratic_equation" title="Quadratic equation">quadratic equation</a>. The solutions to the univariate equation are called the <a href="/wiki/Root_of_a_function" class="mw-redirect" title="Root of a function">roots</a> of the univariate function. The bivariate case in terms of variables x and y has the form <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x,y)=ax^{2}+by^{2}+cxy+dx+ey+f\,\!}"> <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> <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> <mi>x</mi> <mi>y</mi> <mo>+</mo> <mi>d</mi> <mi>x</mi> <mo>+</mo> <mi>e</mi> <mi>y</mi> <mo>+</mo> <mi>f</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x,y)=ax^{2}+by^{2}+cxy+dx+ey+f\,\!}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc22169897f4341363e969b77360efb72bd277ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:40.676ex; height:3.176ex;" alt="{\displaystyle f(x,y)=ax^{2}+by^{2}+cxy+dx+ey+f\,\!}"></dd></dl> with at least one of a, b, c not equal to zero, and an equation setting this function equal to zero gives rise to a <a href="/wiki/Conic_section" title="Conic section">conic section</a> (a <a href="/wiki/Circle" title="Circle">circle</a> or other <a href="/wiki/Ellipse" title="Ellipse">ellipse</a>, a <a href="/wiki/Parabola" title="Parabola">parabola</a>, or a <a href="/wiki/Hyperbola" title="Hyperbola">hyperbola</a>). In general there can be an arbitrarily large number of variables, in which case the resulting <a href="/wiki/Surface_(geometry)" class="mw-redirect" title="Surface (geometry)">surface</a> is called a <a href="/wiki/Quadric" title="Quadric">quadric</a>, but the highest degree term must be of degree 2, such as x2, xy, yz, etc.</dd> <dt id="quadratic_polynomial"><dfn><a href="/wiki/Quadratic_polynomial" class="mw-redirect" title="Quadratic polynomial">quadratic polynomial</a></dfn></dt> <dd>.</dd> <dt id="quotient_rule"><dfn><a href="/wiki/Quotient_rule" title="Quotient rule">quotient rule</a></dfn></dt> <dd>A formula for finding the derivative of a function that is the ratio of two functions.</dd> </dl> <div class="mw-heading mw-heading2"><h2 id="R">R</h2>[<a href="/w/index.php?title=Glossary_of_calculus&action=edit&section=17" title="Edit section: R">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1228772891"> <dl class="glossary"> <dt id="radian"><dfn><a href="/wiki/Radian" title="Radian">radian</a></dfn></dt> <dd>Is the <a href="/wiki/International_System_of_Units" title="International System of Units">SI unit</a> for measuring <a href="/wiki/Angle" title="Angle">angles</a>, and is the standard unit of angular measure used in many areas of <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>. The length of an arc of a <a href="/wiki/Unit_circle" title="Unit circle">unit circle</a> is numerically equal to the measurement in radians of the <a href="/wiki/Angle" title="Angle">angle</a> that it <a href="https://en.wiktionary.org/wiki/subtend" class="extiw" title="wikt:subtend">subtends</a>; one radian is just under 57.3 <a href="/wiki/Degree_(angle)" title="Degree (angle)">degrees</a> (expansion at <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A072097" class="extiw" title="oeis:A072097">A072097</a>). The unit was formerly an <a href="/wiki/SI_supplementary_unit" class="mw-redirect" title="SI supplementary unit">SI supplementary unit</a>, but this category was abolished in 1995 and the radian is now considered an <a href="/wiki/SI_derived_unit" title="SI derived unit">SI derived unit</a>.<a href="#cite_note-83">[79]</a> Separately, the SI unit of <a href="/wiki/Solid_angle" title="Solid angle">solid angle</a> measurement is the <a href="/wiki/Steradian" title="Steradian">steradian</a> .</dd> <dt id="ratio_test"><dfn><a href="/wiki/Ratio_test" title="Ratio test">ratio test</a></dfn></dt> <dd>.</dd> <dt id="reciprocal_function"><dfn><a href="/wiki/Reciprocal_function" class="mw-redirect" title="Reciprocal function">reciprocal function</a></dfn></dt> <dd>.</dd> <dt id="reciprocal_rule"><dfn><a href="/wiki/Reciprocal_rule" title="Reciprocal rule">reciprocal rule</a></dfn></dt> <dd>.</dd> <dt id="riemann_integral"><dfn><a href="/wiki/Riemann_integral" title="Riemann integral">Riemann integral</a></dfn></dt> <dd>.</dd> <dt id="related_rates"><dfn><a href="/wiki/Related_rates" title="Related rates">related rates</a></dfn></dt> <dd>.</dd> <dt id="removable_discontinuity"><dfn><a href="/wiki/Classification_of_discontinuities#Removable_discontinuity" title="Classification of discontinuities">removable discontinuity</a></dfn></dt> <dd>.</dd> <dt id="rolle's_theorem"><dfn><a href="/wiki/Rolle%27s_theorem" title="Rolle's theorem">Rolle's theorem</a></dfn></dt> <dd>.</dd> <dt id="root_test"><dfn><a href="/wiki/Root_test" title="Root test">root test</a></dfn></dt> <dd>.</dd> </dl> <div class="mw-heading mw-heading2"><h2 id="S">S</h2>[<a href="/w/index.php?title=Glossary_of_calculus&action=edit&section=18" title="Edit section: S">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1228772891"> <dl class="glossary"> <dt id="scalar"><dfn><a href="/wiki/Scalar_(mathematics)" title="Scalar (mathematics)">scalar</a></dfn></dt> <dd>.</dd> <dt id="secant_line"><dfn><a href="/wiki/Secant_line" title="Secant line">secant line</a></dfn></dt> <dd>.</dd> <dt id="second-degree_polynomial"><dfn><a href="/wiki/Second-degree_polynomial" class="mw-redirect" title="Second-degree polynomial">second-degree polynomial</a></dfn></dt> <dd>.</dd> <dt id="second_derivative"><dfn><a href="/wiki/Second_derivative" title="Second derivative">second derivative</a></dfn></dt> <dd>.</dd> <dt id="second_derivative_test"><dfn><a href="/wiki/Second_derivative#Second_derivative_test" title="Second derivative">second derivative test</a></dfn></dt> <dd>.</dd> <dt id="second-order_differential_equation"><dfn><a href="/wiki/Differential_equation#Equation_order" title="Differential equation">second-order differential equation</a></dfn></dt> <dd>.</dd> <dt id="series"><dfn><a href="/wiki/Series_(mathematics)" title="Series (mathematics)">series</a></dfn></dt> <dd>.</dd> <dt id="shell_integration"><dfn><a href="/wiki/Shell_integration" title="Shell integration">shell integration</a></dfn></dt> <dd>.</dd> <dt id="simpson's_rule"><dfn><a href="/wiki/Simpson%27s_rule" title="Simpson's rule">Simpson's rule</a></dfn></dt> <dd>.</dd> <dt id="sine"><dfn><a href="/wiki/Sine" class="mw-redirect" title="Sine">sine</a></dfn></dt> <dd>.</dd> <dt id="sine_wave"><dfn><a href="/wiki/Sine_wave" title="Sine wave">sine wave</a></dfn></dt> <dd>.</dd> <dt id="slope_field"><dfn><a href="/wiki/Slope_field" title="Slope field">slope field</a></dfn></dt> <dd>.</dd> <dt id="squeeze_theorem"><dfn><a href="/wiki/Squeeze_theorem" title="Squeeze theorem">squeeze theorem</a></dfn></dt> <dd>.</dd> <dt id="sum_rule_in_differentiation"><dfn><a href="/wiki/Sum_rule_in_differentiation" class="mw-redirect" title="Sum rule in differentiation">sum rule in differentiation</a></dfn></dt> <dd>.</dd> <dt id="sum_rule_in_integration"><dfn><a href="/wiki/Sum_rule_in_integration" class="mw-redirect" title="Sum rule in integration">sum rule in integration</a></dfn></dt> <dd>.</dd> <dt id="summation"><dfn><a href="/wiki/Summation" title="Summation">summation</a></dfn></dt> <dd>.</dd> <dt id="supplementary_angle"><dfn><a href="/wiki/Supplementary_angle" class="mw-redirect" title="Supplementary angle">supplementary angle</a></dfn></dt> <dd>.</dd> <dt id="surface_area"><dfn><a href="/wiki/Surface_area" title="Surface area">surface area</a></dfn></dt> <dd>.</dd> <dt id="system_of_linear_equations"><dfn><a href="/wiki/System_of_linear_equations" title="System of linear equations">system of linear equations</a></dfn></dt> <dd>.</dd> </dl> <div class="mw-heading mw-heading2"><h2 id="T">T</h2>[<a href="/w/index.php?title=Glossary_of_calculus&action=edit&section=19" title="Edit section: T">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1228772891"> <dl class="glossary"> <dt id="table_of_integrals"><dfn><a href="/wiki/List_of_integrals" class="mw-redirect" title="List of integrals">table of integrals</a></dfn></dt> <dd>.</dd> <dt id="taylor_series"><dfn><a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a></dfn></dt> <dd>.</dd> <dt id="taylor's_theorem"><dfn><a href="/wiki/Taylor%27s_theorem" title="Taylor's theorem">Taylor's theorem</a></dfn></dt> <dd>.</dd> <dt id="tangent"><dfn><a href="/wiki/Tangent" title="Tangent">tangent</a></dfn></dt> <dd>.</dd> <dt id="third-degree_polynomial"><dfn><a href="/wiki/Degree_of_a_polynomial" title="Degree of a polynomial">third-degree polynomial</a></dfn></dt> <dd>.</dd> <dt id="third_derivative"><dfn><a href="/wiki/Third_derivative" title="Third derivative">third derivative</a></dfn></dt> <dd>.</dd> <dt id="toroid"><dfn><a href="/wiki/Toroid" title="Toroid">toroid</a></dfn></dt> <dd>.</dd> <dt id="total_differential"><dfn><a href="/wiki/Total_differential" class="mw-redirect" title="Total differential">total differential</a></dfn></dt> <dd>.</dd> <dt id="trigonometric_functions"><dfn><a href="/wiki/Trigonometric_functions" title="Trigonometric functions">trigonometric functions</a></dfn></dt> <dd>.</dd> <dt id="trigonometric_identities"><dfn><a href="/wiki/List_of_trigonometric_identities" title="List of trigonometric identities">trigonometric identities</a></dfn></dt> <dd>.</dd> <dt id="trigonometric_integral"><dfn><a href="/wiki/Trigonometric_integral" title="Trigonometric integral">trigonometric integral</a></dfn></dt> <dd>.</dd> <dt id="trigonometric_substitution"><dfn><a href="/wiki/Trigonometric_substitution" title="Trigonometric substitution">trigonometric substitution</a></dfn></dt> <dd>.</dd> <dt id="trigonometry"><dfn><a href="/wiki/Trigonometry" title="Trigonometry">trigonometry</a></dfn></dt> <dd>.</dd> <dt id="triple_integral"><dfn><a href="/wiki/Triple_integral" class="mw-redirect" title="Triple integral">triple integral</a></dfn></dt> <dd>.</dd> </dl> <div class="mw-heading mw-heading2"><h2 id="U">U</h2>[<a href="/w/index.php?title=Glossary_of_calculus&action=edit&section=20" title="Edit section: U">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1228772891"> <dl class="glossary"> <dt id="upper_bound"><dfn><a href="/wiki/Upper_and_lower_bounds#Bounds_of_functions" title="Upper and lower bounds">upper bound</a></dfn></dt> <dd>.</dd> </dl> <div class="mw-heading mw-heading2"><h2 id="V">V</h2>[<a href="/w/index.php?title=Glossary_of_calculus&action=edit&section=21" title="Edit section: V">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1228772891"> <dl class="glossary"> <dt id="variable"><dfn><a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variable</a></dfn></dt> <dd>.</dd> <dt id="vector"><dfn><a href="/wiki/Vector_(mathematics_and_physics)" title="Vector (mathematics and physics)">vector</a></dfn></dt> <dd>.</dd> <dt id="vector_calculus"><dfn><a href="/wiki/Vector_calculus" title="Vector calculus">vector calculus</a></dfn></dt> <dd>.</dd> </dl> <div class="mw-heading mw-heading2"><h2 id="W">W</h2>[<a href="/w/index.php?title=Glossary_of_calculus&action=edit&section=22" title="Edit section: W">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1228772891"> <dl class="glossary"> <dt id="washer"><dfn><a href="/wiki/Disc_integration#Washer_method" title="Disc integration">washer</a></dfn></dt> <dd>.</dd> <dt id="washer_method"><dfn><a href="/wiki/Disc_integration#Washer_method" title="Disc integration">washer method</a></dfn></dt> <dd>.</dd> </dl> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Glossary_of_calculus&action=edit&section=23" title="Edit section: See also">edit</a>]</div> <ul><li><a href="/wiki/Outline_of_calculus" title="Outline of calculus">Outline of calculus</a></li> <li><a href="/wiki/Glossary_of_areas_of_mathematics" title="Glossary of areas of mathematics">Glossary of areas of mathematics</a></li> <li><a href="/wiki/Glossary_of_astronomy" title="Glossary of astronomy">Glossary of astronomy</a></li> <li><a href="/wiki/Glossary_of_biology" title="Glossary of biology">Glossary of biology</a></li> <li><a href="/wiki/Glossary_of_botanical_terms" title="Glossary of botanical terms">Glossary of botany</a></li> <li><a href="/wiki/Glossary_of_chemistry_terms" title="Glossary of chemistry terms">Glossary of chemistry</a></li> <li><a href="/wiki/Glossary_of_ecology" title="Glossary of ecology">Glossary of ecology</a></li> <li><a href="/wiki/Glossary_of_engineering" title="Glossary of engineering">Glossary of engineering</a></li> <li><a href="/wiki/Glossary_of_physics" title="Glossary of physics">Glossary of physics</a></li> <li><a href="/wiki/Glossary_of_probability_and_statistics" title="Glossary of probability and statistics">Glossary of probability and statistics</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Glossary_of_calculus&action=edit&section=24" title="Edit section: References">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-2"><a href="#cite_ref-2">^</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFStewart2008" class="citation book cs1"><a href="/wiki/James_Stewart_(mathematician)" title="James Stewart (mathematician)">Stewart, James</a> (2008). <a rel="nofollow" class="external text" href="https://archive.org/details/calculusearlytra00stew_1">Calculus: Early Transcendentals</a> (6th ed.). <a href="/wiki/Brooks/Cole" class="mw-redirect" title="Brooks/Cole">Brooks/Cole</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-495-01166-8" title="Special:BookSources/978-0-495-01166-8"><bdi>978-0-495-01166-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus%3A+Early+Transcendentals&rft.edition=6th&rft.pub=Brooks%2FCole&rft.date=2008&rft.isbn=978-0-495-01166-8&rft.aulast=Stewart&rft.aufirst=James&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fcalculusearlytra00stew_1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGlossary+of+calculus" class="Z3988"> </li> <li id="cite_note-3"><a href="#cite_ref-3">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLarsonEdwards2009" class="citation book cs1"><a href="/wiki/Ron_Larson_(mathematician)" class="mw-redirect" title="Ron Larson (mathematician)">Larson, Ron</a>; Edwards, Bruce H. (2009). Calculus (9th ed.). <a href="/wiki/Brooks/Cole" class="mw-redirect" title="Brooks/Cole">Brooks/Cole</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-547-16702-2" title="Special:BookSources/978-0-547-16702-2"><bdi>978-0-547-16702-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus&rft.edition=9th&rft.pub=Brooks%2FCole&rft.date=2009&rft.isbn=978-0-547-16702-2&rft.aulast=Larson&rft.aufirst=Ron&rft.au=Edwards%2C+Bruce+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGlossary+of+calculus" class="Z3988"> </li> <li id="cite_note-4"><a href="#cite_ref-4">^</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/*/http://rowdy.msudenver.edu/~talmanl/PDFs/APCalculus/OnAsymptotes.pdf">"Asymptotes" by Louis A. 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Retrieved 2012-05-18.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Introduction+%C3%A0+l%27Analyse+des+lignes+Courbes+alg%C3%A9briques&rft.place=Geneva&rft.pages=656-659&rft.pub=Europeana&rft.date=1750&rft.au=Cramer%2C+Gabriel&rft_id=https%3A%2F%2Fwww.europeana.eu%2Fresolve%2Frecord%2F03486%2FE71FE3799CEC1F8E2B76962513829D2E36B63015&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGlossary+of+calculus" class="Z3988"> </li> <li id="cite_note-23"><a href="#cite_ref-23">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKosinski2001" class="citation journal cs1">Kosinski, A. A. (2001). "Cramer's Rule is due to Cramer". Mathematics Magazine. 74 (4): 310–312. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2691101">10.2307/2691101</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2691101">2691101</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+Magazine&rft.atitle=Cramer%27s+Rule+is+due+to+Cramer&rft.volume=74&rft.issue=4&rft.pages=310-312&rft.date=2001&rft_id=info%3Adoi%2F10.2307%2F2691101&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2691101%23id-name%3DJSTOR&rft.aulast=Kosinski&rft.aufirst=A.+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGlossary+of+calculus" class="Z3988"> </li> <li id="cite_note-24"><a href="#cite_ref-24">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMacLaurin1748" class="citation book cs1">MacLaurin, Colin (1748). <a rel="nofollow" class="external text" href="https://archive.org/details/atreatisealgebr03maclgoog">A Treatise of Algebra, in Three Parts</a>. Printed for A. Millar & J. Nourse.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Treatise+of+Algebra%2C+in+Three+Parts.&rft.pub=Printed+for+A.+Millar+%26+J.+Nourse&rft.date=1748&rft.aulast=MacLaurin&rft.aufirst=Colin&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fatreatisealgebr03maclgoog&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGlossary+of+calculus" class="Z3988"> </li> <li id="cite_note-25"><a href="#cite_ref-25">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBoyer1968" class="citation book cs1"><a href="/wiki/Carl_Benjamin_Boyer" title="Carl Benjamin Boyer">Boyer, Carl B.</a> (1968). A History of Mathematics (2nd ed.). Wiley. p. 431.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Mathematics&rft.pages=431&rft.edition=2nd&rft.pub=Wiley&rft.date=1968&rft.aulast=Boyer&rft.aufirst=Carl+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGlossary+of+calculus" class="Z3988"> </li> <li id="cite_note-26"><a href="#cite_ref-26">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKatz2004" class="citation book cs1">Katz, Victor (2004). A History of Mathematics (Brief ed.). Pearson Education. pp. 378–379.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Mathematics&rft.pages=378-379&rft.edition=Brief&rft.pub=Pearson+Education&rft.date=2004&rft.aulast=Katz&rft.aufirst=Victor&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGlossary+of+calculus" class="Z3988"> </li> <li id="cite_note-27"><a href="#cite_ref-27">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHedman1999" class="citation journal cs1">Hedman, Bruce A. (1999). <a rel="nofollow" class="external text" href="http://professorhedman.com/Cramers.Rule.pdf">"An Earlier Date for "Cramer's Rule""</a> (PDF). Historia Mathematica. 26 (4): 365–368. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1006%2Fhmat.1999.2247">10.1006/hmat.1999.2247</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:121056843">121056843</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Historia+Mathematica&rft.atitle=An+Earlier+Date+for+%22Cramer%27s+Rule%22&rft.volume=26&rft.issue=4&rft.pages=365-368&rft.date=1999&rft_id=info%3Adoi%2F10.1006%2Fhmat.1999.2247&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121056843%23id-name%3DS2CID&rft.aulast=Hedman&rft.aufirst=Bruce+A.&rft_id=http%3A%2F%2Fprofessorhedman.com%2FCramers.Rule.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGlossary+of+calculus" class="Z3988"> </li> <li id="cite_note-28"><a href="#cite_ref-28">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStewart2008" class="citation book cs1"><a href="/wiki/James_Stewart_(mathematician)" title="James Stewart (mathematician)">Stewart, James</a> (2008). <a rel="nofollow" class="external text" href="https://archive.org/details/calculusearlytra00stew_1">Calculus: Early Transcendentals</a> (6th ed.). <a href="/wiki/Brooks/Cole" class="mw-redirect" title="Brooks/Cole">Brooks/Cole</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-495-01166-8" title="Special:BookSources/978-0-495-01166-8"><bdi>978-0-495-01166-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus%3A+Early+Transcendentals&rft.edition=6th&rft.pub=Brooks%2FCole&rft.date=2008&rft.isbn=978-0-495-01166-8&rft.aulast=Stewart&rft.aufirst=James&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fcalculusearlytra00stew_1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGlossary+of+calculus" class="Z3988"> </li> <li id="cite_note-29"><a href="#cite_ref-29">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLarsonEdwards2009" class="citation book cs1"><a href="/wiki/Ron_Larson_(mathematician)" class="mw-redirect" title="Ron Larson (mathematician)">Larson, Ron</a>; Edwards, Bruce H. (2009). Calculus (9th ed.). <a href="/wiki/Brooks/Cole" class="mw-redirect" title="Brooks/Cole">Brooks/Cole</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-547-16702-2" title="Special:BookSources/978-0-547-16702-2"><bdi>978-0-547-16702-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus&rft.edition=9th&rft.pub=Brooks%2FCole&rft.date=2009&rft.isbn=978-0-547-16702-2&rft.aulast=Larson&rft.aufirst=Ron&rft.au=Edwards%2C+Bruce+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGlossary+of+calculus" class="Z3988"> </li> <li id="cite_note-30"><a href="#cite_ref-30">^</a> Douglas C. Giancoli (2000). [Physics for Scientists and Engineers with Modern Physics (3rd Edition)]. Prentice Hall. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-13-021517-1" title="Special:BookSources/0-13-021517-1">0-13-021517-1</a> </li> <li id="cite_note-31"><a href="#cite_ref-31">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.merriam-webster.com/dictionary/differential%20calculus">"Definition of DIFFERENTIAL CALCULUS"</a>. www.merriam-webster.com. Retrieved 2018-09-26.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.merriam-webster.com&rft.atitle=Definition+of+DIFFERENTIAL+CALCULUS&rft_id=https%3A%2F%2Fwww.merriam-webster.com%2Fdictionary%2Fdifferential%2520calculus&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGlossary+of+calculus" class="Z3988"> </li> <li id="cite_note-32"><a href="#cite_ref-32">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.merriam-webster.com/dictionary/integral%20calculus">"Integral Calculus - Definition of Integral calculus by Merriam-Webster"</a>. www.merriam-webster.com. Retrieved 2018-05-01.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.merriam-webster.com&rft.atitle=Integral+Calculus+-+Definition+of+Integral+calculus+by+Merriam-Webster&rft_id=https%3A%2F%2Fwww.merriam-webster.com%2Fdictionary%2Fintegral%2520calculus&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGlossary+of+calculus" class="Z3988"> </li> <li id="cite_note-33"><a href="#cite_ref-33">^</a> Démonstration d’un théorème d’Abel. Journal de mathématiques pures et appliquées 2nd series, tome 7 (1862), <a rel="nofollow" class="external text" href="http://portail.mathdoc.fr/JMPA/afficher_notice.php?id=JMPA_1862_2_7_A43_0">p. 253-255</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110721011902/http://portail.mathdoc.fr/JMPA/afficher_notice.php?id=JMPA_1862_2_7_A43_0">Archived</a> 2011-07-21 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. </li> <li id="cite_note-Stewart-35"><a href="#cite_ref-Stewart_35-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStewart2008" class="citation book cs1"><a href="/wiki/James_Stewart_(mathematician)" title="James Stewart (mathematician)">Stewart, James</a> (2008). <a rel="nofollow" class="external text" href="https://archive.org/details/calculusearlytra00stew_1">Calculus: Early Transcendentals</a> (6th ed.). Brooks Cole Cengage Learning. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-495-01166-8" title="Special:BookSources/978-0-495-01166-8"><bdi>978-0-495-01166-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus%3A+Early+Transcendentals&rft.edition=6th&rft.pub=Brooks+Cole+Cengage+Learning&rft.date=2008&rft.isbn=978-0-495-01166-8&rft.aulast=Stewart&rft.aufirst=James&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fcalculusearlytra00stew_1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGlossary+of+calculus" class="Z3988"> </li> <li id="cite_note-36"><a href="#cite_ref-36">^</a> <a href="/wiki/Oxford_English_Dictionary" title="Oxford English Dictionary">Oxford English Dictionary</a>, 2nd ed.: <a rel="nofollow" class="external text" href="https://web.archive.org/web/20131006101640/http://oxforddictionaries.com/definition/english/natural-logarithm">natural logarithm</a> </li> <li id="cite_note-37"><a href="#cite_ref-37">^</a> <a href="/wiki/Encyclopedic_Dictionary_of_Mathematics" title="Encyclopedic Dictionary of Mathematics">Encyclopedic Dictionary of Mathematics</a> 142.D </li> <li id="cite_note-38"><a href="#cite_ref-38">^</a> <a href="#CITEREFButcher2003">Butcher 2003</a>, p. 45; <a href="#CITEREFHairerNørsettWanner1993">Hairer, Nørsett & Wanner 1993</a>, p. 35 </li> <li id="cite_note-39"><a href="#cite_ref-39">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStewart2008" class="citation book cs1"><a href="/wiki/James_Stewart_(mathematician)" title="James Stewart (mathematician)">Stewart, James</a> (2008). <a rel="nofollow" class="external text" href="https://archive.org/details/calculusearlytra00stew_1">Calculus: Early Transcendentals</a> (6th ed.). <a href="/wiki/Brooks/Cole" class="mw-redirect" title="Brooks/Cole">Brooks/Cole</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-495-01166-8" title="Special:BookSources/978-0-495-01166-8"><bdi>978-0-495-01166-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus%3A+Early+Transcendentals&rft.edition=6th&rft.pub=Brooks%2FCole&rft.date=2008&rft.isbn=978-0-495-01166-8&rft.aulast=Stewart&rft.aufirst=James&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fcalculusearlytra00stew_1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGlossary+of+calculus" class="Z3988"> </li> <li id="cite_note-40"><a href="#cite_ref-40">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLarsonEdwards2009" class="citation book cs1"><a href="/wiki/Ron_Larson_(mathematician)" class="mw-redirect" title="Ron Larson (mathematician)">Larson, Ron</a>; Edwards, Bruce H. (2009). Calculus (9th ed.). <a href="/wiki/Brooks/Cole" class="mw-redirect" title="Brooks/Cole">Brooks/Cole</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-547-16702-2" title="Special:BookSources/978-0-547-16702-2"><bdi>978-0-547-16702-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus&rft.edition=9th&rft.pub=Brooks%2FCole&rft.date=2009&rft.isbn=978-0-547-16702-2&rft.aulast=Larson&rft.aufirst=Ron&rft.au=Edwards%2C+Bruce+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGlossary+of+calculus" class="Z3988"> </li> <li id="cite_note-41"><a href="#cite_ref-41">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFThomasWeirHass2010" class="citation book cs1"><a href="/wiki/George_B._Thomas" title="George B. Thomas">Thomas, George B.</a>; Weir, Maurice D.; <a href="/wiki/Joel_Hass" title="Joel Hass">Hass, Joel</a> (2010). Thomas' Calculus: Early Transcendentals (12th ed.). <a href="/wiki/Addison-Wesley" title="Addison-Wesley">Addison-Wesley</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-321-58876-0" title="Special:BookSources/978-0-321-58876-0"><bdi>978-0-321-58876-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Thomas%27+Calculus%3A+Early+Transcendentals&rft.edition=12th&rft.pub=Addison-Wesley&rft.date=2010&rft.isbn=978-0-321-58876-0&rft.aulast=Thomas&rft.aufirst=George+B.&rft.au=Weir%2C+Maurice+D.&rft.au=Hass%2C+Joel&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGlossary+of+calculus" class="Z3988"> </li> <li id="cite_note-42"><a href="#cite_ref-42">^</a> (<a href="#CITEREFArbogast1800">Arbogast 1800</a>). </li> <li id="cite_note-43"><a href="#cite_ref-43">^</a> According to <a href="#CITEREFCraik2005">Craik (2005</a>, pp. 120–122): see also the analysis of Arbogast's work by <a href="#CITEREFJohnson2002">Johnson (2002</a>, p. 230). </li> <li id="cite_note-45"><a href="#cite_ref-45">^</a> William F. Kern, James R. Bland, Solid Mensuration with proofs, 1938, p. 67 </li> <li id="cite_note-MacLane-46"><a href="#cite_ref-MacLane_46-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMacLaneBirkhoff1967" class="citation book cs1"><a href="/wiki/Saunders_MacLane" class="mw-redirect" title="Saunders MacLane">MacLane, Saunders</a>; <a href="/wiki/Garrett_Birkhoff" title="Garrett Birkhoff">Birkhoff, Garrett</a> (1967). <a rel="nofollow" class="external text" href="https://archive.org/details/algebra00macl">Algebra</a> (First ed.). New York: Macmillan. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/algebra00macl/page/1">1–13</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebra&rft.place=New+York&rft.pages=1-13&rft.edition=First&rft.pub=Macmillan&rft.date=1967&rft.aulast=MacLane&rft.aufirst=Saunders&rft.au=Birkhoff%2C+Garrett&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Falgebra00macl&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGlossary+of+calculus" class="Z3988"> </li> <li id="cite_note-47"><a href="#cite_ref-47">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSpivak1980" class="citation cs2">Spivak, Michael (1980), Calculus (2nd ed.), Houston, Texas: Publish or Perish Inc.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus&rft.place=Houston%2C+Texas&rft.edition=2nd&rft.pub=Publish+or+Perish+Inc.&rft.date=1980&rft.aulast=Spivak&rft.aufirst=Michael&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGlossary+of+calculus" class="Z3988"> </li> <li id="cite_note-48"><a href="#cite_ref-48">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOlver2000" class="citation book cs1">Olver, Peter J. (2000). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=sI2bAxgLMXYC&pg=PA318">Applications of Lie Groups to Differential Equations</a>. Springer. pp. 318–319. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780387950006" title="Special:BookSources/9780387950006"><bdi>9780387950006</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Applications+of+Lie+Groups+to+Differential+Equations&rft.pages=318-319&rft.pub=Springer&rft.date=2000&rft.isbn=9780387950006&rft.aulast=Olver&rft.aufirst=Peter+J.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DsI2bAxgLMXYC%26pg%3DPA318&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGlossary+of+calculus" class="Z3988"> </li> <li id="cite_note-49"><a href="#cite_ref-49">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStewart2008" class="citation book cs1"><a href="/wiki/James_Stewart_(mathematician)" title="James Stewart (mathematician)">Stewart, James</a> (2008). <a rel="nofollow" class="external text" href="https://archive.org/details/calculusearlytra00stew_1">Calculus: Early Transcendentals</a> (6th ed.). <a href="/wiki/Brooks/Cole" class="mw-redirect" title="Brooks/Cole">Brooks/Cole</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-495-01166-8" title="Special:BookSources/978-0-495-01166-8"><bdi>978-0-495-01166-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus%3A+Early+Transcendentals&rft.edition=6th&rft.pub=Brooks%2FCole&rft.date=2008&rft.isbn=978-0-495-01166-8&rft.aulast=Stewart&rft.aufirst=James&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fcalculusearlytra00stew_1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGlossary+of+calculus" class="Z3988"> </li> <li id="cite_note-50"><a href="#cite_ref-50">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLarsonEdwards2009" class="citation book cs1"><a href="/wiki/Ron_Larson_(mathematician)" class="mw-redirect" title="Ron Larson (mathematician)">Larson, Ron</a>; Edwards, Bruce H. (2009). Calculus (9th ed.). <a href="/wiki/Brooks/Cole" class="mw-redirect" title="Brooks/Cole">Brooks/Cole</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-547-16702-2" title="Special:BookSources/978-0-547-16702-2"><bdi>978-0-547-16702-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus&rft.edition=9th&rft.pub=Brooks%2FCole&rft.date=2009&rft.isbn=978-0-547-16702-2&rft.aulast=Larson&rft.aufirst=Ron&rft.au=Edwards%2C+Bruce+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGlossary+of+calculus" class="Z3988"> </li> <li id="cite_note-51"><a href="#cite_ref-51">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFThomasWeirHass2010" class="citation book cs1"><a href="/wiki/George_B._Thomas" title="George B. Thomas">Thomas, George B.</a>; Weir, Maurice D.; <a href="/wiki/Joel_Hass" title="Joel Hass">Hass, Joel</a> (2010). Thomas' Calculus: Early Transcendentals (12th ed.). <a href="/wiki/Addison-Wesley" title="Addison-Wesley">Addison-Wesley</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-321-58876-0" title="Special:BookSources/978-0-321-58876-0"><bdi>978-0-321-58876-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Thomas%27+Calculus%3A+Early+Transcendentals&rft.edition=12th&rft.pub=Addison-Wesley&rft.date=2010&rft.isbn=978-0-321-58876-0&rft.aulast=Thomas&rft.aufirst=George+B.&rft.au=Weir%2C+Maurice+D.&rft.au=Hass%2C+Joel&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGlossary+of+calculus" class="Z3988"> </li> <li id="cite_note-52"><a href="#cite_ref-52">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStewart2008" class="citation book cs1"><a href="/wiki/James_Stewart_(mathematician)" title="James Stewart (mathematician)">Stewart, James</a> (2008). <a rel="nofollow" class="external text" href="https://archive.org/details/calculusearlytra00stew_1">Calculus: Early Transcendentals</a> (6th ed.). <a href="/wiki/Brooks/Cole" class="mw-redirect" title="Brooks/Cole">Brooks/Cole</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-495-01166-8" title="Special:BookSources/978-0-495-01166-8"><bdi>978-0-495-01166-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus%3A+Early+Transcendentals&rft.edition=6th&rft.pub=Brooks%2FCole&rft.date=2008&rft.isbn=978-0-495-01166-8&rft.aulast=Stewart&rft.aufirst=James&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fcalculusearlytra00stew_1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGlossary+of+calculus" class="Z3988"> </li> <li id="cite_note-53"><a href="#cite_ref-53">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLarsonEdwards2009" class="citation book cs1"><a href="/wiki/Ron_Larson_(mathematician)" class="mw-redirect" title="Ron Larson (mathematician)">Larson, Ron</a>; Edwards, Bruce H. (2009). Calculus (9th ed.). <a href="/wiki/Brooks/Cole" class="mw-redirect" title="Brooks/Cole">Brooks/Cole</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-547-16702-2" title="Special:BookSources/978-0-547-16702-2"><bdi>978-0-547-16702-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus&rft.edition=9th&rft.pub=Brooks%2FCole&rft.date=2009&rft.isbn=978-0-547-16702-2&rft.aulast=Larson&rft.aufirst=Ron&rft.au=Edwards%2C+Bruce+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGlossary+of+calculus" class="Z3988"> </li> <li id="cite_note-54"><a href="#cite_ref-54">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFThomasWeirHass2010" class="citation book cs1"><a href="/wiki/George_B._Thomas" title="George B. Thomas">Thomas, George B.</a>; Weir, Maurice D.; <a href="/wiki/Joel_Hass" title="Joel Hass">Hass, Joel</a> (2010). Thomas' Calculus: Early Transcendentals (12th ed.). <a href="/wiki/Addison-Wesley" title="Addison-Wesley">Addison-Wesley</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-321-58876-0" title="Special:BookSources/978-0-321-58876-0"><bdi>978-0-321-58876-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Thomas%27+Calculus%3A+Early+Transcendentals&rft.edition=12th&rft.pub=Addison-Wesley&rft.date=2010&rft.isbn=978-0-321-58876-0&rft.aulast=Thomas&rft.aufirst=George+B.&rft.au=Weir%2C+Maurice+D.&rft.au=Hass%2C+Joel&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGlossary+of+calculus" class="Z3988"> </li> <li id="cite_note-55"><a href="#cite_ref-55">^</a> Chang, Yu-sung, "<a rel="nofollow" class="external text" href="http://demonstrations.wolfram.com/GoldenSpiral/">Golden Spiral</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20190728084311/http://demonstrations.wolfram.com/GoldenSpiral/">Archived</a> 2019-07-28 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>", <a href="/wiki/The_Wolfram_Demonstrations_Project" class="mw-redirect" title="The Wolfram Demonstrations Project">The Wolfram Demonstrations Project</a>. </li> <li id="cite_note-56"><a href="#cite_ref-56">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFErdős1932" class="citation cs2 cs1-prop-foreign-lang-source"><a href="/wiki/Paul_Erd%C5%91s" title="Paul Erdős">Erdős, P.</a> (1932), <a rel="nofollow" class="external text" href="https://www.renyi.hu/~p_erdos/1932-02.pdf">"Egy Kürschák-féle elemi számelméleti tétel általánosítása"</a> [Generalization of an elementary number-theoretic theorem of Kürschák] (PDF), Mat. Fiz. Lapok (in Hungarian), 39: 17–24</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mat.+Fiz.+Lapok&rft.atitle=Egy+K%C3%BCrsch%C3%A1k-f%C3%A9le+elemi+sz%C3%A1melm%C3%A9leti+t%C3%A9tel+%C3%A1ltal%C3%A1nos%C3%ADt%C3%A1sa&rft.volume=39&rft.pages=17-24&rft.date=1932&rft.aulast=Erd%C5%91s&rft.aufirst=P.&rft_id=https%3A%2F%2Fwww.renyi.hu%2F~p_erdos%2F1932-02.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGlossary+of+calculus" class="Z3988">. As cited by <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGraham2013" class="citation cs2"><a href="/wiki/Ronald_Graham" title="Ronald Graham">Graham, Ronald L.</a> (2013), "Paul Erdős and Egyptian fractions", Erdős centennial, Bolyai Soc. Math. Stud., vol. 25, János Bolyai Math. Soc., Budapest, pp. 289–309, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-642-39286-3_9">10.1007/978-3-642-39286-3_9</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-39285-6" title="Special:BookSources/978-3-642-39285-6"><bdi>978-3-642-39285-6</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=3203600">3203600</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Paul+Erd%C5%91s+and+Egyptian+fractions&rft.btitle=Erd%C5%91s+centennial&rft.series=Bolyai+Soc.+Math.+Stud.&rft.pages=289-309&rft.pub=J%C3%A1nos+Bolyai+Math.+Soc.%2C+Budapest&rft.date=2013&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D3203600%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2F978-3-642-39286-3_9&rft.isbn=978-3-642-39285-6&rft.aulast=Graham&rft.aufirst=Ronald+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGlossary+of+calculus" class="Z3988">. </li> <li id="cite_note-58"><a href="#cite_ref-58">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFUno_Ingard1988" class="citation book cs1">Uno Ingard, K. (1988). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=SGVfGIewvxkC&pg=PA38">"Chapter 2"</a>. Fundamentals of Waves and Oscillations. Cambridge University Press. p. 38. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-33957-X" title="Special:BookSources/0-521-33957-X"><bdi>0-521-33957-X</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+2&rft.btitle=Fundamentals+of+Waves+and+Oscillations&rft.pages=38&rft.pub=Cambridge+University+Press&rft.date=1988&rft.isbn=0-521-33957-X&rft.aulast=Uno+Ingard&rft.aufirst=K.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DSGVfGIewvxkC%26pg%3DPA38&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGlossary+of+calculus" class="Z3988"> </li> <li id="cite_note-59"><a href="#cite_ref-59">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSinha2008" class="citation book cs1">Sinha, K.C. (2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=mqdzqbPYiAUC&pg=SA11-PA2">A Text Book of Mathematics Class XI</a> (Second ed.). 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Retrieved 2014-10-30.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=World+Wide+Words&rft.atitle=World+Wide+Words%3A+Vulgar+fractions&rft_id=http%3A%2F%2Fwww.worldwidewords.org%2Fqa%2Fqa-vul1.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGlossary+of+calculus" class="Z3988"> </li> <li id="cite_note-62"><a href="#cite_ref-62">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. 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Retrieved 2014-10-30.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Math+Forum+%E2%80%93+Ask+Dr.+Math%3ACan+Negative+Fractions+Also+Be+Proper+or+Improper%3F&rft.date=2004-03-31&rft.au=Laurel&rft_id=http%3A%2F%2Fmathforum.org%2Flibrary%2Fdrmath%2Fview%2F65128.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGlossary+of+calculus" class="Z3988"> </li> <li id="cite_note-64"><a href="#cite_ref-64">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20120415053421/http://www.necompact.org/ea/gle_support/Math/resources_number/prop_fraction.htm">"New England Compact Math Resources"</a>. Archived from <a rel="nofollow" class="external text" href="http://www.necompact.org/ea/gle_support/Math/resources_number/prop_fraction.htm">the original</a> on 2012-04-15. 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Retrieved 2014-07-29.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=New+comprehensive+mathematics+for+%27O%27+level&rft.place=Cheltenham&rft.pages=5&rft.edition=2nd+ed.%2C+reprinted.&rft.pub=Thornes&rft.date=1986&rft.isbn=978-0-85950-159-0&rft.aulast=Greer&rft.aufirst=A.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DwX2dxeDahAwC%26pg%3DPA5&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGlossary+of+calculus" class="Z3988"> </li> <li id="cite_note-Brook_Taylor_biography,_St._Andrews-66"><a href="#cite_ref-Brook_Taylor_biography,_St._Andrews_66-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www-history.mcs.st-andrews.ac.uk/Biographies/Taylor.html">"Brook Taylor"</a>. History.MCS.St-Andrews.ac.uk. Retrieved May 25, 2018.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=History.MCS.St-Andrews.ac.uk&rft.atitle=Brook+Taylor&rft_id=http%3A%2F%2Fwww-history.mcs.st-andrews.ac.uk%2FBiographies%2FTaylor.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGlossary+of+calculus" class="Z3988"> </li> <li id="cite_note-Brook_Taylor_biography,_Stetson-67"><a href="#cite_ref-Brook_Taylor_biography,_Stetson_67-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20180103003304/http://www2.stetson.edu/~efriedma/periodictable/html/Tl.html">"Brook Taylor"</a>. Stetson.edu. Archived from <a rel="nofollow" class="external text" href="https://www2.stetson.edu/~efriedma/periodictable/html/Tl.html">the original</a> on January 3, 2018. Retrieved May 25, 2018.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Stetson.edu&rft.atitle=Brook+Taylor&rft_id=https%3A%2F%2Fwww2.stetson.edu%2F~efriedma%2Fperiodictable%2Fhtml%2FTl.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGlossary+of+calculus" class="Z3988"> </li> <li id="cite_note-68"><a href="#cite_ref-68">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/BolzanosTheorem.html">"Bolzano's Theorem"</a>. <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Bolzano%27s+Theorem&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FBolzanosTheorem.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGlossary+of+calculus" class="Z3988"> </li> <li id="cite_note-69"><a href="#cite_ref-69">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTaczanowski1978" class="citation journal cs1">Taczanowski, Stefan (October 1978). "On the optimization of some geometric parameters in 14 MeV neutron activation analysis". Nuclear Instruments and Methods. 155 (3): 543–546. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1978NucIM.155..543T">1978NucIM.155..543T</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0029-554X%2878%2990541-4">10.1016/0029-554X(78)90541-4</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nuclear+Instruments+and+Methods&rft.atitle=On+the+optimization+of+some+geometric+parameters+in+14+MeV+neutron+activation+analysis&rft.volume=155&rft.issue=3&rft.pages=543-546&rft.date=1978-10&rft_id=info%3Adoi%2F10.1016%2F0029-554X%2878%2990541-4&rft_id=info%3Abibcode%2F1978NucIM.155..543T&rft.aulast=Taczanowski&rft.aufirst=Stefan&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGlossary+of+calculus" class="Z3988"> </li> <li id="cite_note-70"><a href="#cite_ref-70">^</a> Hazewinkel, Michiel (1994) [1987]. Encyclopaedia of Mathematics (unabridged reprint ed.). Kluwer Academic Publishers / Springer Science & Business Media. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-155608010-4" title="Special:BookSources/978-155608010-4">978-155608010-4</a>.[<a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">page needed</a>] </li> <li id="cite_note-71"><a href="#cite_ref-71">^</a> Ebner, Dieter (2005-07-25). Preparatory Course in Mathematics (PDF) (6 ed.). Department of Physics, University of Konstanz. Archived (PDF) from the original on 2017-07-26. Retrieved 2017-07-26.[<a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">page needed</a>] </li> <li id="cite_note-72"><a href="#cite_ref-72">^</a> Mejlbro, Leif (2010-11-11). Stability, Riemann Surfaces, Conformal Mappings - Complex Functions Theory (PDF) (1 ed.). Ventus Publishing ApS / Bookboon. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-87-7681-702-2" title="Special:BookSources/978-87-7681-702-2">978-87-7681-702-2</a>. Archived (PDF) from the original on 2017-07-26. Retrieved 2017-07-26.[<a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">page needed</a>] </li> <li id="cite_note-73"><a href="#cite_ref-73">^</a> Durán, Mario (2012). Mathematical methods for wave propagation in science and engineering. 1: Fundamentals (1 ed.). Ediciones UC. p. 88. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-956141314-6" title="Special:BookSources/978-956141314-6">978-956141314-6</a>.[<a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">page needed</a>] </li> <li id="cite_note-74"><a href="#cite_ref-74">^</a> Hall, Arthur Graham; Frink, Fred Goodrich (January 1909). "Chapter II. The Acute Angle [14] Inverse trigonometric functions". Written at Ann Arbor, Michigan, USA. Trigonometry. Part I: Plane Trigonometry. New York, USA: Henry Holt and Company / Norwood Press / J. S. Cushing Co. - Berwick & Smith Co., Norwood, Massachusetts, USA. p. 15. Retrieved 2017-08-12. […] α = arcsin m: It is frequently read "arc-sinem" or "anti-sine m," since two mutually inverse functions are said each to be the anti-function of the other. […] A similar symbolic relation holds for the other trigonometric functions. […] This notation is universally used in Europe and is fast gaining ground in this country. A less desirable symbol, α = sin-1m, is still found in English and American texts. The notation α = inv sin m is perhaps better still on account of its general applicability. […] </li> <li id="cite_note-75"><a href="#cite_ref-75">^</a> Klein, Christian Felix (1924) [1902]. Elementarmathematik vom höheren Standpunkt aus: Arithmetik, Algebra, Analysis (in German). 1 (3rd ed.). Berlin: J. Springer. </li> <li id="cite_note-76"><a href="#cite_ref-76">^</a> Klein, Christian Felix (2004) [1932]. Elementary Mathematics from an Advanced Standpoint: Arithmetic, Algebra, Analysis. Translated by Hedrick, E. R.; Noble, C. A. (Translation of 3rd German ed.). Dover Publications, Inc. / The Macmillan Company. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-48643480-3" title="Special:BookSources/978-0-48643480-3">978-0-48643480-3</a>. Retrieved 2017-08-13. </li> <li id="cite_note-77"><a href="#cite_ref-77">^</a> Dörrie, Heinrich (1965). Triumph der Mathematik. Translated by Antin, David. Dover Publications. p. 69. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-61348-2" title="Special:BookSources/978-0-486-61348-2">978-0-486-61348-2</a>. </li> <li id="cite_note-78"><a href="#cite_ref-78">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLay2006" class="citation book cs1">Lay, David C. (2006). <a rel="nofollow" class="external text" href="https://archive.org/details/studyguidetoline0000layd">Linear Algebra and Its Applications</a> (3rd ed.). <a href="/wiki/Addison%E2%80%93Wesley" class="mw-redirect" title="Addison–Wesley">Addison–Wesley</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-321-28713-4" title="Special:BookSources/0-321-28713-4"><bdi>0-321-28713-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebra+and+Its+Applications&rft.edition=3rd&rft.pub=Addison%E2%80%93Wesley&rft.date=2006&rft.isbn=0-321-28713-4&rft.aulast=Lay&rft.aufirst=David+C.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fstudyguidetoline0000layd&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGlossary+of+calculus" class="Z3988"> </li> <li id="cite_note-79"><a href="#cite_ref-79">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStrang2006" class="citation book cs1"><a href="/wiki/Gilbert_Strang" title="Gilbert Strang">Strang, Gilbert</a> (2006). Linear Algebra and Its Applications (4th ed.). <a href="/wiki/Brooks_Cole" class="mw-redirect" title="Brooks Cole">Brooks Cole</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-03-010567-6" title="Special:BookSources/0-03-010567-6"><bdi>0-03-010567-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebra+and+Its+Applications&rft.edition=4th&rft.pub=Brooks+Cole&rft.date=2006&rft.isbn=0-03-010567-6&rft.aulast=Strang&rft.aufirst=Gilbert&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGlossary+of+calculus" class="Z3988"> </li> <li id="cite_note-80"><a href="#cite_ref-80">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAxler2002" class="citation book cs1">Axler, Sheldon (2002). Linear Algebra Done Right (2nd ed.). <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-98258-2" title="Special:BookSources/0-387-98258-2"><bdi>0-387-98258-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebra+Done+Right&rft.edition=2nd&rft.pub=Springer&rft.date=2002&rft.isbn=0-387-98258-2&rft.aulast=Axler&rft.aufirst=Sheldon&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGlossary+of+calculus" class="Z3988"> </li> <li id="cite_note-81"><a href="#cite_ref-81">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMortimer2005" class="citation book cs1">Mortimer, Robert G. (2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=nGoSv5tmATsC">Mathematics for physical chemistry</a> (3rd ed.). <a href="/wiki/Academic_Press" title="Academic Press">Academic Press</a>. p. 9. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-12-508347-5" title="Special:BookSources/0-12-508347-5"><bdi>0-12-508347-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+for+physical+chemistry&rft.pages=9&rft.edition=3rd&rft.pub=Academic+Press&rft.date=2005&rft.isbn=0-12-508347-5&rft.aulast=Mortimer&rft.aufirst=Robert+G.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DnGoSv5tmATsC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGlossary+of+calculus" class="Z3988"> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=nGoSv5tmATsC&pg=PA9">Extract of page 9</a> </li> <li id="cite_note-wolfram-82"><a href="#cite_ref-wolfram_82-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/QuadraticEquation.html">"Quadratic Equation -- from Wolfram MathWorld"</a>. Retrieved January 6, 2013.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Quadratic+Equation+--+from+Wolfram+MathWorld&rft_id=http%3A%2F%2Fmathworld.wolfram.com%2FQuadraticEquation.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGlossary+of+calculus" class="Z3988"> </li> <li id="cite_note-83"><a href="#cite_ref-83">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.bipm.org/en/CGPM/db/20/8/">"Resolution 8 of the CGPM at its 20th Meeting (1995)"</a>. <a href="/wiki/Bureau_International_des_Poids_et_Mesures" class="mw-redirect" title="Bureau International des Poids et Mesures">Bureau International des Poids et Mesures</a>. Retrieved 2014-09-23.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Resolution+8+of+the+CGPM+at+its+20th+Meeting+%281995%29&rft.pub=Bureau+International+des+Poids+et+Mesures&rft_id=http%3A%2F%2Fwww.bipm.org%2Fen%2FCGPM%2Fdb%2F20%2F8%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGlossary+of+calculus" class="Z3988"> </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="Works_cited">Works cited</h3>[<a href="/w/index.php?title=Glossary_of_calculus&action=edit&section=25" title="Edit section: Works cited">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFApostol1967" class="citation cs2"><a href="/wiki/Tom_M._Apostol" title="Tom M. Apostol">Apostol, T</a> (1967), Calculus, Vol. 1 (2nd ed.), Jon Wiley & Sons</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus%2C+Vol.+1&rft.edition=2nd&rft.pub=Jon+Wiley+%26+Sons&rft.date=1967&rft.aulast=Apostol&rft.aufirst=T&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGlossary+of+calculus" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFArbogast1800" class="citation cs2 cs1-prop-foreign-lang-source"><a href="/wiki/Louis_Fran%C3%A7ois_Antoine_Arbogast" title="Louis François Antoine Arbogast">Arbogast, L. F. A.</a> (1800), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=YoPq8uCy5Y8C">Du calcul des derivations</a> [On the calculus of derivatives] (in French), Strasbourg: Levrault, pp. xxiii+404</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Du+calcul+des+derivations&rft.place=Strasbourg&rft.pages=xxiii%2B404&rft.pub=Levrault&rft.date=1800&rft.aulast=Arbogast&rft.aufirst=L.+F.+A.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DYoPq8uCy5Y8C&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGlossary+of+calculus" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFButcher2003" class="citation book cs1"><a href="/wiki/John_C._Butcher" title="John C. Butcher">Butcher, John C.</a> (2003). Numerical Methods for Ordinary Differential Equations. New York: <a href="/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-96758-3" title="Special:BookSources/978-0-471-96758-3"><bdi>978-0-471-96758-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Numerical+Methods+for+Ordinary+Differential+Equations&rft.place=New+York&rft.pub=John+Wiley+%26+Sons&rft.date=2003&rft.isbn=978-0-471-96758-3&rft.aulast=Butcher&rft.aufirst=John+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGlossary+of+calculus" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCraik2005" class="citation cs2">Craik, Alex D. D. (February 2005), "Prehistory of Faà di Bruno's Formula", <a href="/wiki/American_Mathematical_Monthly" class="mw-redirect" title="American Mathematical Monthly">American Mathematical Monthly</a>, 112 (2): 217–234, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F30037410">10.2307/30037410</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/30037410">30037410</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2121322">2121322</a>, <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1088.01008">1088.01008</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Mathematical+Monthly&rft.atitle=Prehistory+of+Fa%C3%A0+di+Bruno%27s+Formula&rft.volume=112&rft.issue=2&rft.pages=217-234&rft.date=2005-02&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A1088.01008%23id-name%3DZbl&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2121322%23id-name%3DMR&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F30037410%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F30037410&rft.aulast=Craik&rft.aufirst=Alex+D.+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGlossary+of+calculus" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHairerNørsettWanner1993" class="citation book cs1">Hairer, Ernst; Nørsett, Syvert Paul; Wanner, Gerhard (1993). Solving ordinary differential equations I: Nonstiff problems. Berlin, New York: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-56670-0" title="Special:BookSources/978-3-540-56670-0"><bdi>978-3-540-56670-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Solving+ordinary+differential+equations+I%3A+Nonstiff+problems&rft.place=Berlin%2C+New+York&rft.pub=Springer-Verlag&rft.date=1993&rft.isbn=978-3-540-56670-0&rft.aulast=Hairer&rft.aufirst=Ernst&rft.au=N%C3%B8rsett%2C+Syvert+Paul&rft.au=Wanner%2C+Gerhard&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGlossary+of+calculus" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJohnson2002" class="citation cs2">Johnson, Warren P. (March 2002), <a rel="nofollow" class="external text" href="https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Johnson217-234.pdf">"The Curious History of Faà di Bruno's Formula"</a> (PDF), <a href="/wiki/American_Mathematical_Monthly" class="mw-redirect" title="American Mathematical Monthly">American Mathematical Monthly</a>, 109 (3): 217–234, <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.109.4135">10.1.1.109.4135</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2695352">10.2307/2695352</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2695352">2695352</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1903577">1903577</a>, <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1024.01010">1024.01010</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Mathematical+Monthly&rft.atitle=The+Curious+History+of+Fa%C3%A0+di+Bruno%27s+Formula&rft.volume=109&rft.issue=3&rft.pages=217-234&rft.date=2002-03&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A1024.01010%23id-name%3DZbl&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2695352%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F2695352&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.109.4135%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1903577%23id-name%3DMR&rft.aulast=Johnson&rft.aufirst=Warren+P.&rft_id=https%3A%2F%2Fwww.maa.org%2Fsites%2Fdefault%2Ffiles%2Fpdf%2Fupload_library%2F22%2FFord%2FJohnson217-234.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGlossary+of+calculus" class="Z3988">.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Glossary_of_calculus&action=edit&section=26" title="Edit section: Notes">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-34"><a href="#cite_ref-34">^</a> The term scalar product is often also used more generally to mean a <a href="/wiki/Symmetric_bilinear_form" title="Symmetric bilinear form">symmetric bilinear form</a>, for example for a <a href="/wiki/Pseudo-Euclidean_space" title="Pseudo-Euclidean space">pseudo-Euclidean space</a>.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] </li> <li id="cite_note-57"><a href="#cite_ref-57">^</a> j is usually used in Engineering contexts where i has other meanings (such as electrical current) </li> </ol></div></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> Antiderivatives are also called general integrals, and sometimes integrals. The latter term is generic, and refers not only to indefinite integrals (antiderivatives), but also to <a href="/wiki/Definite_integral" class="mw-redirect" title="Definite integral">definite integrals</a>. When the word integral is used without additional specification, the reader is supposed to deduce from the context whether it refers to a definite or indefinite integral. Some authors define the indefinite integral of a function as the set of its infinitely many possible antiderivatives. Others define it as an arbitrarily selected element of that set. Wikipedia adopts the latter approach.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] </li> <li id="cite_note-44"><a href="#cite_ref-44">^</a> The symbol J is commonly used instead of the intuitive I in order to avoid confusion with other concepts identified by similar I–like <a href="/wiki/Glyph" title="Glyph">glyphs</a>, e.g. <a href="/wiki/Identity_(mathematics)" title="Identity (mathematics)">identities</a>. </li> </ol></div></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist reflist-lower-alpha"> </div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output 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integral">Daniell integral</a></li> <li><a href="/wiki/Darboux_integral" title="Darboux integral">Darboux integral</a></li> <li><a href="/wiki/Henstock%E2%80%93Kurzweil_integral" title="Henstock–Kurzweil integral">Henstock–Kurzweil integral</a></li> <li><a href="/wiki/Haar_measure" title="Haar measure">Haar integral</a></li> <li><a href="/wiki/Hellinger_integral" title="Hellinger integral">Hellinger integral</a></li> <li><a href="/wiki/Khinchin_integral" title="Khinchin integral">Khinchin integral</a></li> <li><a href="/wiki/Kolmogorov_integral" title="Kolmogorov integral">Kolmogorov integral</a></li> <li><a href="/wiki/Lebesgue%E2%80%93Stieltjes_integration" title="Lebesgue–Stieltjes integration">Lebesgue–Stieltjes integral</a></li> <li><a href="/wiki/Pettis_integral" title="Pettis integral">Pettis integral</a></li> <li><a href="/wiki/Pfeffer_integral" title="Pfeffer integral">Pfeffer integral</a></li> <li><a href="/wiki/Riemann%E2%80%93Stieltjes_integral" title="Riemann–Stieltjes integral">Riemann–Stieltjes integral</a></li> <li><a href="/wiki/Regulated_integral" title="Regulated integral">Regulated integral</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Integration techniques</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Integration_by_substitution" title="Integration by substitution">Substitution</a> <ul><li><a href="/wiki/Trigonometric_substitution" title="Trigonometric substitution">Trigonometric</a></li> <li><a href="/wiki/Euler_substitution" title="Euler substitution">Euler</a></li> <li><a href="/wiki/Weierstrass_substitution" class="mw-redirect" title="Weierstrass substitution">Weierstrass</a></li></ul></li> <li><a href="/wiki/Integration_by_parts" title="Integration by parts">By parts</a></li> <li><a href="/wiki/Integration_by_partial_fractions" class="mw-redirect" title="Integration by partial fractions">Partial fractions</a></li> <li><a href="/wiki/Integration_using_Euler%27s_formula" title="Integration using Euler's formula">Euler's formula</a></li> <li><a href="/wiki/Integral_of_inverse_functions" title="Integral of inverse functions">Inverse functions</a></li> <li><a href="/wiki/Order_of_integration_(calculus)" title="Order of integration (calculus)">Changing order</a></li> <li><a href="/wiki/Integration_by_reduction_formulae" title="Integration by reduction formulae">Reduction formulas</a></li> <li><a href="/wiki/Integration_using_parametric_derivatives" title="Integration using parametric derivatives">Parametric derivatives</a></li> <li><a href="/wiki/Leibniz_integral_rule#Evaluating_definite_integrals" title="Leibniz integral rule">Differentiation under the integral sign</a></li> <li><a href="/wiki/Laplace_transform#Evaluating_improper_integrals" title="Laplace transform">Laplace transform</a></li> <li><a href="/wiki/Contour_integration" title="Contour integration">Contour integration</a></li> <li><a href="/wiki/Laplace%27s_method" title="Laplace's method">Laplace's method</a></li> <li><a href="/wiki/Numerical_integration" title="Numerical integration">Numerical integration</a> <ul><li><a href="/wiki/Simpson%27s_rule" title="Simpson's rule">Simpson's rule</a></li> <li><a href="/wiki/Trapezoidal_rule" title="Trapezoidal rule">Trapezoidal rule</a></li></ul></li> <li><a href="/wiki/Risch_algorithm" title="Risch algorithm">Risch algorithm</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Improper_integral" title="Improper integral">Improper integrals</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Gaussian_integral" title="Gaussian integral">Gaussian integral</a></li> <li><a href="/wiki/Dirichlet_integral" title="Dirichlet integral">Dirichlet integral</a></li> <li>Fermi–Dirac integral <ul><li><a href="/wiki/Complete_Fermi%E2%80%93Dirac_integral" title="Complete Fermi–Dirac integral">complete</a></li> <li><a href="/wiki/Incomplete_Fermi%E2%80%93Dirac_integral" title="Incomplete Fermi–Dirac integral">incomplete</a></li></ul></li> <li><a href="/wiki/Bose%E2%80%93Einstein_integral" class="mw-redirect" title="Bose–Einstein integral">Bose–Einstein integral</a></li> <li><a href="/wiki/Frullani_integral" title="Frullani integral">Frullani integral</a></li> <li><a href="/wiki/Common_integrals_in_quantum_field_theory" title="Common integrals in quantum field theory">Common integrals in quantum field theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Stochastic_integral" class="mw-redirect" title="Stochastic integral">Stochastic integrals</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/It%C3%B4_calculus" title="Itô calculus">Itô integral</a></li> <li><a href="/wiki/Russo%E2%80%93Vallois_integral" title="Russo–Vallois integral">Russo–Vallois integral</a></li> <li><a href="/wiki/Stratonovich_integral" title="Stratonovich integral">Stratonovich integral</a></li> <li><a href="/wiki/Skorokhod_integral" title="Skorokhod integral">Skorokhod integral</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Miscellaneous</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Basel_problem" title="Basel problem">Basel problem</a></li> <li><a href="/wiki/Euler%E2%80%93Maclaurin_formula" title="Euler–Maclaurin formula">Euler–Maclaurin formula</a></li> <li><a href="/wiki/Gabriel%27s_horn" title="Gabriel's horn">Gabriel's horn</a></li> <li><a href="/wiki/Integration_Bee" title="Integration Bee">Integration Bee</a></li> <li><a href="/wiki/Proof_that_22/7_exceeds_%CF%80" title="Proof that 22/7 exceeds π">Proof that 22/7 exceeds π</a></li> <li>Volumes <ul><li><a href="/wiki/Disc_integration" title="Disc integration">Washers</a></li> <li><a href="/wiki/Shell_integration" title="Shell integration">Shells</a></li></ul></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Infinitesimals" style="padding:3px"><table class="nowraplinks mw-collapsible expanded navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Infinitesimals" title="Template:Infinitesimals"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Infinitesimals" title="Template talk:Infinitesimals"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Infinitesimals" title="Special:EditPage/Template:Infinitesimals"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Infinitesimals" style="font-size:114%;margin:0 4em"><a href="/wiki/Infinitesimal" title="Infinitesimal">Infinitesimals</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">History</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Adequality" title="Adequality">Adequality</a></li> <li><a href="/wiki/Leibniz%27s_notation" title="Leibniz's notation">Leibniz's notation</a></li> <li><a href="/wiki/Integral_symbol" title="Integral symbol">Integral symbol</a></li> <li><a href="/wiki/Criticism_of_nonstandard_analysis" title="Criticism of nonstandard analysis">Criticism of nonstandard analysis</a></li> <li><a href="/wiki/The_Analyst" title="The Analyst">The Analyst</a></li> <li><a href="/wiki/The_Method_of_Mechanical_Theorems" title="The Method of Mechanical Theorems">The Method of Mechanical Theorems</a></li> <li><a href="/wiki/Cavalieri%27s_principle" title="Cavalieri's principle">Cavalieri's principle</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="6" style="width:1px;padding:0 0 0 2px"><div><a href="/wiki/File:German_integral.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b5/German_integral.gif/50px-German_integral.gif" decoding="async" width="50" height="92" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b5/German_integral.gif/75px-German_integral.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/b/b5/German_integral.gif 2x" data-file-width="84" data-file-height="155" /></a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related branches</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Nonstandard_analysis" title="Nonstandard analysis">Nonstandard analysis</a></li> <li><a href="/wiki/Nonstandard_calculus" title="Nonstandard calculus">Nonstandard calculus</a></li> <li><a href="/wiki/Internal_set_theory" title="Internal set theory">Internal set theory</a></li> <li><a href="/wiki/Synthetic_differential_geometry" title="Synthetic differential geometry">Synthetic differential geometry</a></li> <li><a href="/wiki/Smooth_infinitesimal_analysis" title="Smooth infinitesimal analysis">Smooth infinitesimal analysis</a></li> <li><a href="/wiki/Constructive_nonstandard_analysis" title="Constructive nonstandard analysis">Constructive nonstandard analysis</a></li> <li><a href="/wiki/Infinitesimal_strain_theory" title="Infinitesimal strain theory">Infinitesimal strain theory (physics)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Formalizations</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Differential_(mathematics)" title="Differential (mathematics)">Differentials</a></li> <li><a href="/wiki/Hyperreal_number" title="Hyperreal number">Hyperreal numbers</a></li> <li><a href="/wiki/Dual_number" title="Dual number">Dual numbers</a></li> <li><a href="/wiki/Surreal_number" title="Surreal number">Surreal numbers</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Individual concepts</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Standard_part_function" title="Standard part function">Standard part function</a></li> <li><a href="/wiki/Transfer_principle" title="Transfer principle">Transfer principle</a></li> <li><a href="/wiki/Hyperinteger" title="Hyperinteger">Hyperinteger</a></li> <li><a href="/wiki/Increment_theorem" title="Increment theorem">Increment theorem</a></li> <li><a href="/wiki/Monad_(nonstandard_analysis)" title="Monad (nonstandard analysis)">Monad</a></li> <li><a href="/wiki/Internal_set" title="Internal set">Internal set</a></li> <li><a href="/wiki/Levi-Civita_field" title="Levi-Civita field">Levi-Civita field</a></li> <li><a href="/wiki/Hyperfinite_set" title="Hyperfinite set">Hyperfinite set</a></li> <li><a href="/wiki/Law_of_continuity" title="Law of continuity">Law of continuity</a></li> <li><a href="/wiki/Overspill" title="Overspill">Overspill</a></li> <li><a href="/wiki/Microcontinuity" title="Microcontinuity">Microcontinuity</a></li> <li><a href="/wiki/Transcendental_law_of_homogeneity" title="Transcendental law of homogeneity">Transcendental law of homogeneity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Mathematicians</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a></li> <li><a href="/wiki/Abraham_Robinson" title="Abraham Robinson">Abraham Robinson</a></li> <li><a href="/wiki/Pierre_de_Fermat" title="Pierre de Fermat">Pierre de Fermat</a></li> <li><a href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Augustin-Louis Cauchy</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Textbooks</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0;font-style:italic;"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Analyse_des_Infiniment_Petits_pour_l%27Intelligence_des_Lignes_Courbes" title="Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes">Analyse des Infiniment Petits</a></li> <li><a href="/wiki/Elementary_Calculus:_An_Infinitesimal_Approach" title="Elementary Calculus: An Infinitesimal Approach">Elementary Calculus</a></li> <li><a href="/wiki/Cours_d%27Analyse" title="Cours d'Analyse">Cours d'Analyse</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Calculus" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Calculus_topics" title="Template:Calculus topics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Calculus_topics" title="Template talk:Calculus topics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Calculus_topics" title="Special:EditPage/Template:Calculus topics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Calculus" style="font-size:114%;margin:0 4em"><a href="/wiki/Calculus" title="Calculus">Calculus</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Precalculus" title="Precalculus">Precalculus</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Binomial_theorem" title="Binomial theorem">Binomial theorem</a></li> <li><a href="/wiki/Concave_function" title="Concave function">Concave function</a></li> <li><a href="/wiki/Continuous_function" title="Continuous function">Continuous function</a></li> <li><a href="/wiki/Factorial" title="Factorial">Factorial</a></li> <li><a href="/wiki/Finite_difference" title="Finite difference">Finite difference</a></li> <li><a href="/wiki/Free_variables_and_bound_variables" title="Free variables and bound variables">Free variables and bound variables</a></li> <li><a href="/wiki/Graph_of_a_function" title="Graph of a function">Graph of a function</a></li> <li><a href="/wiki/Linear_function" title="Linear function">Linear function</a></li> <li><a href="/wiki/Radian" title="Radian">Radian</a></li> <li><a href="/wiki/Rolle%27s_theorem" title="Rolle's theorem">Rolle's theorem</a></li> <li><a href="/wiki/Secant_line" title="Secant line">Secant</a></li> <li><a href="/wiki/Slope" title="Slope">Slope</a></li> <li><a href="/wiki/Tangent" title="Tangent">Tangent</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Limit_(mathematics)" title="Limit (mathematics)">Limits</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Indeterminate_form" title="Indeterminate form">Indeterminate form</a></li> <li><a href="/wiki/Limit_of_a_function" title="Limit of a function">Limit of a function</a> <ul><li><a href="/wiki/One-sided_limit" title="One-sided limit">One-sided limit</a></li></ul></li> <li><a href="/wiki/Limit_of_a_sequence" title="Limit of a sequence">Limit of a sequence</a></li> <li><a href="/wiki/Order_of_approximation" title="Order of approximation">Order of approximation</a></li> <li><a href="/wiki/(%CE%B5,_%CE%B4)-definition_of_limit" class="mw-redirect" title="(ε, δ)-definition of limit">(ε, δ)-definition of limit</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Differential_calculus" title="Differential calculus">Differential calculus</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Derivative" title="Derivative">Derivative</a></li> <li><a href="/wiki/Second_derivative" title="Second derivative">Second derivative</a></li> <li><a href="/wiki/Partial_derivative" title="Partial derivative">Partial derivative</a></li> <li><a href="/wiki/Differential_(mathematics)" title="Differential (mathematics)">Differential</a></li> <li><a href="/wiki/Differential_operator" title="Differential operator">Differential operator</a></li> <li><a href="/wiki/Mean_value_theorem" title="Mean value theorem">Mean value theorem</a></li> <li><a href="/wiki/Notation_for_differentiation" title="Notation for differentiation">Notation</a> <ul><li><a href="/wiki/Leibniz%27s_notation" title="Leibniz's notation">Leibniz's notation</a></li> <li><a href="/wiki/Newton%27s_notation_for_differentiation" class="mw-redirect" title="Newton's notation for differentiation">Newton's notation</a></li></ul></li> <li><a href="/wiki/Differentiation_rules" title="Differentiation rules">Rules of differentiation</a> <ul><li><a href="/wiki/Linearity_of_differentiation" title="Linearity of differentiation">linearity</a></li> <li><a href="/wiki/Power_rule" title="Power rule">Power</a></li> <li><a href="/wiki/Sum_rule_in_differentiation" class="mw-redirect" title="Sum rule in differentiation">Sum</a></li> <li><a href="/wiki/Chain_rule" title="Chain rule">Chain</a></li> <li><a href="/wiki/L%27H%C3%B4pital%27s_rule" title="L'Hôpital's rule">L'Hôpital's</a></li> <li><a href="/wiki/Product_rule" title="Product rule">Product</a> <ul><li><a href="/wiki/General_Leibniz_rule" title="General Leibniz rule">General Leibniz's rule</a></li></ul></li> <li><a href="/wiki/Quotient_rule" title="Quotient rule">Quotient</a></li></ul></li> <li>Other techniques <ul><li><a href="/wiki/Implicit_differentiation" class="mw-redirect" title="Implicit differentiation">Implicit differentiation</a></li> <li><a href="/wiki/Inverse_functions_and_differentiation" class="mw-redirect" title="Inverse functions and differentiation">Inverse functions and differentiation</a></li> <li><a href="/wiki/Logarithmic_derivative" title="Logarithmic derivative">Logarithmic derivative</a></li> <li><a href="/wiki/Related_rates" title="Related rates">Related rates</a></li></ul></li> <li><a href="/wiki/Stationary_point" title="Stationary point">Stationary points</a> <ul><li><a href="/wiki/First_derivative_test" class="mw-redirect" title="First derivative test">First derivative test</a></li> <li><a href="/wiki/Second_derivative_test" class="mw-redirect" title="Second derivative test">Second derivative test</a></li> <li><a href="/wiki/Extreme_value_theorem" title="Extreme value theorem">Extreme value theorem</a></li> <li><a href="/wiki/Maximum_and_minimum" title="Maximum and minimum">Maximum and minimum</a></li></ul></li> <li>Further applications <ul><li><a href="/wiki/Newton%27s_method" title="Newton's method">Newton's method</a></li> <li><a href="/wiki/Taylor%27s_theorem" title="Taylor's theorem">Taylor's theorem</a></li></ul></li> <li><a href="/wiki/Differential_equation" title="Differential equation">Differential equation</a> <ul><li><a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">Ordinary differential equation</a></li> <li><a href="/wiki/Partial_differential_equation" title="Partial differential equation">Partial differential equation</a></li> <li><a href="/wiki/Stochastic_differential_equation" title="Stochastic differential equation">Stochastic differential equation</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Integral_calculus" class="mw-redirect" title="Integral calculus">Integral calculus</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Antiderivative" title="Antiderivative">Antiderivative</a></li> <li><a href="/wiki/Arc_length" title="Arc length">Arc length</a></li> <li><a href="/wiki/Riemann_integral" title="Riemann integral">Riemann integral</a></li> <li><a href="/wiki/Integral#Properties" title="Integral">Basic properties</a></li> <li><a href="/wiki/Constant_of_integration" title="Constant of integration">Constant of integration</a></li> <li><a href="/wiki/Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus">Fundamental theorem of calculus</a> <ul><li><a href="/wiki/Leibniz_integral_rule" title="Leibniz integral rule">Differentiating under the integral sign</a></li></ul></li> <li><a href="/wiki/Integration_by_parts" title="Integration by parts">Integration by parts</a></li> <li><a href="/wiki/Integration_by_substitution" title="Integration by substitution">Integration by substitution</a> <ul><li><a href="/wiki/Trigonometric_substitution" title="Trigonometric substitution">trigonometric</a></li> <li><a href="/wiki/Euler_substitution" title="Euler substitution">Euler</a></li> <li><a href="/wiki/Tangent_half-angle_substitution" title="Tangent half-angle substitution">Tangent half-angle substitution</a></li></ul></li> <li><a href="/wiki/Partial_fractions_in_integration" class="mw-redirect" title="Partial fractions in integration">Partial fractions in integration</a> <ul><li><a href="/wiki/Quadratic_integral" title="Quadratic integral">Quadratic integral</a></li></ul></li> <li><a href="/wiki/Trapezoidal_rule" title="Trapezoidal rule">Trapezoidal rule</a></li> <li>Volumes <ul><li><a href="/wiki/Disc_integration" title="Disc integration">Washer method</a></li> <li><a href="/wiki/Shell_integration" title="Shell integration">Shell method</a></li></ul></li> <li><a href="/wiki/Integral_equation" title="Integral equation">Integral equation</a></li> <li><a href="/wiki/Integro-differential_equation" title="Integro-differential equation">Integro-differential equation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Vector_calculus" title="Vector calculus">Vector calculus</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>Derivatives <ul><li><a href="/wiki/Curl_(mathematics)" title="Curl (mathematics)">Curl</a></li> <li><a href="/wiki/Directional_derivative" title="Directional derivative">Directional derivative</a></li> <li><a href="/wiki/Divergence" title="Divergence">Divergence</a></li> <li><a href="/wiki/Gradient" title="Gradient">Gradient</a></li> <li><a href="/wiki/Laplace_operator" title="Laplace operator">Laplacian</a></li></ul></li> <li>Basic theorems <ul><li><a href="/wiki/Fundamental_Theorem_of_Line_Integrals" class="mw-redirect" title="Fundamental Theorem of Line Integrals">Line integrals</a></li> <li><a href="/wiki/Green%27s_theorem" title="Green's theorem">Green's</a></li> <li><a href="/wiki/Stokes%27_theorem" title="Stokes' theorem">Stokes'</a></li> <li><a href="/wiki/Divergence_theorem" title="Divergence theorem">Gauss'</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Multivariable_calculus" title="Multivariable calculus">Multivariable calculus</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Divergence_theorem" title="Divergence theorem">Divergence theorem</a></li> <li><a href="/wiki/Geometric_calculus" title="Geometric calculus">Geometric</a></li> <li><a href="/wiki/Hessian_matrix" title="Hessian matrix">Hessian matrix</a></li> <li><a href="/wiki/Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian matrix and determinant</a></li> <li><a href="/wiki/Lagrange_multiplier" title="Lagrange multiplier">Lagrange multiplier</a></li> <li><a href="/wiki/Line_integral" title="Line integral">Line integral</a></li> <li><a href="/wiki/Matrix_calculus" title="Matrix calculus">Matrix</a></li> <li><a href="/wiki/Multiple_integral" title="Multiple integral">Multiple integral</a></li> <li><a href="/wiki/Partial_derivative" title="Partial derivative">Partial derivative</a></li> <li><a href="/wiki/Surface_integral" title="Surface integral">Surface integral</a></li> <li><a href="/wiki/Volume_integral" title="Volume integral">Volume integral</a></li> <li>Advanced topics <ul><li><a href="/wiki/Differential_form" title="Differential form">Differential forms</a></li> <li><a href="/wiki/Exterior_derivative" title="Exterior derivative">Exterior derivative</a></li> <li><a href="/wiki/Generalized_Stokes%27_theorem" class="mw-redirect" title="Generalized Stokes' theorem">Generalized Stokes' theorem</a></li> <li><a href="/wiki/Tensor_calculus" class="mw-redirect" title="Tensor calculus">Tensor calculus</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Sequences and series</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Arithmetico-geometric_sequence" title="Arithmetico-geometric sequence">Arithmetico-geometric sequence</a></li> <li>Types of series <ul><li><a href="/wiki/Alternating_series" title="Alternating series">Alternating</a></li> <li><a href="/wiki/Binomial_series" title="Binomial series">Binomial</a></li> <li><a href="/wiki/Fourier_series" title="Fourier series">Fourier</a></li> <li><a href="/wiki/Geometric_series" title="Geometric series">Geometric</a></li> <li><a href="/wiki/Harmonic_series_(mathematics)" title="Harmonic series (mathematics)">Harmonic</a></li> <li><a href="/wiki/Infinite_series" class="mw-redirect" title="Infinite series">Infinite</a></li> <li><a href="/wiki/Power_series" title="Power series">Power</a> <ul><li><a href="/wiki/Maclaurin_series" class="mw-redirect" title="Maclaurin series">Maclaurin</a></li> <li><a href="/wiki/Taylor_series" title="Taylor series">Taylor</a></li></ul></li> <li><a href="/wiki/Telescoping_series" title="Telescoping series">Telescoping</a></li></ul></li> <li>Tests of convergence <ul><li><a href="/wiki/Abel%27s_test" title="Abel's test">Abel's</a></li> <li><a href="/wiki/Alternating_series_test" title="Alternating series test">Alternating series</a></li> <li><a href="/wiki/Cauchy_condensation_test" title="Cauchy condensation test">Cauchy condensation</a></li> <li><a href="/wiki/Direct_comparison_test" title="Direct comparison test">Direct comparison</a></li> <li><a href="/wiki/Dirichlet%27s_test" title="Dirichlet's test">Dirichlet's</a></li> <li><a href="/wiki/Integral_test_for_convergence" title="Integral test for convergence">Integral</a></li> <li><a href="/wiki/Limit_comparison_test" title="Limit comparison test">Limit comparison</a></li> <li><a href="/wiki/Ratio_test" title="Ratio test">Ratio</a></li> <li><a href="/wiki/Root_test" title="Root test">Root</a></li> <li><a href="/wiki/Term_test" class="mw-redirect" title="Term test">Term</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Special functions and numbers</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bernoulli_number" title="Bernoulli number">Bernoulli numbers</a></li> <li><a href="/wiki/E_(mathematical_constant)" title="E (mathematical constant)">e (mathematical constant)</a></li> <li><a href="/wiki/Exponential_function" title="Exponential function">Exponential function</a></li> <li><a href="/wiki/Natural_logarithm" title="Natural logarithm">Natural logarithm</a></li> <li><a href="/wiki/Stirling%27s_approximation" title="Stirling's approximation">Stirling's approximation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/History_of_calculus" title="History of calculus">History of calculus</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Adequality" title="Adequality">Adequality</a></li> <li><a href="/wiki/Brook_Taylor" title="Brook Taylor">Brook Taylor</a></li> <li><a href="/wiki/Colin_Maclaurin" title="Colin Maclaurin">Colin Maclaurin</a></li> <li><a href="/wiki/Generality_of_algebra" title="Generality of algebra">Generality of algebra</a></li> <li><a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a></li> <li><a href="/wiki/Infinitesimal" title="Infinitesimal">Infinitesimal</a></li> <li><a href="/wiki/Infinitesimal_calculus" class="mw-redirect" title="Infinitesimal calculus">Infinitesimal calculus</a></li> <li><a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a></li> <li><a href="/wiki/Fluxion" title="Fluxion">Fluxion</a></li> <li><a href="/wiki/Law_of_Continuity" class="mw-redirect" title="Law of Continuity">Law of Continuity</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a></li> <li><a href="/wiki/Method_of_Fluxions" title="Method of Fluxions">Method of Fluxions</a></li> <li><a href="/wiki/The_Method_of_Mechanical_Theorems" title="The Method of Mechanical Theorems">The Method of Mechanical Theorems</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Lists</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Integrals" scope="row" class="navbox-group" style="width:1%;text-align:left"><a href="/wiki/Lists_of_integrals" title="Lists of integrals">Integrals</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_integrals_of_rational_functions" title="List of integrals of rational functions">rational functions</a></li> <li><a href="/wiki/List_of_integrals_of_irrational_functions" title="List of integrals of irrational functions">irrational functions</a></li> <li><a href="/wiki/List_of_integrals_of_exponential_functions" title="List of integrals of exponential functions">exponential functions</a></li> <li><a href="/wiki/List_of_integrals_of_logarithmic_functions" title="List of integrals of logarithmic functions">logarithmic functions</a></li> <li><a href="/wiki/List_of_integrals_of_hyperbolic_functions" title="List of integrals of hyperbolic functions">hyperbolic functions</a> <ul><li><a href="/wiki/List_of_integrals_of_inverse_hyperbolic_functions" title="List of integrals of inverse hyperbolic functions">inverse</a></li></ul></li> <li><a href="/wiki/List_of_integrals_of_trigonometric_functions" title="List of integrals of trigonometric functions">trigonometric functions</a> <ul><li><a href="/wiki/List_of_integrals_of_inverse_trigonometric_functions" title="List of integrals of inverse trigonometric functions">inverse</a></li> <li><a href="/wiki/Integral_of_the_secant_function" title="Integral of the secant function">Secant</a></li> <li><a href="/wiki/Integral_of_secant_cubed" title="Integral of secant cubed">Secant cubed</a></li></ul></li></ul> </div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_limits" title="List of limits">List of limits</a></li> <li><a href="/wiki/Differentiation_rules" title="Differentiation rules">List of derivatives</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Miscellaneous topics</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>Complex calculus <ul><li><a href="/wiki/Contour_integral" class="mw-redirect" title="Contour integral">Contour integral</a></li></ul></li> <li>Differential geometry <ul><li><a href="/wiki/Manifold" title="Manifold">Manifold</a></li> <li><a href="/wiki/Curvature" title="Curvature">Curvature</a></li> <li><a href="/wiki/Differential_geometry_of_curves" class="mw-redirect" title="Differential geometry of curves">of curves</a></li> <li><a href="/wiki/Differential_geometry_of_surfaces" title="Differential geometry of surfaces">of surfaces</a></li> <li><a href="/wiki/Tensor" title="Tensor">Tensor</a></li></ul></li> <li><a href="/wiki/Euler%E2%80%93Maclaurin_formula" title="Euler–Maclaurin formula">Euler–Maclaurin formula</a></li> <li><a href="/wiki/Gabriel%27s_horn" title="Gabriel's horn">Gabriel's horn</a></li> <li><a href="/wiki/Integration_Bee" title="Integration Bee">Integration Bee</a></li> <li><a href="/wiki/Proof_that_22/7_exceeds_%CF%80" title="Proof that 22/7 exceeds π">Proof that 22/7 exceeds π</a></li> <li><a href="/wiki/Regiomontanus%27_angle_maximization_problem" title="Regiomontanus' angle maximization problem">Regiomontanus' angle maximization problem</a></li> <li><a href="/wiki/Steinmetz_solid" title="Steinmetz solid">Steinmetz solid</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" 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