Trajtimi klasik i tensorëve

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aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Trajtimi klasik i tensorëve</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 60 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-60" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">60 gjuhë</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D9%88%D8%AA%D8%B1" title="موتر – arabisht" lang="ar" hreflang="ar" data-title="موتر" data-language-autonym="العربية" data-language-local-name="arabisht" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Tenzor" title="Tenzor – azerbajxhanisht" lang="az" hreflang="az" data-title="Tenzor" data-language-autonym="Azərbaycanca" data-language-local-name="azerbajxhanisht" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BD%D0%B7%D0%BE%D1%80" title="Тензор – bullgarisht" lang="bg" hreflang="bg" data-title="Тензор" data-language-autonym="Български" data-language-local-name="bullgarisht" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%9F%E0%A7%87%E0%A6%A8%E0%A7%8D%E0%A6%B8%E0%A6%B0" title="টেন্সর – bengalisht" lang="bn" hreflang="bn" data-title="টেন্সর" data-language-autonym="বাংলা" data-language-local-name="bengalisht" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Tensor" title="Tensor – katalonisht" lang="ca" hreflang="ca" data-title="Tensor" data-language-autonym="Català" data-language-local-name="katalonisht" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Tenzor" title="Tenzor – çekisht" lang="cs" hreflang="cs" data-title="Tenzor" data-language-autonym="Čeština" data-language-local-name="çekisht" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BD%D0%B7%D0%BE%D1%80" title="Тензор – çuvashisht" lang="cv" hreflang="cv" data-title="Тензор" data-language-autonym="Чӑвашла" data-language-local-name="çuvashisht" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Tensor" title="Tensor – gjermanisht" lang="de" hreflang="de" data-title="Tensor" data-language-autonym="Deutsch" data-language-local-name="gjermanisht" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A4%CE%B1%CE%BD%CF%85%CF%83%CF%84%CE%AE%CF%82" title="Τανυστής – greqisht" lang="el" hreflang="el" data-title="Τανυστής" data-language-autonym="Ελληνικά" data-language-local-name="greqisht" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Tensor" title="Tensor – anglisht" lang="en" hreflang="en" data-title="Tensor" data-language-autonym="English" data-language-local-name="anglisht" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Tensoro" title="Tensoro – esperanto" lang="eo" hreflang="eo" data-title="Tensoro" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Tensor" title="Tensor – spanjisht" lang="es" hreflang="es" data-title="Tensor" data-language-autonym="Español" data-language-local-name="spanjisht" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Tensor" title="Tensor – estonisht" lang="et" hreflang="et" data-title="Tensor" data-language-autonym="Eesti" data-language-local-name="estonisht" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Tentsore" title="Tentsore – baskisht" lang="eu" hreflang="eu" data-title="Tentsore" data-language-autonym="Euskara" data-language-local-name="baskisht" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D8%A7%D9%86%D8%B3%D9%88%D8%B1" title="تانسور – persisht" lang="fa" hreflang="fa" data-title="تانسور" data-language-autonym="فارسی" data-language-local-name="persisht" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Tensori" title="Tensori – finlandisht" lang="fi" hreflang="fi" data-title="Tensori" data-language-autonym="Suomi" data-language-local-name="finlandisht" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Tenseur" title="Tenseur – frëngjisht" lang="fr" hreflang="fr" data-title="Tenseur" data-language-autonym="Français" data-language-local-name="frëngjisht" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Tensor" title="Tensor – galicisht" lang="gl" hreflang="gl" data-title="Tensor" data-language-autonym="Galego" data-language-local-name="galicisht" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%98%D7%A0%D7%96%D7%95%D7%A8" title="טנזור – hebraisht" lang="he" hreflang="he" data-title="טנזור" data-language-autonym="עברית" data-language-local-name="hebraisht" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%A6%E0%A4%BF%E0%A4%B6" title="प्रदिश – indisht" lang="hi" hreflang="hi" data-title="प्रदिश" data-language-autonym="हिन्दी" data-language-local-name="indisht" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Tenzor" title="Tenzor – kroatisht" lang="hr" hreflang="hr" data-title="Tenzor" data-language-autonym="Hrvatski" data-language-local-name="kroatisht" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Tenzor" title="Tenzor – hungarisht" lang="hu" hreflang="hu" data-title="Tenzor" data-language-autonym="Magyar" data-language-local-name="hungarisht" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%8F%D5%A5%D5%B6%D5%A6%D5%B8%D6%80" title="Տենզոր – armenisht" lang="hy" hreflang="hy" data-title="Տենզոր" data-language-autonym="Հայերեն" data-language-local-name="armenisht" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Tensor" title="Tensor – indonezisht" lang="id" hreflang="id" data-title="Tensor" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonezisht" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Tensore" title="Tensore – italisht" lang="it" hreflang="it" data-title="Tensore" data-language-autonym="Italiano" data-language-local-name="italisht" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%86%E3%83%B3%E3%82%BD%E3%83%AB" title="テンソル – japonisht" lang="ja" hreflang="ja" data-title="テンソル" data-language-autonym="日本語" data-language-local-name="japonisht" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BD%D0%B7%D0%BE%D1%80" title="Тензор – kazakisht" lang="kk" hreflang="kk" data-title="Тензор" data-language-autonym="Қазақша" data-language-local-name="kazakisht" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%85%90%EC%84%9C" title="텐서 – koreanisht" lang="ko" hreflang="ko" data-title="텐서" data-language-autonym="한국어" data-language-local-name="koreanisht" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BD%D0%B7%D0%BE%D1%80" title="Тензор – kirgizisht" lang="ky" hreflang="ky" data-title="Тензор" data-language-autonym="Кыргызча" data-language-local-name="kirgizisht" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Tensor_(mathematica)" title="Tensor (mathematica) – latinisht" lang="la" hreflang="la" data-title="Tensor (mathematica)" data-language-autonym="Latina" data-language-local-name="latinisht" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Tenzorius" title="Tenzorius – lituanisht" lang="lt" hreflang="lt" data-title="Tenzorius" data-language-autonym="Lietuvių" data-language-local-name="lituanisht" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Tenzors" title="Tenzors – letonisht" lang="lv" hreflang="lv" data-title="Tenzors" data-language-autonym="Latviešu" data-language-local-name="letonisht" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BD%D0%B7%D0%BE%D1%80" title="Тензор – maqedonisht" lang="mk" hreflang="mk" data-title="Тензор" data-language-autonym="Македонски" data-language-local-name="maqedonisht" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%AA%E0%B5%8D%E0%B4%B0%E0%B4%A6%E0%B4%BF%E0%B4%B6%E0%B4%82" title="പ്രദിശം – malajalamisht" lang="ml" hreflang="ml" data-title="പ്രദിശം" data-language-autonym="മലയാളം" data-language-local-name="malajalamisht" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Tensor" title="Tensor – malajisht" lang="ms" hreflang="ms" data-title="Tensor" data-language-autonym="Bahasa Melayu" data-language-local-name="malajisht" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%90%E1%80%AC%E1%80%A1%E1%80%AF%E1%80%B6" title="တာအုံ – birmanisht" lang="my" hreflang="my" data-title="တာအုံ" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="birmanisht" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Tensor" title="Tensor – holandisht" lang="nl" hreflang="nl" data-title="Tensor" data-language-autonym="Nederlands" data-language-local-name="holandisht" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Tensor" title="Tensor – norvegjishte nynorsk" lang="nn" hreflang="nn" data-title="Tensor" data-language-autonym="Norsk nynorsk" data-language-local-name="norvegjishte nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Tensor" title="Tensor – norvegjishte letrare" lang="nb" hreflang="nb" data-title="Tensor" data-language-autonym="Norsk bokmål" data-language-local-name="norvegjishte letrare" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%9F%E0%A9%88%E0%A8%82%E0%A8%B8%E0%A8%B0" title="ਟੈਂਸਰ – punxhabisht" lang="pa" hreflang="pa" data-title="ਟੈਂਸਰ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="punxhabisht" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Tensor" title="Tensor – polonisht" lang="pl" hreflang="pl" data-title="Tensor" data-language-autonym="Polski" data-language-local-name="polonisht" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Tensor" title="Tensor – portugalisht" lang="pt" hreflang="pt" data-title="Tensor" data-language-autonym="Português" data-language-local-name="portugalisht" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Tensor" title="Tensor – rumanisht" lang="ro" hreflang="ro" data-title="Tensor" data-language-autonym="Română" data-language-local-name="rumanisht" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BD%D0%B7%D0%BE%D1%80" title="Тензор – rusisht" lang="ru" hreflang="ru" data-title="Тензор" data-language-autonym="Русский" data-language-local-name="rusisht" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Tenzor" title="Tenzor – serbo-kroatisht" lang="sh" hreflang="sh" data-title="Tenzor" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbo-kroatisht" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Tensor" title="Tensor – Simple English" lang="en-simple" hreflang="en-simple" data-title="Tensor" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Tenzor" title="Tenzor – sllovakisht" lang="sk" hreflang="sk" data-title="Tenzor" data-language-autonym="Slovenčina" data-language-local-name="sllovakisht" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Tenzor" title="Tenzor – sllovenisht" lang="sl" hreflang="sl" data-title="Tenzor" data-language-autonym="Slovenščina" data-language-local-name="sllovenisht" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BD%D0%B7%D0%BE%D1%80" title="Тензор – serbisht" lang="sr" hreflang="sr" data-title="Тензор" data-language-autonym="Српски / srpski" data-language-local-name="serbisht" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Tensor" title="Tensor – suedisht" lang="sv" hreflang="sv" data-title="Tensor" data-language-autonym="Svenska" data-language-local-name="suedisht" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-szl mw-list-item"><a href="https://szl.wikipedia.org/wiki/Tensor" title="Tensor – Silesian" lang="szl" hreflang="szl" data-title="Tensor" data-language-autonym="Ślůnski" data-language-local-name="Silesian" class="interlanguage-link-target"><span>Ślůnski</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Tensor" title="Tensor – Tagalog" lang="tl" hreflang="tl" data-title="Tensor" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Tens%C3%B6r" title="Tensör – turqisht" lang="tr" hreflang="tr" data-title="Tensör" data-language-autonym="Türkçe" data-language-local-name="turqisht" 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Tensorët na lejojnë që të shprehim <a href="/w/index.php?title=Ligjet_fizike&action=edit&redlink=1" class="new" title="Ligjet fizike (nuk është shkruar akoma)">ligjet fizike</a> në një formë që është e aplikueshme në çdo <a href="/w/index.php?title=Sistem_kordinativ&action=edit&redlink=1" class="new" title="Sistem kordinativ (nuk është shkruar akoma)">sistem kordinativ</a>. Për këtë arsye, ato përdoren shumë në <a href="/w/index.php?title=Mekanik%C3%ABn_e_vazhduar&action=edit&redlink=1" class="new" title="Mekanikën e vazhduar (nuk është shkruar akoma)">mekanikën e vazhduar</a> dhe në <a href="/wiki/Teorin%C3%AB_e_relativitetit" class="mw-redirect" title="Teorinë e relativitetit">teorinë e relativitetit</a>. </p><p>Një tensor është një transformim multi-dimensional <a href="/w/index.php?title=Invariant_(matematika)&action=edit&redlink=1" class="new" title="Invariant (matematika) (nuk është shkruar akoma)">invariant</a> , që merr forma të ndryshme nga një sistem kordinativ në një tjetër. Ai merr formën: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{\left[j_{1},j_{2},j_{3},...j_{m}\right]}^{\left[i_{1},i_{2},i_{3},...i_{n}\right]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mrow> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mrow> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\left[j_{1},j_{2},j_{3},...j_{m}\right]}^{\left[i_{1},i_{2},i_{3},...i_{n}\right]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/266718dd5b5bfef0e3f85d053833d38f943b4a13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:11.789ex; height:4.176ex;" alt="{\displaystyle T_{\left[j_{1},j_{2},j_{3},...j_{m}\right]}^{\left[i_{1},i_{2},i_{3},...i_{n}\right]}}"></span></dd></dl> <p>Sistemi i ri kordinativ paraqitet nga 'me vizë'(<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {x}}^{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {x}}^{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78d5efe788abd6350c79b617bfc418dc53f67850" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.129ex; height:2.676ex;" alt="{\displaystyle {\bar {x}}^{i}}"></span>), dhe sistemi i vjetër kordinativ është pa vizë(<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be8cd88951c7c0e3b181f956531b0a878bbed203" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.129ex; height:2.676ex;" alt="{\displaystyle x^{i}}"></span>). </p><p>Indekset e larta [<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i_{1},i_{2},i_{3},...i_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i_{1},i_{2},i_{3},...i_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1243135bde1b3567736236ff9fdf272bbe9369af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.795ex; height:2.509ex;" alt="{\displaystyle i_{1},i_{2},i_{3},...i_{n}}"></span>] janë komponent <a href="/w/index.php?title=Kovarianca_dhe_kontravarianca_e_vektor%C3%ABve&action=edit&redlink=1" class="new" title="Kovarianca dhe kontravarianca e vektorëve (nuk është shkruar akoma)">kontravariantë</a>, dhe indekset e poshtme [<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j_{1},j_{2},j_{3},...j_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j_{1},j_{2},j_{3},...j_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc2d4d08e2a6d8353881964df511ebdb58e96f95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:14.444ex; height:2.509ex;" alt="{\displaystyle j_{1},j_{2},j_{3},...j_{n}}"></span>] janë komponentët <a href="/w/index.php?title=Kovarianca_dhe_kontravarianca_e_vektor%C3%ABve&action=edit&redlink=1" class="new" title="Kovarianca dhe kontravarianca e vektorëve (nuk është shkruar akoma)">kovariantë</a>. </p><p><br /> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Tensorët_kontravariantë_dhe_kovariantë"><span id="Tensor.C3.ABt_kontravariant.C3.AB_dhe_kovariant.C3.AB"></span>Tensorët kontravariantë dhe kovariantë</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Trajtimi_klasik_i_tensor%C3%ABve&veaction=edit&section=1" title="Redakto pjesën: Tensorët kontravariantë dhe kovariantë" class="mw-editsection-visualeditor"><span>Redakto</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Trajtimi_klasik_i_tensor%C3%ABve&action=edit&section=1" title="Edit section's source code: Tensorët kontravariantë dhe kovariantë"><span>Redakto nëpërmjet kodit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Një tensor kontravariant i rendit të 1(<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T^{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T^{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2620595d032ff3ee5c2f60aceea0a78817949a09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.52ex; height:2.676ex;" alt="{\displaystyle T^{i}}"></span>) përcaktohet si: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {T}}^{i}=T^{r}{\frac {\partial {\bar {x}}^{i}}{\partial x^{r}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>T</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {T}}^{i}=T^{r}{\frac {\partial {\bar {x}}^{i}}{\partial x^{r}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54dce9735db5cfbf591542f90ee7bd2bab246f35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.47ex; height:5.843ex;" alt="{\displaystyle {\bar {T}}^{i}=T^{r}{\frac {\partial {\bar {x}}^{i}}{\partial x^{r}}}.}"></span></dd></dl> <p>Një tensor kovariant i rendit të 1(<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8dd1c50cb9436474f83624c3f679ccf3eebbfef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.157ex; height:2.509ex;" alt="{\displaystyle T_{i}}"></span>) përcaktohet si: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {T}}_{i}=T_{r}{\frac {\partial x^{r}}{\partial {\bar {x}}^{i}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>T</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msup> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {T}}_{i}=T_{r}{\frac {\partial x^{r}}{\partial {\bar {x}}^{i}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5db285dff1c1bb5042f729f0a82a10911775029d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:13.108ex; height:5.676ex;" alt="{\displaystyle {\bar {T}}_{i}=T_{r}{\frac {\partial x^{r}}{\partial {\bar {x}}^{i}}}.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Tensorët_e_pergjithshëm"><span id="Tensor.C3.ABt_e_pergjithsh.C3.ABm"></span>Tensorët e pergjithshëm</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Trajtimi_klasik_i_tensor%C3%ABve&veaction=edit&section=2" title="Redakto pjesën: Tensorët e pergjithshëm" class="mw-editsection-visualeditor"><span>Redakto</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Trajtimi_klasik_i_tensor%C3%ABve&action=edit&section=2" title="Edit section's source code: Tensorët e pergjithshëm"><span>Redakto nëpërmjet kodit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Nje tensor i pergjithshem më shumë-rendor është thjesht një <a href="/w/index.php?title=Prodhimi_tensorial&action=edit&redlink=1" class="new" title="Prodhimi tensorial (nuk është shkruar akoma)">prodhim tensorial</a> i tensorëve të rendit të parë: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{\left[j_{1},j_{2},...j_{q}\right]}^{\left[i_{1},i_{2},...i_{p}\right]}=T^{i_{1}}\otimes T^{i_{2}}...\otimes T^{i_{p}}\otimes T_{j_{1}}\otimes T_{j_{2}}...\otimes T_{j_{q}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mrow> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mrow> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> </mrow> </msubsup> <mo>=</mo> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> <mo>⊗</mo> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msup> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>⊗</mo> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mrow> </msup> <mo>⊗</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo>⊗</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>⊗</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\left[j_{1},j_{2},...j_{q}\right]}^{\left[i_{1},i_{2},...i_{p}\right]}=T^{i_{1}}\otimes T^{i_{2}}...\otimes T^{i_{p}}\otimes T_{j_{1}}\otimes T_{j_{2}}...\otimes T_{j_{q}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cabd7423c048e3c5181008c60229bf18fd06203f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:50.462ex; height:4.843ex;" alt="{\displaystyle T_{\left[j_{1},j_{2},...j_{q}\right]}^{\left[i_{1},i_{2},...i_{p}\right]}=T^{i_{1}}\otimes T^{i_{2}}...\otimes T^{i_{p}}\otimes T_{j_{1}}\otimes T_{j_{2}}...\otimes T_{j_{q}}}"></span></dd></dl> <p>e tillë që: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {T}}_{\left[j_{1},j_{2},...j_{q}\right]}^{\left[i_{1},i_{2},...i_{p}\right]}=T_{\left[s_{1},s_{2},...s_{q}\right]}^{\left[r_{1},r_{2},...r_{p}\right]}{\frac {\partial {\bar {x}}^{i_{1}}}{\partial x^{r_{1}}}}{\frac {\partial {\bar {x}}^{i_{2}}}{\partial x^{r_{2}}}}...{\frac {\partial {\bar {x}}^{i_{p}}}{\partial x^{r_{p}}}}{\frac {\partial x^{s_{1}}}{\partial {\bar {x}}^{j_{1}}}}{\frac {\partial x^{s_{2}}}{\partial {\bar {x}}^{j_{2}}}}...{\frac {\partial x^{s_{q}}}{\partial {\bar {x}}^{j_{q}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>T</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mrow> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mrow> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mrow> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msup> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> 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mathvariant="normal">∂</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msup> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> </mrow> </msup> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {T}}_{\left[j_{1},j_{2},...j_{q}\right]}^{\left[i_{1},i_{2},...i_{p}\right]}=T_{\left[s_{1},s_{2},...s_{q}\right]}^{\left[r_{1},r_{2},...r_{p}\right]}{\frac {\partial {\bar {x}}^{i_{1}}}{\partial x^{r_{1}}}}{\frac {\partial {\bar {x}}^{i_{2}}}{\partial x^{r_{2}}}}...{\frac {\partial {\bar {x}}^{i_{p}}}{\partial x^{r_{p}}}}{\frac {\partial x^{s_{1}}}{\partial {\bar {x}}^{j_{1}}}}{\frac {\partial x^{s_{2}}}{\partial {\bar {x}}^{j_{2}}}}...{\frac {\partial x^{s_{q}}}{\partial {\bar {x}}^{j_{q}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75de8980bd933eba054a6ad815f6829d41861400" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:61.739ex; height:6.176ex;" alt="{\displaystyle {\bar {T}}_{\left[j_{1},j_{2},...j_{q}\right]}^{\left[i_{1},i_{2},...i_{p}\right]}=T_{\left[s_{1},s_{2},...s_{q}\right]}^{\left[r_{1},r_{2},...r_{p}\right]}{\frac {\partial {\bar {x}}^{i_{1}}}{\partial x^{r_{1}}}}{\frac {\partial {\bar {x}}^{i_{2}}}{\partial x^{r_{2}}}}...{\frac {\partial {\bar {x}}^{i_{p}}}{\partial x^{r_{p}}}}{\frac {\partial x^{s_{1}}}{\partial {\bar {x}}^{j_{1}}}}{\frac {\partial x^{s_{2}}}{\partial {\bar {x}}^{j_{2}}}}...{\frac {\partial x^{s_{q}}}{\partial {\bar {x}}^{j_{q}}}}.}"></span></dd></dl> <p>Kjo zakonisht quhet <b>ligji i transformimit tensorial</b>. </p> <div class="mw-heading mw-heading2"><h2 id="Shikoni_gjithashtu">Shikoni gjithashtu</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Trajtimi_klasik_i_tensor%C3%ABve&veaction=edit&section=3" title="Redakto pjesën: Shikoni gjithashtu" class="mw-editsection-visualeditor"><span>Redakto</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Trajtimi_klasik_i_tensor%C3%ABve&action=edit&section=3" title="Edit section's source code: Shikoni gjithashtu"><span>Redakto nëpërmjet kodit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/w/index.php?title=Derivati_i_tensorit&action=edit&redlink=1" class="new" title="Derivati i tensorit (nuk është shkruar akoma)">Derivati i tensorit</a></li> <li><a href="/w/index.php?title=Diferencimi_absolut&action=edit&redlink=1" class="new" title="Diferencimi absolut (nuk është shkruar akoma)">Diferencimi absolut</a></li> <li><a href="/w/index.php?title=Kurbatura&action=edit&redlink=1" class="new" title="Kurbatura (nuk është shkruar akoma)">Kurbatura</a></li> <li><a href="/w/index.php?title=Gjeometria_Rimaniane&action=edit&redlink=1" class="new" title="Gjeometria Rimaniane (nuk është shkruar akoma)">Gjeometria Rimaniane</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Lexime_të_mëtejshme"><span id="Lexime_t.C3.AB_m.C3.ABtejshme"></span>Lexime të mëtejshme</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Trajtimi_klasik_i_tensor%C3%ABve&veaction=edit&section=4" title="Redakto pjesën: Lexime të mëtejshme" class="mw-editsection-visualeditor"><span>Redakto</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Trajtimi_klasik_i_tensor%C3%ABve&action=edit&section=4" title="Edit section's source code: Lexime të mëtejshme"><span>Redakto nëpërmjet kodit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Schaum's Outline of Tensor Calculus</li> <li>Synge and Schild, Tensor Calculus, Toronto Press: Toronto, 1949</li></ul> <p><br /> </p></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Marrë nga "<a dir="ltr" href="https://sq.wikipedia.org/w/index.php?title=Trajtimi_klasik_i_tensorëve&oldid=1423553">https://sq.wikipedia.org/w/index.php?title=Trajtimi_klasik_i_tensorëve&oldid=1423553</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Speciale:Kategorit%C3%AB" title="Speciale:Kategoritë">Kategoritë</a>: <ul><li><a href="/wiki/Kategoria:Tensor%C3%ABt" title="Kategoria:Tensorët">Tensorët</a></li><li><a href="/wiki/Kategoria:Fizik%C3%AB" title="Kategoria:Fizikë">Fizikë</a></li><li><a href="/wiki/Kategoria:Matematik%C3%AB" 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