Differential form - Wikipedia

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class="vector-toc-list"> <li id="toc-Integration_and_orientation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Integration_and_orientation"> <div class="vector-toc-text"> 2.1 Integration and orientation </div> </a> <ul id="toc-Integration_and_orientation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Multi-index_notation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Multi-index_notation"> <div class="vector-toc-text"> 2.2 Multi-index notation </div> </a> <ul id="toc-Multi-index_notation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_exterior_derivative" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_exterior_derivative"> <div class="vector-toc-text"> 2.3 The exterior derivative </div> </a> <ul id="toc-The_exterior_derivative-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Differential_calculus" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Differential_calculus"> <div class="vector-toc-text"> 2.4 Differential calculus </div> </a> <ul id="toc-Differential_calculus-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Intrinsic_definitions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Intrinsic_definitions"> <div class="vector-toc-text"> 3 Intrinsic definitions </div> </a> <ul id="toc-Intrinsic_definitions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Operations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Operations"> <div class="vector-toc-text"> 4 Operations </div> </a> <button aria-controls="toc-Operations-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Operations subsection </button> <ul id="toc-Operations-sublist" class="vector-toc-list"> <li id="toc-Exterior_product" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Exterior_product"> <div class="vector-toc-text"> 4.1 Exterior product </div> </a> <ul id="toc-Exterior_product-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Riemannian_manifold" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Riemannian_manifold"> <div class="vector-toc-text"> 4.2 Riemannian manifold </div> </a> <ul id="toc-Riemannian_manifold-sublist" class="vector-toc-list"> <li id="toc-Vector_field_structures" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Vector_field_structures"> <div class="vector-toc-text"> 4.2.1 Vector field structures </div> </a> <ul id="toc-Vector_field_structures-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Exterior_differential_complex" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Exterior_differential_complex"> <div class="vector-toc-text"> 4.3 Exterior differential complex </div> </a> <ul id="toc-Exterior_differential_complex-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Pullback" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Pullback"> <div class="vector-toc-text"> 5 Pullback </div> </a> <ul id="toc-Pullback-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Integration" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Integration"> <div class="vector-toc-text"> 6 Integration </div> </a> <button aria-controls="toc-Integration-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Integration subsection </button> <ul id="toc-Integration-sublist" class="vector-toc-list"> <li id="toc-Integration_on_Euclidean_space" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Integration_on_Euclidean_space"> <div class="vector-toc-text"> 6.1 Integration on Euclidean space </div> </a> <ul id="toc-Integration_on_Euclidean_space-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Integration_over_chains" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Integration_over_chains"> <div class="vector-toc-text"> 6.2 Integration over chains </div> </a> <ul id="toc-Integration_over_chains-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Integration_using_partitions_of_unity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Integration_using_partitions_of_unity"> <div class="vector-toc-text"> 6.3 Integration using partitions of unity </div> </a> <ul id="toc-Integration_using_partitions_of_unity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Integration_along_fibers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Integration_along_fibers"> <div class="vector-toc-text"> 6.4 Integration along fibers </div> </a> <ul id="toc-Integration_along_fibers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Stokes's_theorem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Stokes's_theorem"> <div class="vector-toc-text"> 6.5 Stokes's theorem </div> </a> <ul id="toc-Stokes's_theorem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relation_with_measures" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relation_with_measures"> <div class="vector-toc-text"> 6.6 Relation with measures </div> </a> <ul id="toc-Relation_with_measures-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Currents" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Currents"> <div class="vector-toc-text"> 6.7 Currents </div> </a> <ul id="toc-Currents-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Applications_in_physics" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applications_in_physics"> <div class="vector-toc-text"> 7 Applications in physics </div> </a> <ul id="toc-Applications_in_physics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Applications_in_geometric_measure_theory" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applications_in_geometric_measure_theory"> <div class="vector-toc-text"> 8 Applications in geometric measure theory </div> </a> <ul id="toc-Applications_in_geometric_measure_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 9 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 10 Notes </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 11 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 12 External links </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header 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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 23 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-23" aria-hidden="true" > 23 languages </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/%D8%B4%D9%83%D9%84_%D8%AA%D9%81%D8%A7%D8%B6%D9%84%D9%8A" title="شكل تفاضلي – Arabic" lang="ar" hreflang="ar" data-title="شكل تفاضلي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%94%D0%B8%D1%84%D0%B5%D1%80%D0%B5%D0%BD%D1%86%D0%B8%D0%B0%D0%BB%D0%BD%D0%B0_%D1%84%D0%BE%D1%80%D0%BC%D0%B0" title="Диференциална форма – Bulgarian" lang="bg" hreflang="bg" data-title="Диференциална форма" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Forma_diferencial" title="Forma diferencial – Catalan" lang="ca" hreflang="ca" data-title="Forma diferencial" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Diferenci%C3%A1ln%C3%AD_forma" title="Diferenciální forma – Czech" lang="cs" hreflang="cs" data-title="Diferenciální forma" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Differentialform" title="Differentialform – Danish" lang="da" hreflang="da" data-title="Differentialform" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Differentialform" title="Differentialform – German" lang="de" hreflang="de" data-title="Differentialform" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Forma_diferencial" title="Forma diferencial – Spanish" lang="es" hreflang="es" data-title="Forma diferencial" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Diferenciala_formo" title="Diferenciala formo – Esperanto" lang="eo" hreflang="eo" data-title="Diferenciala formo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Forma_diferentzial" title="Forma diferentzial – Basque" lang="eu" hreflang="eu" data-title="Forma diferentzial" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Forme_diff%C3%A9rentielle" title="Forme différentielle – French" lang="fr" hreflang="fr" data-title="Forme différentielle" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%AF%B8%EB%B6%84_%ED%98%95%EC%8B%9D" title="미분 형식 – Korean" lang="ko" hreflang="ko" data-title="미분 형식" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Forma_differenziale" title="Forma differenziale – Italian" lang="it" hreflang="it" data-title="Forma differenziale" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%AA%D7%91%D7%A0%D7%99%D7%AA_%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99%D7%AA" title="תבנית דיפרנציאלית – Hebrew" lang="he" hreflang="he" data-title="תבנית דיפרנציאלית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Differentiaalvorm" title="Differentiaalvorm – Dutch" lang="nl" hreflang="nl" data-title="Differentiaalvorm" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%BE%AE%E5%88%86%E5%BD%A2%E5%BC%8F" title="微分形式 – Japanese" lang="ja" hreflang="ja" data-title="微分形式" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Differensialform" title="Differensialform – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Differensialform" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Forma_r%C3%B3%C5%BCniczkowa" title="Forma różniczkowa – Polish" lang="pl" hreflang="pl" data-title="Forma różniczkowa" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Forma_diferencial" title="Forma diferencial – Portuguese" lang="pt" hreflang="pt" data-title="Forma diferencial" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%94%D0%B8%D1%84%D1%84%D0%B5%D1%80%D0%B5%D0%BD%D1%86%D0%B8%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0%D1%8F_%D1%84%D0%BE%D1%80%D0%BC%D0%B0" title="Дифференциальная форма – Russian" lang="ru" hreflang="ru" data-title="Дифференциальная форма" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Differentialform" title="Differentialform – Swedish" lang="sv" hreflang="sv" data-title="Differentialform" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%94%D0%B8%D1%84%D0%B5%D1%80%D0%B5%D0%BD%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0_%D1%84%D0%BE%D1%80%D0%BC%D0%B0" title="Диференціальна форма – Ukrainian" lang="uk" hreflang="uk" data-title="Диференціальна форма" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%BE%AE%E5%88%86%E5%BD%A2%E5%BC%8F" title="微分形式 – Cantonese" lang="yue" hreflang="yue" data-title="微分形式" data-language-autonym="粵語" data-language-local-name="Cantonese" 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class="mw-redirect" title="Exterior calculus">Exterior calculus</a>)</div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Expression that may be integrated over a region</div> In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, differential forms provide a unified approach to define <a href="/wiki/Integrand" class="mw-redirect" title="Integrand">integrands</a> over curves, surfaces, solids, and higher-dimensional <a href="/wiki/Manifold" title="Manifold">manifolds</a>. The modern notion of differential forms was pioneered by <a href="/wiki/%C3%89lie_Cartan" title="Élie Cartan">Élie Cartan</a>. It has many applications, especially in geometry, topology and physics. For instance, the expression f(x) dx is an example of a <a href="/wiki/1-form" class="mw-redirect" title="1-form">1-form</a>, and can be <a href="/wiki/Integral" title="Integral">integrated</a> over an interval [a, b] contained in the domain of f: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{a}^{b}f(x)\,dx.}"> <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> <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 \int _{a}^{b}f(x)\,dx.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/249ee3fdbced31dfc328ff357f67bb134ca66b8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.786ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}f(x)\,dx.}"></dd></dl> Similarly, the expression f(x, y, z) dx ∧ dy + g(x, y, z) dz ∧ dx + h(x, y, z) dy ∧ dz is a 2-form that can be integrated over a <a href="/wiki/Surface_(mathematics)" title="Surface (mathematics)">surface</a> S: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{S}(f(x,y,z)\,dx\wedge dy+g(x,y,z)\,dz\wedge dx+h(x,y,z)\,dy\wedge dz).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>∧</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>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>z</mi> <mo>∧</mo> <mi>d</mi> <mi>x</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>y</mi> <mo>∧</mo> <mi>d</mi> <mi>z</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{S}(f(x,y,z)\,dx\wedge dy+g(x,y,z)\,dz\wedge dx+h(x,y,z)\,dy\wedge dz).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8a5fd1cbe66af28e60e92c64607746ca87c92e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:60.158ex; height:5.676ex;" alt="{\displaystyle \int _{S}(f(x,y,z)\,dx\wedge dy+g(x,y,z)\,dz\wedge dx+h(x,y,z)\,dy\wedge dz).}"></dd></dl> The symbol ∧ denotes the <a href="/wiki/Exterior_product" class="mw-redirect" title="Exterior product">exterior product</a>, sometimes called the wedge product, of two differential forms. Likewise, a 3-form f(x, y, z) dx ∧ dy ∧ dz represents a <a href="/wiki/Volume_element" title="Volume element">volume element</a> that can be integrated over a region of space. In general, a k-form is an object that may be integrated over a k-dimensional manifold, and is <a href="/wiki/Homogeneous_polynomial" title="Homogeneous polynomial">homogeneous</a> of degree k in the coordinate differentials <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dx,dy,\ldots .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>x</mi> <mo>,</mo> <mi>d</mi> <mi>y</mi> <mo>,</mo> <mo>…</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dx,dy,\ldots .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d68be24160926d9778b06bf8150ce584165353c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.742ex; height:2.509ex;" alt="{\displaystyle dx,dy,\ldots .}"> On an n-dimensional manifold, the top-dimensional form (n-form) is called a <a href="/wiki/Volume_form" title="Volume form">volume form</a>. The differential forms form an <a href="/wiki/Alternating_algebra" title="Alternating algebra">alternating algebra</a>. This implies that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dy\wedge dx=-dx\wedge dy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>y</mi> <mo>∧</mo> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mi>d</mi> <mi>x</mi> <mo>∧</mo> <mi>d</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dy\wedge dx=-dx\wedge dy}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9eca305d04c05506e4b8445007f535639968cfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.906ex; height:2.509ex;" alt="{\displaystyle dy\wedge dx=-dx\wedge dy}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dx\wedge dx=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>x</mi> <mo>∧</mo> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dx\wedge dx=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/593aa821aee8e3b74928300debf298c2f95db58d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.581ex; height:2.176ex;" alt="{\displaystyle dx\wedge dx=0.}"> This alternating property reflects the <a href="/wiki/Orientation_(mathematics)" class="mw-redirect" title="Orientation (mathematics)">orientation</a> of the domain of integration. The <a href="/wiki/Exterior_derivative" title="Exterior derivative">exterior derivative</a> is an operation on differential forms that, given a k-form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }">, produces a (k+1)-form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\varphi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>φ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\varphi .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/874e9f3ab5a6881add7f4324139b4bd83be29d95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.383ex; height:2.676ex;" alt="{\displaystyle d\varphi .}"> This operation extends the <a href="/wiki/Differential_of_a_function" title="Differential of a function">differential of a function</a> (a function can be considered as a 0-form, and its differential is <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> <mi>d</mi> <mi>x</mi> </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/73394585726b10ca394a1c1aea3fef82c85feb05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.422ex; height:3.009ex;" alt="{\displaystyle df(x)=f'(x)dx}">). This allows expressing the <a href="/wiki/Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus">fundamental theorem of calculus</a>, the <a href="/wiki/Divergence_theorem" title="Divergence theorem">divergence theorem</a>, <a href="/wiki/Green%27s_theorem" title="Green's theorem">Green's theorem</a>, and <a href="/wiki/Stokes%27_theorem" title="Stokes' theorem">Stokes' theorem</a> as special cases of a single general result, the <a href="/wiki/Generalized_Stokes_theorem" title="Generalized Stokes theorem">generalized Stokes theorem</a>. Differential 1-forms are naturally dual to <a href="/wiki/Vector_field" title="Vector field">vector fields</a> on a <a href="/wiki/Differentiable_manifold" title="Differentiable manifold">differentiable manifold</a>, and the pairing between vector fields and 1-forms is extended to arbitrary differential forms by the <a href="/wiki/Interior_product" title="Interior product">interior product</a>. The algebra of differential forms along with the exterior derivative defined on it is preserved by the <a href="/wiki/Pullback_(differential_geometry)" title="Pullback (differential geometry)">pullback</a> under smooth functions between two manifolds. This feature allows geometrically invariant information to be moved from one space to another via the pullback, provided that the information is expressed in terms of differential forms. As an example, the <a href="/wiki/Change_of_variables_formula" class="mw-redirect" title="Change of variables formula">change of variables formula</a> for integration becomes a simple statement that an integral is preserved under pullback. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2>[<a href="/w/index.php?title=Differential_form&action=edit&section=1" title="Edit section: History">edit</a>]</div> Differential forms are part of the field of differential geometry, influenced by linear algebra. Although the notion of a differential is quite old, the initial attempt at an algebraic organization of differential forms is usually credited to <a href="/wiki/%C3%89lie_Cartan" title="Élie Cartan">Élie Cartan</a> with reference to his 1899 paper.<a href="#cite_note-1">[1]</a> Some aspects of the <a href="/wiki/Exterior_algebra" title="Exterior algebra">exterior algebra</a> of differential forms appears in <a href="/wiki/Hermann_Grassmann" title="Hermann Grassmann">Hermann Grassmann</a>'s 1844 work, Die Lineale Ausdehnungslehre, ein neuer Zweig der Mathematik (The Theory of Linear Extension, a New Branch of Mathematics). <div class="mw-heading mw-heading2"><h2 id="Concept">Concept</h2>[<a href="/w/index.php?title=Differential_form&action=edit&section=2" title="Edit section: Concept">edit</a>]</div> Differential forms provide an approach to <a href="/wiki/Multivariable_calculus" title="Multivariable calculus">multivariable calculus</a> that is independent of <a href="/wiki/Coordinate" class="mw-redirect" title="Coordinate">coordinates</a>. <div class="mw-heading mw-heading3"><h3 id="Integration_and_orientation">Integration and orientation</h3>[<a href="/w/index.php?title=Differential_form&action=edit&section=3" title="Edit section: Integration and orientation">edit</a>]</div> A differential k-form can be integrated over an oriented <a href="/wiki/Manifold_(mathematics)" class="mw-redirect" title="Manifold (mathematics)">manifold</a> of dimension k. A differential 1-form can be thought of as measuring an infinitesimal oriented length, or 1-dimensional oriented density. A differential 2-form can be thought of as measuring an infinitesimal oriented area, or 2-dimensional oriented density. And so on. Integration of differential forms is well-defined only on <a href="/wiki/Orientability" title="Orientability">oriented</a> <a href="/wiki/Manifold_(mathematics)" class="mw-redirect" title="Manifold (mathematics)">manifolds</a>. An example of a 1-dimensional manifold is an interval [a, b], and intervals can be given an orientation: they are positively oriented if a < b, and negatively oriented otherwise. If a < b then the integral of the differential 1-form f(x) dx over the interval [a, b] (with its natural positive orientation) is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{a}^{b}f(x)\,dx}"> <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> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}f(x)\,dx}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac02adeed584466d53dee65f3228ad66939eb58b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.139ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}f(x)\,dx}"></dd></dl> which is the negative of the integral of the same differential form over the same interval, when equipped with the opposite orientation. That is: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{b}^{a}f(x)\,dx=-\int _{a}^{b}f(x)\,dx.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</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> <mo>−</mo> <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 \int _{b}^{a}f(x)\,dx=-\int _{a}^{b}f(x)\,dx.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fa0c45bba6b7237d507c281d2686600ac6e7e1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:28.383ex; height:6.343ex;" alt="{\displaystyle \int _{b}^{a}f(x)\,dx=-\int _{a}^{b}f(x)\,dx.}"></dd></dl> This gives a geometrical context to the <a href="/wiki/Integral#Conventions" title="Integral">conventions</a> for one-dimensional integrals, that the sign changes when the orientation of the interval is reversed. A standard explanation of this in one-variable integration theory is that, when the limits of integration are in the opposite order (b < a), the increment dx is negative in the direction of integration. More generally, an m-form is an oriented density that can be integrated over an m-dimensional oriented manifold. (For example, a 1-form can be integrated over an oriented curve, a 2-form can be integrated over an oriented surface, etc.) If M is an oriented m-dimensional manifold, and M′ is the same manifold with opposite orientation and ω is an m-form, then one has: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{M}\omega =-\int _{M'}\omega \,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mi>ω</mi> <mo>=</mo> <mo>−</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>M</mi> <mo>′</mo> </msup> </mrow> </msub> <mi>ω</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{M}\omega =-\int _{M'}\omega \,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e34c2752dd1295f6cb6c32c04cb05a27fe0cdfd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.068ex; height:5.676ex;" alt="{\displaystyle \int _{M}\omega =-\int _{M'}\omega \,.}"></dd></dl> These conventions correspond to interpreting the integrand as a differential form, integrated over a <a href="/wiki/Chain_(algebraic_topology)" title="Chain (algebraic topology)">chain</a>. In <a href="/wiki/Measure_theory" class="mw-redirect" title="Measure theory">measure theory</a>, by contrast, one interprets the integrand as a function f with respect to a measure μ and integrates over a subset A, without any notion of orientation; one writes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \int _{A}f\,d\mu =\int _{[a,b]}f\,d\mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mi>f</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>μ</mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mrow> </msub> <mi>f</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \int _{A}f\,d\mu =\int _{[a,b]}f\,d\mu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/511cfc5df1d72c9db9323c6d97c4cfc3548bea8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:19.278ex; height:3.676ex;" alt="{\textstyle \int _{A}f\,d\mu =\int _{[a,b]}f\,d\mu }"> to indicate integration over a subset A. This is a minor distinction in one dimension, but becomes subtler on higher-dimensional manifolds; see <a href="#Relation_with_measures">below</a> for details. Making the notion of an oriented density precise, and thus of a differential form, involves the <a href="/wiki/Exterior_algebra" title="Exterior algebra">exterior algebra</a>. The differentials of a set of coordinates, dx1, ..., dxn can be used as a basis for all 1-forms. Each of these represents a <a href="/wiki/Covector" class="mw-redirect" title="Covector">covector</a> at each point on the manifold that may be thought of as measuring a small displacement in the corresponding coordinate direction. A general 1-form is a linear combination of these differentials at every point on the manifold: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{1}\,dx^{1}+\cdots +f_{n}\,dx^{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{1}\,dx^{1}+\cdots +f_{n}\,dx^{n},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b845acf139ac10d16db75a95d81f6d19c283f932" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.74ex; height:3.009ex;" alt="{\displaystyle f_{1}\,dx^{1}+\cdots +f_{n}\,dx^{n},}"></dd></dl> where the fk = fk(x1, ... , xn) are functions of all the coordinates. A differential 1-form is integrated along an oriented curve as a line integral. The expressions dxi ∧ dxj, where i < j can be used as a basis at every point on the manifold for all 2-forms. This may be thought of as an infinitesimal oriented square parallel to the xi–xj-plane. A general 2-form is a linear combination of these at every point on the manifold: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum _{1\leq i<j\leq n}f_{i,j}\,dx^{i}\wedge dx^{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>≤</mo> <mi>i</mi> <mo><</mo> <mi>j</mi> <mo>≤</mo> <mi>n</mi> </mrow> </munder> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>∧</mo> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{1\leq i<j\leq n}f_{i,j}\,dx^{i}\wedge dx^{j}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1104e81454c88e5e0c0c16e63cab911da6af12b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:22.806ex; height:3.509ex;" alt="{\textstyle \sum _{1\leq i<j\leq n}f_{i,j}\,dx^{i}\wedge dx^{j}}">, and it is integrated just like a surface integral. A fundamental operation defined on differential forms is the <a href="/wiki/Exterior_product" class="mw-redirect" title="Exterior product">exterior product</a> (the symbol is the <a href="/wiki/Wedge_(symbol)" title="Wedge (symbol)">wedge</a> ∧). This is similar to the <a href="/wiki/Cross_product" title="Cross product">cross product</a> from vector calculus, in that it is an alternating product. For instance, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dx^{1}\wedge dx^{2}=-dx^{2}\wedge dx^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>∧</mo> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>∧</mo> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dx^{1}\wedge dx^{2}=-dx^{2}\wedge dx^{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46e07cd976f309e282a7fe8fc43efa5cd9e618e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:24.471ex; height:2.843ex;" alt="{\displaystyle dx^{1}\wedge dx^{2}=-dx^{2}\wedge dx^{1}}"></dd></dl> because the square whose first side is dx1 and second side is dx2 is to be regarded as having the opposite orientation as the square whose first side is dx2 and whose second side is dx1. This is why we only need to sum over expressions dxi ∧ dxj, with i < j; for example: a(dxi ∧ dxj) + b(dxj ∧ dxi) = (a − b) dxi ∧ dxj. The exterior product allows higher-degree differential forms to be built out of lower-degree ones, in much the same way that the <a href="/wiki/Cross_product" title="Cross product">cross product</a> in vector calculus allows one to compute the area vector of a parallelogram from vectors pointing up the two sides. Alternating also implies that dxi ∧ dxi = 0, in the same way that the cross product of parallel vectors, whose magnitude is the area of the parallelogram spanned by those vectors, is zero. In higher dimensions, dxi1 ∧ ⋅⋅⋅ ∧ dxim = 0 if any two of the indices i1, ..., im are equal, in the same way that the "volume" enclosed by a <a href="/wiki/Parallelepiped#Parallelotope" title="Parallelepiped">parallelotope</a> whose edge vectors are <a href="/wiki/Linear_independence" title="Linear independence">linearly dependent</a> is zero. <div class="mw-heading mw-heading3"><h3 id="Multi-index_notation">Multi-index notation</h3>[<a href="/w/index.php?title=Differential_form&action=edit&section=4" title="Edit section: Multi-index notation">edit</a>]</div> A common notation for the wedge product of elementary k-forms is so called <a href="/wiki/Multi-index_notation" title="Multi-index notation">multi-index notation</a>: in an n-dimensional context, for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I=(i_{1},i_{2},\ldots ,i_{k}),1\leq i_{1}<i_{2}<\cdots <i_{k}\leq n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>=</mo> <mo stretchy="false">(</mo> <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> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mn>1</mn> <mo>≤</mo> <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> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>≤</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I=(i_{1},i_{2},\ldots ,i_{k}),1\leq i_{1}<i_{2}<\cdots <i_{k}\leq n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22321587b6fcbbb6df12ad4ee29b20e57821bf06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.307ex; height:2.843ex;" alt="{\displaystyle I=(i_{1},i_{2},\ldots ,i_{k}),1\leq i_{1}<i_{2}<\cdots <i_{k}\leq n}">, we define <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle dx^{I}:=dx^{i_{1}}\wedge \cdots \wedge dx^{i_{k}}=\bigwedge _{i\in I}dx^{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msup> <mo>:=</mo> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> <mo>∧</mo> <mo>⋯</mo> <mo>∧</mo> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msup> <mo>=</mo> <munder> <mo>⋀</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle dx^{I}:=dx^{i_{1}}\wedge \cdots \wedge dx^{i_{k}}=\bigwedge _{i\in I}dx^{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cc974ca67fffe1d9129d932c495c04aa319d83f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:35.113ex; height:3.176ex;" alt="{\textstyle dx^{I}:=dx^{i_{1}}\wedge \cdots \wedge dx^{i_{k}}=\bigwedge _{i\in I}dx^{i}}">.<a href="#cite_note-2">[2]</a> Another useful notation is obtained by defining the set of all strictly increasing multi-indices of length k, in a space of dimension n, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {J}}_{k,n}:=\{I=(i_{1},\ldots ,i_{k}):1\leq i_{1}<i_{2}<\cdots <i_{k}\leq n\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">J</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <mi>I</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>:</mo> <mn>1</mn> <mo>≤</mo> <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> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>≤</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {J}}_{k,n}:=\{I=(i_{1},\ldots ,i_{k}):1\leq i_{1}<i_{2}<\cdots <i_{k}\leq n\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec06bf7dd41053a257d829949b55b648e36871dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:53.498ex; height:3.009ex;" alt="{\displaystyle {\mathcal {J}}_{k,n}:=\{I=(i_{1},\ldots ,i_{k}):1\leq i_{1}<i_{2}<\cdots <i_{k}\leq n\}}">. Then locally (wherever the coordinates apply), <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{dx^{I}\}_{I\in {\mathcal {J}}_{k,n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msup> <msub> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> <mo>∈</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">J</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{dx^{I}\}_{I\in {\mathcal {J}}_{k,n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5ffe1d0e9cdb3b922c902f16f816a62b6b1608a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:11.234ex; height:3.509ex;" alt="{\displaystyle \{dx^{I}\}_{I\in {\mathcal {J}}_{k,n}}}"> spans the space of differential k-forms in a manifold M of dimension n, when viewed as a module over the ring C∞(M) of smooth functions on M. By calculating the size of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {J}}_{k,n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">J</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {J}}_{k,n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62fd0fe22a036366a00d79844fd94819e2102dda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.108ex; height:2.843ex;" alt="{\displaystyle {\mathcal {J}}_{k,n}}"> combinatorially, the module of k-forms on an n-dimensional manifold, and in general space of k-covectors on an n-dimensional vector space, is n choose k: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle |{\mathcal {J}}_{k,n}|={\binom {n}{k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">J</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <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> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle |{\mathcal {J}}_{k,n}|={\binom {n}{k}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b141636d214173e51087b43c3ca7ae3be41e782a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.616ex; height:3.176ex;" alt="{\textstyle |{\mathcal {J}}_{k,n}|={\binom {n}{k}}}">. This also demonstrates that there are no nonzero differential forms of degree greater than the dimension of the underlying manifold. <div class="mw-heading mw-heading3"><h3 id="The_exterior_derivative">The exterior derivative</h3>[<a href="/w/index.php?title=Differential_form&action=edit&section=5" title="Edit section: The exterior derivative">edit</a>]</div> In addition to the exterior product, there is also the <a href="/wiki/Exterior_derivative" title="Exterior derivative">exterior derivative</a> operator d. The exterior derivative of a differential form is a generalization of the <a href="/wiki/Differential_of_a_function" title="Differential of a function">differential of a function</a>, in the sense that the exterior derivative of f ∈ C∞(M) = Ω0(M) is exactly the differential of f. When generalized to higher forms, if ω = f dxI is a simple k-form, then its exterior derivative dω is a (k + 1)-form defined by taking the differential of the coefficient functions: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\omega =\sum _{i=1}^{n}{\frac {\partial f}{\partial x^{i}}}\,dx^{i}\wedge dx^{I}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>ω</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>∧</mo> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\omega =\sum _{i=1}^{n}{\frac {\partial f}{\partial x^{i}}}\,dx^{i}\wedge dx^{I}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55d98b7900aa873823e3d88d0c34c5ab3022aa57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:24.354ex; height:6.843ex;" alt="{\displaystyle d\omega =\sum _{i=1}^{n}{\frac {\partial f}{\partial x^{i}}}\,dx^{i}\wedge dx^{I}.}"></dd></dl> with extension to general k-forms through linearity: if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \tau =\sum _{I\in {\mathcal {J}}_{k,n}}a_{I}\,dx^{I}\in \Omega ^{k}(M)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>τ</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> <mo>∈</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">J</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msup> <mo>∈</mo> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \tau =\sum _{I\in {\mathcal {J}}_{k,n}}a_{I}\,dx^{I}\in \Omega ^{k}(M)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/216b25d141ff4db0ebacb2cdb14837d140e99d72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:28.587ex; height:3.676ex;" alt="{\textstyle \tau =\sum _{I\in {\mathcal {J}}_{k,n}}a_{I}\,dx^{I}\in \Omega ^{k}(M)}">, then its exterior derivative is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\tau =\sum _{I\in {\mathcal {J}}_{k,n}}\left(\sum _{j=1}^{n}{\frac {\partial a_{I}}{\partial x^{j}}}\,dx^{j}\right)\wedge dx^{I}\in \Omega ^{k+1}(M)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>τ</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> <mo>∈</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">J</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>∧</mo> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msup> <mo>∈</mo> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\tau =\sum _{I\in {\mathcal {J}}_{k,n}}\left(\sum _{j=1}^{n}{\frac {\partial a_{I}}{\partial x^{j}}}\,dx^{j}\right)\wedge dx^{I}\in \Omega ^{k+1}(M)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af228d587f8071a18c142615d23b4ecb0b2d63c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:44.833ex; height:8.009ex;" alt="{\displaystyle d\tau =\sum _{I\in {\mathcal {J}}_{k,n}}\left(\sum _{j=1}^{n}{\frac {\partial a_{I}}{\partial x^{j}}}\,dx^{j}\right)\wedge dx^{I}\in \Omega ^{k+1}(M)}"></dd></dl> In R3, with the <a href="/wiki/Hodge_star_operator" title="Hodge star operator">Hodge star operator</a>, the exterior derivative corresponds to <a href="/wiki/Gradient" title="Gradient">gradient</a>, <a href="/wiki/Curl_(mathematics)" title="Curl (mathematics)">curl</a>, and <a href="/wiki/Divergence" title="Divergence">divergence</a>, although this correspondence, like the cross product, does not generalize to higher dimensions, and should be treated with some caution. The exterior derivative itself applies in an arbitrary finite number of dimensions, and is a flexible and powerful tool with wide application in <a href="/wiki/Differential_geometry" title="Differential geometry">differential geometry</a>, <a href="/wiki/Differential_topology" title="Differential topology">differential topology</a>, and many areas in physics. Of note, although the above definition of the exterior derivative was defined with respect to local coordinates, it can be defined in an entirely coordinate-free manner, as an <a href="/wiki/Derivation_(differential_algebra)" title="Derivation (differential algebra)">antiderivation</a> of degree 1 on the <a href="/wiki/Exterior_algebra" title="Exterior algebra">exterior algebra</a> of differential forms. The benefit of this more general approach is that it allows for a natural coordinate-free approach to integrate on <a href="/wiki/Manifold" title="Manifold">manifolds</a>. It also allows for a natural generalization of the <a href="/wiki/Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus">fundamental theorem of calculus</a>, called the (generalized) <a href="/wiki/Stokes%27_theorem" title="Stokes' theorem">Stokes' theorem</a>, which is a central result in the theory of integration on manifolds. <div class="mw-heading mw-heading3"><h3 id="Differential_calculus">Differential calculus</h3>[<a href="/w/index.php?title=Differential_form&action=edit&section=6" title="Edit section: Differential calculus">edit</a>]</div> Let U be an <a href="/wiki/Open_set" title="Open set">open set</a> in Rn. A differential 0-form ("zero-form") is defined to be a <a href="/wiki/Smooth_function" class="mw-redirect" title="Smooth function">smooth function</a> f on U – the set of which is denoted C∞(U). If v is any vector in Rn, then f has a <a href="/wiki/Directional_derivative" title="Directional derivative">directional derivative</a> ∂v f, which is another function on U whose value at a point p ∈ U is the rate of change (at p) of f in the v direction: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\partial _{\mathbf {v} }f)(p)=\left.{\frac {d}{dt}}f(p+t\mathbf {v} )\right|_{t=0}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> </msub> <mi>f</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>+</mo> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>=</mo> <mn>0</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\partial _{\mathbf {v} }f)(p)=\left.{\frac {d}{dt}}f(p+t\mathbf {v} )\right|_{t=0}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d23a223c412bdd6b5b9f65410a80206dd0ab0192" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:28.09ex; height:5.843ex;" alt="{\displaystyle (\partial _{\mathbf {v} }f)(p)=\left.{\frac {d}{dt}}f(p+t\mathbf {v} )\right|_{t=0}.}"></dd></dl> (This notion can be extended pointwise to the case that v is a <a href="/wiki/Vector_field" title="Vector field">vector field</a> on U by evaluating v at the point p in the definition.) In particular, if v = ej is the jth <a href="/wiki/Coordinate_vector" title="Coordinate vector">coordinate vector</a> then ∂v f is the <a href="/wiki/Partial_derivative" title="Partial derivative">partial derivative</a> of f with respect to the jth coordinate vector, i.e., ∂f / ∂xj, where x1, x2, ..., xn are the coordinate vectors in U. By their very definition, partial derivatives depend upon the choice of coordinates: if new coordinates y1, y2, ..., yn are introduced, then <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial f}{\partial x^{j}}}=\sum _{i=1}^{n}{\frac {\partial y^{i}}{\partial x^{j}}}{\frac {\partial f}{\partial y^{i}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>y</mi> <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>j</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial f}{\partial x^{j}}}=\sum _{i=1}^{n}{\frac {\partial y^{i}}{\partial x^{j}}}{\frac {\partial f}{\partial y^{i}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e86cfd6843c91b0a094cc7ad1a741a1a06c0909" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.389ex; height:6.843ex;" alt="{\displaystyle {\frac {\partial f}{\partial x^{j}}}=\sum _{i=1}^{n}{\frac {\partial y^{i}}{\partial x^{j}}}{\frac {\partial f}{\partial y^{i}}}.}"></dd></dl> The first idea leading to differential forms is the observation that ∂v f (p) is a <a href="/wiki/Linear_function" title="Linear function">linear function</a> of v: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}(\partial _{\mathbf {v} +\mathbf {w} }f)(p)&=(\partial _{\mathbf {v} }f)(p)+(\partial _{\mathbf {w} }f)(p)\\(\partial _{c\mathbf {v} }f)(p)&=c(\partial _{\mathbf {v} }f)(p)\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> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> </mrow> </msub> <mi>f</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> </msub> <mi>f</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> </mrow> </msub> <mi>f</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> </msub> <mi>f</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>c</mi> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> </msub> <mi>f</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(\partial _{\mathbf {v} +\mathbf {w} }f)(p)&=(\partial _{\mathbf {v} }f)(p)+(\partial _{\mathbf {w} }f)(p)\\(\partial _{c\mathbf {v} }f)(p)&=c(\partial _{\mathbf {v} }f)(p)\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41c4ca739864c70f15f2998d2ec1eb607e7d0ca8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:35.295ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}(\partial _{\mathbf {v} +\mathbf {w} }f)(p)&=(\partial _{\mathbf {v} }f)(p)+(\partial _{\mathbf {w} }f)(p)\\(\partial _{c\mathbf {v} }f)(p)&=c(\partial _{\mathbf {v} }f)(p)\end{aligned}}}"></dd></dl> for any vectors v, w and any real number c. At each point p, this <a href="/wiki/Linear_map" title="Linear map">linear map</a> from Rn to R is denoted dfp and called the <a href="/wiki/Derivative" title="Derivative">derivative</a> or <a href="/wiki/Differential_of_a_function" title="Differential of a function">differential</a> of f at p. Thus dfp(v) = ∂v f (p). Extended over the whole set, the object df can be viewed as a function that takes a vector field on U, and returns a real-valued function whose value at each point is the derivative along the vector field of the function f. Note that at each p, the differential dfp is not a real number, but a linear functional on tangent vectors, and a prototypical example of a differential <a href="/wiki/1-form" class="mw-redirect" title="1-form">1-form</a>. Since any vector v is a <a href="/wiki/Linear_combination" title="Linear combination">linear combination</a> Σ vjej of its <a href="/wiki/Euclidean_vector#Decomposition" title="Euclidean vector">components</a>, df is uniquely determined by dfp(ej) for each j and each p ∈ U, which are just the partial derivatives of f on U. Thus df provides a way of encoding the partial derivatives of f. It can be decoded by noticing that the coordinates x1, x2, ..., xn are themselves functions on U, and so define differential 1-forms dx1, dx2, ..., dxn. Let f = xi. Since ∂xi / ∂xj = δij, the <a href="/wiki/Kronecker_delta_function" class="mw-redirect" title="Kronecker delta function">Kronecker delta function</a>, it follows that <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle df=\sum _{i=1}^{n}{\frac {\partial f}{\partial x^{i}}}\,dx^{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>f</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle df=\sum _{i=1}^{n}{\frac {\partial f}{\partial x^{i}}}\,dx^{i}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18d54e1d89b068cef9858a9cc2350fad62cd741e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:17.998ex; height:6.843ex;" alt="{\displaystyle df=\sum _{i=1}^{n}{\frac {\partial f}{\partial x^{i}}}\,dx^{i}.}"></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(*)</td></tr></tbody></table> The meaning of this expression is given by evaluating both sides at an arbitrary point p: on the right hand side, the sum is defined "<a href="/wiki/Pointwise" title="Pointwise">pointwise</a>", so that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle df_{p}=\sum _{i=1}^{n}{\frac {\partial f}{\partial x^{i}}}(p)(dx^{i})_{p}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle df_{p}=\sum _{i=1}^{n}{\frac {\partial f}{\partial x^{i}}}(p)(dx^{i})_{p}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08036ae5562d74daa5fde37bd9169bdce438e8bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:24.378ex; height:6.843ex;" alt="{\displaystyle df_{p}=\sum _{i=1}^{n}{\frac {\partial f}{\partial x^{i}}}(p)(dx^{i})_{p}.}"></dd></dl> Applying both sides to ej, the result on each side is the jth partial derivative of f at p. Since p and j were arbitrary, this proves the formula <a href="#math_*">(*)</a>. More generally, for any smooth functions gi and hi on U, we define the differential 1-form α = Σi gi dhi pointwise by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{p}=\sum _{i}g_{i}(p)(dh_{i})_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>d</mi> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{p}=\sum _{i}g_{i}(p)(dh_{i})_{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01434a640cebe6ca8ebd6e38956c498822310d40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.498ex; height:5.509ex;" alt="{\displaystyle \alpha _{p}=\sum _{i}g_{i}(p)(dh_{i})_{p}}"></dd></dl> for each p ∈ U. Any differential 1-form arises this way, and by using <a href="#math_*">(*)</a> it follows that any differential 1-form α on U may be expressed in coordinates as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =\sum _{i=1}^{n}f_{i}\,dx^{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =\sum _{i=1}^{n}f_{i}\,dx^{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b88eb5c2fedd9a3c3c24030d1c805f460f7a32f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:13.999ex; height:6.843ex;" alt="{\displaystyle \alpha =\sum _{i=1}^{n}f_{i}\,dx^{i}}"></dd></dl> for some smooth functions fi on U. The second idea leading to differential forms arises from the following question: given a differential 1-form α on U, when does there exist a function f on U such that α = df? The above expansion reduces this question to the search for a function f whose partial derivatives ∂f / ∂xi are equal to n given functions fi. For n > 1, such a function does not always exist: any smooth function f satisfies <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial ^{2}f}{\partial x^{i}\,\partial x^{j}}}={\frac {\partial ^{2}f}{\partial x^{j}\,\partial x^{i}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial ^{2}f}{\partial x^{i}\,\partial x^{j}}}={\frac {\partial ^{2}f}{\partial x^{j}\,\partial x^{i}}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b11659d7978d50713246fa58cf41e038831f0426" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:20.202ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial ^{2}f}{\partial x^{i}\,\partial x^{j}}}={\frac {\partial ^{2}f}{\partial x^{j}\,\partial x^{i}}},}"></dd></dl> so it will be impossible to find such an f unless <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial f_{j}}{\partial x^{i}}}-{\frac {\partial f_{i}}{\partial x^{j}}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial f_{j}}{\partial x^{i}}}-{\frac {\partial f_{i}}{\partial x^{j}}}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7375dec5415d7c35bfdc84086771d839714ef075" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:15.778ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial f_{j}}{\partial x^{i}}}-{\frac {\partial f_{i}}{\partial x^{j}}}=0}"></dd></dl> for all i and j. The <a href="/wiki/Skew_symmetry" class="mw-redirect" title="Skew symmetry">skew-symmetry</a> of the left hand side in i and j suggests introducing an antisymmetric product ∧ on differential 1-forms, the <a href="/wiki/Exterior_product" class="mw-redirect" title="Exterior product">exterior product</a>, so that these equations can be combined into a single condition <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i,j=1}^{n}{\frac {\partial f_{j}}{\partial x^{i}}}\,dx^{i}\wedge dx^{j}=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>∧</mo> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i,j=1}^{n}{\frac {\partial f_{j}}{\partial x^{i}}}\,dx^{i}\wedge dx^{j}=0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89a8d9a7326d6ffed45c689d2bf06be81faa3673" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:23.151ex; height:7.176ex;" alt="{\displaystyle \sum _{i,j=1}^{n}{\frac {\partial f_{j}}{\partial x^{i}}}\,dx^{i}\wedge dx^{j}=0,}"></dd></dl> where ∧ is defined so that: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dx^{i}\wedge dx^{j}=-dx^{j}\wedge dx^{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>∧</mo> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo>∧</mo> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dx^{i}\wedge dx^{j}=-dx^{j}\wedge dx^{i}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e6564b779e9db06381a8da5b697cf53d775c174" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:24.32ex; height:2.843ex;" alt="{\displaystyle dx^{i}\wedge dx^{j}=-dx^{j}\wedge dx^{i}.}"></dd></dl> This is an example of a differential 2-form. This 2-form is called the <a href="/wiki/Exterior_derivative" title="Exterior derivative">exterior derivative</a> dα of α = Σn j=1 fj dxj. It is given by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\alpha =\sum _{j=1}^{n}df_{j}\wedge dx^{j}=\sum _{i,j=1}^{n}{\frac {\partial f_{j}}{\partial x^{i}}}\,dx^{i}\wedge dx^{j}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>α</mi> <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> <mi>d</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>∧</mo> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>∧</mo> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\alpha =\sum _{j=1}^{n}df_{j}\wedge dx^{j}=\sum _{i,j=1}^{n}{\frac {\partial f_{j}}{\partial x^{i}}}\,dx^{i}\wedge dx^{j}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3944d7e9218f0105e86018a19f6f90c8228fe757" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:40.836ex; height:7.176ex;" alt="{\displaystyle d\alpha =\sum _{j=1}^{n}df_{j}\wedge dx^{j}=\sum _{i,j=1}^{n}{\frac {\partial f_{j}}{\partial x^{i}}}\,dx^{i}\wedge dx^{j}.}"></dd></dl> To summarize: dα = 0 is a necessary condition for the existence of a function f with α = df. Differential 0-forms, 1-forms, and 2-forms are special cases of differential forms. For each k, there is a space of differential k-forms, which can be expressed in terms of the coordinates as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i_{1},i_{2}\ldots i_{k}=1}^{n}f_{i_{1}i_{2}\ldots i_{k}}\,dx^{i_{1}}\wedge dx^{i_{2}}\wedge \cdots \wedge dx^{i_{k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <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"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <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"> <mi>k</mi> </mrow> </msub> </mrow> </msub> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> <mo>∧</mo> <mi>d</mi> <msup> <mi>x</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> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i_{1},i_{2}\ldots i_{k}=1}^{n}f_{i_{1}i_{2}\ldots i_{k}}\,dx^{i_{1}}\wedge dx^{i_{2}}\wedge \cdots \wedge dx^{i_{k}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad6a1542e589f0d6b94e2cc2b7554db7618efb95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:40.034ex; height:7.176ex;" alt="{\displaystyle \sum _{i_{1},i_{2}\ldots i_{k}=1}^{n}f_{i_{1}i_{2}\ldots i_{k}}\,dx^{i_{1}}\wedge dx^{i_{2}}\wedge \cdots \wedge dx^{i_{k}}}"></dd></dl> for a collection of functions fi1i2⋅⋅⋅ik. Antisymmetry, which was already present for 2-forms, makes it possible to restrict the sum to those sets of indices for which i1 < i2 < ... < ik−1 < ik. Differential forms can be multiplied together using the exterior product, and for any differential k-form α, there is a differential (k + 1)-form dα called the exterior derivative of α. Differential forms, the exterior product and the exterior derivative are independent of a choice of coordinates. Consequently, they may be defined on any <a href="/wiki/Smooth_manifold" class="mw-redirect" title="Smooth manifold">smooth manifold</a> M. One way to do this is cover M with <a href="/wiki/Coordinate_chart" class="mw-redirect" title="Coordinate chart">coordinate charts</a> and define a differential k-form on M to be a family of differential k-forms on each chart which agree on the overlaps. However, there are more intrinsic definitions which make the independence of coordinates manifest. <div class="mw-heading mw-heading2"><h2 id="Intrinsic_definitions">Intrinsic definitions</h2>[<a href="/w/index.php?title=Differential_form&action=edit&section=7" title="Edit section: Intrinsic definitions">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Exterior_algebra" title="Exterior algebra">Exterior algebra</a></div> Let M be a <a href="/wiki/Smooth_manifold" class="mw-redirect" title="Smooth manifold">smooth manifold</a>. A smooth differential form of degree k is a <a href="/wiki/Section_(fiber_bundle)" title="Section (fiber bundle)">smooth section</a> of the kth <a href="/wiki/Exterior_algebra" title="Exterior algebra">exterior power</a> of the <a href="/wiki/Cotangent_bundle" title="Cotangent bundle">cotangent bundle</a> of M. The set of all differential k-forms on a manifold M is a <a href="/wiki/Vector_space" title="Vector space">vector space</a>, often denoted Ωk(M). The definition of a differential form may be restated as follows. At any point p ∈ M, a k-form β defines an element <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{p}\in {\textstyle \bigwedge }^{k}T_{p}^{*}M,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>⋀</mo> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mi>M</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{p}\in {\textstyle \bigwedge }^{k}T_{p}^{*}M,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/152a0e73c538eaa2072cb86b54cbf550b16953f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.103ex; height:3.509ex;" alt="{\displaystyle \beta _{p}\in {\textstyle \bigwedge }^{k}T_{p}^{*}M,}"></dd></dl> where TpM is the <a href="/wiki/Tangent_space" title="Tangent space">tangent space</a> to M at p and Tp*M is its <a href="/wiki/Dual_space" title="Dual space">dual space</a>. This space is <a href="/wiki/Natural_isomorphism" class="mw-redirect" title="Natural isomorphism">naturally isomorphic</a><a href="#cite_note-3">[3]</a>[<a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="Please specify the isomorphism. The linked MO answer does not give the isomorphism explicitly and is unclear for non-specialists. The definition of the isomorphism needs to be clear to be able to chose between two commonly used definitions of the wedge product. (December 2021)">clarification needed</a>] to the fiber at p of the dual bundle of the kth exterior power of the <a href="/wiki/Tangent_bundle" title="Tangent bundle">tangent bundle</a> of M. That is, β is also a linear functional <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \beta _{p}\colon {\textstyle \bigwedge }^{k}T_{p}M\to \mathbf {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>⋀</mo> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mi>M</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \beta _{p}\colon {\textstyle \bigwedge }^{k}T_{p}M\to \mathbf {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b807751d6ba0965b06cdf8844aec109bacd5e4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.91ex; height:3.509ex;" alt="{\textstyle \beta _{p}\colon {\textstyle \bigwedge }^{k}T_{p}M\to \mathbf {R} }">, i.e. the dual of the kth exterior power is isomorphic to the kth exterior power of the dual: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textstyle \bigwedge }^{k}T_{p}^{*}M\cong {\Big (}{\textstyle \bigwedge }^{k}T_{p}M{\Big )}^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>⋀</mo> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mi>M</mi> <mo>≅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>⋀</mo> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mi>M</mi> <msup> <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"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\textstyle \bigwedge }^{k}T_{p}^{*}M\cong {\Big (}{\textstyle \bigwedge }^{k}T_{p}M{\Big )}^{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23b4b8a1779c19751365f09d19a9d6ebd9a8522f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.053ex; height:4.843ex;" alt="{\displaystyle {\textstyle \bigwedge }^{k}T_{p}^{*}M\cong {\Big (}{\textstyle \bigwedge }^{k}T_{p}M{\Big )}^{*}}"></dd></dl> By the universal property of exterior powers, this is equivalently an <a href="/wiki/Alternating_form" class="mw-redirect" title="Alternating form">alternating</a> <a href="/wiki/Multilinear_map" title="Multilinear map">multilinear map</a>: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{p}\colon \bigoplus _{n=1}^{k}T_{p}M\to \mathbf {R} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>:</mo> <munderover> <mo>⨁</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munderover> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mi>M</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{p}\colon \bigoplus _{n=1}^{k}T_{p}M\to \mathbf {R} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85b2cac7957e7382e777204e16471b133445fd11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.43ex; height:7.343ex;" alt="{\displaystyle \beta _{p}\colon \bigoplus _{n=1}^{k}T_{p}M\to \mathbf {R} .}"></dd></dl> Consequently, a differential k-form may be evaluated against any k-tuple of tangent vectors to the same point p of M. For example, a differential 1-form α assigns to each point p ∈ M a <a href="/wiki/Linear_functional" class="mw-redirect" title="Linear functional">linear functional</a> αp on TpM. In the presence of an <a href="/wiki/Inner_product" class="mw-redirect" title="Inner product">inner product</a> on TpM (induced by a <a href="/wiki/Riemannian_metric" class="mw-redirect" title="Riemannian metric">Riemannian metric</a> on M), αp may be <a href="/wiki/Riesz_representation_theorem" title="Riesz representation theorem">represented</a> as the inner product with a <a href="/wiki/Tangent_vector" title="Tangent vector">tangent vector</a> Xp. Differential 1-forms are sometimes called <a href="/wiki/Covariance_and_contravariance_of_vectors" title="Covariance and contravariance of vectors">covariant vector fields</a>, covector fields, or "dual vector fields", particularly within physics. The exterior algebra may be embedded in the tensor algebra by means of the alternation map. The alternation map is defined as a mapping <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Alt} \colon {\bigotimes }^{k}T^{*}M\to {\bigotimes }^{k}T^{*}M.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Alt</mi> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo>⨂</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>M</mi> <mo stretchy="false">→</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo>⨂</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>M</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Alt} \colon {\bigotimes }^{k}T^{*}M\to {\bigotimes }^{k}T^{*}M.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47d99f1eaddf9b9f5d5e111d286923673eb9935e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:28.221ex; height:4.343ex;" alt="{\displaystyle \operatorname {Alt} \colon {\bigotimes }^{k}T^{*}M\to {\bigotimes }^{k}T^{*}M.}"></dd></dl> For a tensor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }"> at a point p, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Alt} (\tau _{p})(x_{1},\dots ,x_{k})={\frac {1}{k!}}\sum _{\sigma \in S_{k}}\operatorname {sgn}(\sigma )\tau _{p}(x_{\sigma (1)},\dots ,x_{\sigma (k)}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Alt</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">)</mo> <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>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>k</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> <mo>∈</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </munder> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>σ</mi> <mo stretchy="false">)</mo> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Alt} (\tau _{p})(x_{1},\dots ,x_{k})={\frac {1}{k!}}\sum _{\sigma \in S_{k}}\operatorname {sgn}(\sigma )\tau _{p}(x_{\sigma (1)},\dots ,x_{\sigma (k)}),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ae755d5a481cc9ff8a9703675f5d102c9bfd528" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:54.901ex; height:6.843ex;" alt="{\displaystyle \operatorname {Alt} (\tau _{p})(x_{1},\dots ,x_{k})={\frac {1}{k!}}\sum _{\sigma \in S_{k}}\operatorname {sgn}(\sigma )\tau _{p}(x_{\sigma (1)},\dots ,x_{\sigma (k)}),}"></dd></dl> where Sk is the <a href="/wiki/Symmetric_group" title="Symmetric group">symmetric group</a> on k elements. The alternation map is constant on the cosets of the ideal in the tensor algebra generated by the symmetric 2-forms, and therefore descends to an embedding <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Alt} \colon {\textstyle \bigwedge }^{k}T^{*}M\to {\bigotimes }^{k}T^{*}M.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Alt</mi> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>⋀</mo> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>M</mi> <mo stretchy="false">→</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo>⨂</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>M</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Alt} \colon {\textstyle \bigwedge }^{k}T^{*}M\to {\bigotimes }^{k}T^{*}M.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85621bff4f78b8c4b46cef3d725d8d22ecc30b71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:26.646ex; height:4.343ex;" alt="{\displaystyle \operatorname {Alt} \colon {\textstyle \bigwedge }^{k}T^{*}M\to {\bigotimes }^{k}T^{*}M.}"></dd></dl> This map exhibits β as a <a href="/wiki/Antisymmetric_tensor" title="Antisymmetric tensor">totally antisymmetric</a> <a href="/wiki/Covariance_and_contravariance_of_vectors" title="Covariance and contravariance of vectors">covariant</a> <a href="/wiki/Tensor_field" title="Tensor field">tensor field</a> of rank k. The differential forms on M are in one-to-one correspondence with such tensor fields. <div class="mw-heading mw-heading2"><h2 id="Operations">Operations</h2>[<a href="/w/index.php?title=Differential_form&action=edit&section=8" title="Edit section: Operations">edit</a>]</div> As well as the addition and multiplication by scalar operations which arise from the vector space structure, there are several other standard operations defined on differential forms. The most important operations are the <a href="/wiki/Exterior_product" class="mw-redirect" title="Exterior product">exterior product</a> of two differential forms, the <a href="/wiki/Exterior_derivative" title="Exterior derivative">exterior derivative</a> of a single differential form, the <a href="/wiki/Interior_product" title="Interior product">interior product</a> of a differential form and a vector field, the <a href="/wiki/Lie_derivative" title="Lie derivative">Lie derivative</a> of a differential form with respect to a vector field and the <a href="/wiki/Covariant_derivative" title="Covariant derivative">covariant derivative</a> of a differential form with respect to a vector field on a manifold with a defined connection. <div class="mw-heading mw-heading3"><h3 id="Exterior_product">Exterior product</h3>[<a href="/w/index.php?title=Differential_form&action=edit&section=9" title="Edit section: Exterior product">edit</a>]</div> The exterior product of a k-form α and an ℓ-form β, denoted α ∧ β, is a (k + ℓ)-form. At each point p of the manifold M, the forms α and β are elements of an exterior power of the cotangent space at p. When the exterior algebra is viewed as a quotient of the tensor algebra, the exterior product corresponds to the tensor product (modulo the equivalence relation defining the exterior algebra). The antisymmetry inherent in the exterior algebra means that when α ∧ β is viewed as a multilinear functional, it is alternating. However, when the exterior algebra is embedded as a subspace of the tensor algebra by means of the alternation map, the tensor product α ⊗ β is not alternating. There is an explicit formula which describes the exterior product in this situation. The exterior product is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \wedge \beta =\operatorname {Alt} (\alpha \otimes \beta ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>∧</mo> <mi>β</mi> <mo>=</mo> <mi>Alt</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>⊗</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \wedge \beta =\operatorname {Alt} (\alpha \otimes \beta ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64e370f8f91d909d57a35aecb464cb1d58fdcc54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.911ex; height:2.843ex;" alt="{\displaystyle \alpha \wedge \beta =\operatorname {Alt} (\alpha \otimes \beta ).}"></dd></dl> If the embedding of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textstyle \bigwedge }^{n}T^{*}M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>⋀</mo> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\textstyle \bigwedge }^{n}T^{*}M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07980b360534afeefdd711bbf6752f15c01e2f37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.371ex; height:3.009ex;" alt="{\displaystyle {\textstyle \bigwedge }^{n}T^{*}M}"> into <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bigotimes }^{n}T^{*}M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo>⨂</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bigotimes }^{n}T^{*}M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30a31e0cccfa65190633377f7f3b627254143783" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:9.945ex; height:3.843ex;" alt="{\displaystyle {\bigotimes }^{n}T^{*}M}"> is done via the map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n!\operatorname {Alt} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>!</mo> <mi>Alt</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n!\operatorname {Alt} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79f13656f832eee913f507545fd419635992c4d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.723ex; height:2.176ex;" alt="{\displaystyle n!\operatorname {Alt} }"> instead of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Alt} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Alt</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Alt} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cada76b2ad3c1e5cbed0390716541e46b750f73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.295ex; height:2.176ex;" alt="{\displaystyle \operatorname {Alt} }">, the exterior product is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \wedge \beta ={\frac {(k+\ell )!}{k!\ell !}}\operatorname {Alt} (\alpha \otimes \beta ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>∧</mo> <mi>β</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mi>ℓ</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> <mrow> <mi>k</mi> <mo>!</mo> <mi>ℓ</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mi>Alt</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>⊗</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \wedge \beta ={\frac {(k+\ell )!}{k!\ell !}}\operatorname {Alt} (\alpha \otimes \beta ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba3e9590fd24a84e793df92bd1968248d85d98b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:28.612ex; height:5.843ex;" alt="{\displaystyle \alpha \wedge \beta ={\frac {(k+\ell )!}{k!\ell !}}\operatorname {Alt} (\alpha \otimes \beta ).}"></dd></dl> This description is useful for explicit computations. For example, if k = ℓ = 1, then α ∧ β is the 2-form whose value at a point p is the <a href="/wiki/Alternating_bilinear_form" class="mw-redirect" title="Alternating bilinear form">alternating bilinear form</a> defined by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\alpha \wedge \beta )_{p}(v,w)=\alpha _{p}(v)\beta _{p}(w)-\alpha _{p}(w)\beta _{p}(v)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>α</mi> <mo>∧</mo> <mi>β</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>v</mi> <mo>,</mo> <mi>w</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>w</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>w</mi> <mo stretchy="false">)</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\alpha \wedge \beta )_{p}(v,w)=\alpha _{p}(v)\beta _{p}(w)-\alpha _{p}(w)\beta _{p}(v)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9408eb2ea9ad495cc2b6760e0eca2902de04ef70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:42.509ex; height:3.009ex;" alt="{\displaystyle (\alpha \wedge \beta )_{p}(v,w)=\alpha _{p}(v)\beta _{p}(w)-\alpha _{p}(w)\beta _{p}(v)}"></dd></dl> for v, w ∈ TpM. The exterior product is bilinear: If α, β, and γ are any differential forms, and if f is any smooth function, then <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \wedge (\beta +\gamma )=\alpha \wedge \beta +\alpha \wedge \gamma ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>β</mi> <mo>+</mo> <mi>γ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>α</mi> <mo>∧</mo> <mi>β</mi> <mo>+</mo> <mi>α</mi> <mo>∧</mo> <mi>γ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \wedge (\beta +\gamma )=\alpha \wedge \beta +\alpha \wedge \gamma ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d05c888c855b7fffc883b5db15367747f25be21f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.635ex; height:2.843ex;" alt="{\displaystyle \alpha \wedge (\beta +\gamma )=\alpha \wedge \beta +\alpha \wedge \gamma ,}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \wedge (f\cdot \beta )=f\cdot (\alpha \wedge \beta ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo>⋅</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>∧</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \wedge (f\cdot \beta )=f\cdot (\alpha \wedge \beta ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f7e75119f6068ebce781f0ca4ed79d443d3f982" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.084ex; height:2.843ex;" alt="{\displaystyle \alpha \wedge (f\cdot \beta )=f\cdot (\alpha \wedge \beta ).}"></dd></dl> It is skew commutative (also known as graded commutative), meaning that it satisfies a variant of <a href="/wiki/Anticommutativity" class="mw-redirect" title="Anticommutativity">anticommutativity</a> that depends on the degrees of the forms: if α is a k-form and β is an ℓ-form, then <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \wedge \beta =(-1)^{k\ell }\beta \wedge \alpha .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>∧</mo> <mi>β</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>ℓ</mi> </mrow> </msup> <mi>β</mi> <mo>∧</mo> <mi>α</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \wedge \beta =(-1)^{k\ell }\beta \wedge \alpha .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65283427068fc9d8e56fccc4b22a1fd4c2550e47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.104ex; height:3.176ex;" alt="{\displaystyle \alpha \wedge \beta =(-1)^{k\ell }\beta \wedge \alpha .}"></dd></dl> One also has the <a href="/wiki/Differential_graded_algebra" title="Differential graded algebra">graded Leibniz rule</a>:<blockquote><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(\alpha \wedge \beta )=d\alpha \wedge \beta +(-1)^{k}\alpha \wedge d\beta .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo>∧</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>d</mi> <mi>α</mi> <mo>∧</mo> <mi>β</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mi>α</mi> <mo>∧</mo> <mi>d</mi> <mi>β</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(\alpha \wedge \beta )=d\alpha \wedge \beta +(-1)^{k}\alpha \wedge d\beta .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b6c514c33202abff26abff19e89eafdb248f68f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.118ex; height:3.176ex;" alt="{\displaystyle d(\alpha \wedge \beta )=d\alpha \wedge \beta +(-1)^{k}\alpha \wedge d\beta .}"></blockquote> <div class="mw-heading mw-heading3"><h3 id="Riemannian_manifold">Riemannian manifold</h3>[<a href="/w/index.php?title=Differential_form&action=edit&section=10" title="Edit section: Riemannian manifold">edit</a>]</div> On a <a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a>, or more generally a <a href="/wiki/Pseudo-Riemannian_manifold" title="Pseudo-Riemannian manifold">pseudo-Riemannian manifold</a>, the metric defines a fibre-wise isomorphism of the tangent and cotangent bundles. This makes it possible to convert vector fields to covector fields and vice versa. It also enables the definition of additional operations such as the <a href="/wiki/Hodge_star_operator" title="Hodge star operator">Hodge star operator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \star \colon \Omega ^{k}(M)\ {\stackrel {\sim }{\to }}\ \Omega ^{n-k}(M)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋆</mo> <mo>:</mo> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">→</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∼</mo> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \star \colon \Omega ^{k}(M)\ {\stackrel {\sim }{\to }}\ \Omega ^{n-k}(M)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a893ef4120fd98141515bd6e8446247f692d4890" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.983ex; height:3.343ex;" alt="{\displaystyle \star \colon \Omega ^{k}(M)\ {\stackrel {\sim }{\to }}\ \Omega ^{n-k}(M)}"> and the <a href="/wiki/Hodge_star#Codifferential" class="mw-redirect" title="Hodge star">codifferential</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta \colon \Omega ^{k}(M)\rightarrow \Omega ^{k-1}(M)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo>:</mo> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta \colon \Omega ^{k}(M)\rightarrow \Omega ^{k-1}(M)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f579491dfe04e01e72245ada390c0e07c8c9a4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.834ex; height:3.176ex;" alt="{\displaystyle \delta \colon \Omega ^{k}(M)\rightarrow \Omega ^{k-1}(M)}">, which has degree −1 and is <a href="/wiki/Differential_operator#Adjoint_of_an_operator" title="Differential operator">adjoint</a> to the exterior differential d. <div class="mw-heading mw-heading4"><h4 id="Vector_field_structures">Vector field structures</h4>[<a href="/w/index.php?title=Differential_form&action=edit&section=11" title="Edit section: Vector field structures">edit</a>]</div> On a pseudo-Riemannian manifold, 1-forms can be identified with vector fields; vector fields have additional distinct algebraic structures, which are listed here for context and to avoid confusion. Firstly, each (co)tangent space generates a <a href="/wiki/Clifford_algebra" title="Clifford algebra">Clifford algebra</a>, where the product of a (co)vector with itself is given by the value of a quadratic form – in this case, the natural one induced by the <a href="/wiki/Metric_tensor" title="Metric tensor">metric</a>. This algebra is distinct from the <a href="/wiki/Exterior_algebra" title="Exterior algebra">exterior algebra</a> of differential forms, which can be viewed as a Clifford algebra where the quadratic form vanishes (since the exterior product of any vector with itself is zero). Clifford algebras are thus non-anticommutative ("quantum") deformations of the exterior algebra. They are studied in <a href="/wiki/Geometric_algebra" title="Geometric algebra">geometric algebra</a>. Another alternative is to consider vector fields as derivations. The (noncommutative) algebra of <a href="/wiki/Differential_operator" title="Differential operator">differential operators</a> they generate is the <a href="/wiki/Weyl_algebra" title="Weyl algebra">Weyl algebra</a> and is a noncommutative ("quantum") deformation of the symmetric algebra in the vector fields. <div class="mw-heading mw-heading3"><h3 id="Exterior_differential_complex">Exterior differential complex</h3>[<a href="/w/index.php?title=Differential_form&action=edit&section=12" title="Edit section: Exterior differential complex">edit</a>]</div> One important property of the exterior derivative is that d2 = 0. This means that the exterior derivative defines a <a href="/wiki/Cochain_complex" class="mw-redirect" title="Cochain complex">cochain complex</a>: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\ \to \ \Omega ^{0}(M)\ {\stackrel {d}{\to }}\ \Omega ^{1}(M)\ {\stackrel {d}{\to }}\ \Omega ^{2}(M)\ {\stackrel {d}{\to }}\ \Omega ^{3}(M)\ \to \ \cdots \ \to \ \Omega ^{n}(M)\ \to \ 0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mtext> </mtext> <mo stretchy="false">→</mo> <mtext> </mtext> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">→</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">→</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">→</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo stretchy="false">→</mo> <mtext> </mtext> <mo>⋯</mo> <mtext> </mtext> <mo stretchy="false">→</mo> <mtext> </mtext> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo stretchy="false">→</mo> <mtext> </mtext> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\ \to \ \Omega ^{0}(M)\ {\stackrel {d}{\to }}\ \Omega ^{1}(M)\ {\stackrel {d}{\to }}\ \Omega ^{2}(M)\ {\stackrel {d}{\to }}\ \Omega ^{3}(M)\ \to \ \cdots \ \to \ \Omega ^{n}(M)\ \to \ 0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd2f3b559dcc67537affdd40a6d3754b4ec7e6e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:71.109ex; height:4.176ex;" alt="{\displaystyle 0\ \to \ \Omega ^{0}(M)\ {\stackrel {d}{\to }}\ \Omega ^{1}(M)\ {\stackrel {d}{\to }}\ \Omega ^{2}(M)\ {\stackrel {d}{\to }}\ \Omega ^{3}(M)\ \to \ \cdots \ \to \ \Omega ^{n}(M)\ \to \ 0.}"></dd></dl> This complex is called the de Rham complex, and its <a href="/wiki/Cohomology" title="Cohomology">cohomology</a> is by definition the <a href="/wiki/De_Rham_cohomology" title="De Rham cohomology">de Rham cohomology</a> of M. By the <a href="/wiki/Poincar%C3%A9_lemma" title="Poincaré lemma">Poincaré lemma</a>, the de Rham complex is locally <a href="/wiki/Exact_sequence" title="Exact sequence">exact</a> except at Ω0(M). The kernel at Ω0(M) is the space of <a href="/wiki/Locally_constant_function" title="Locally constant function">locally constant functions</a> on M. Therefore, the complex is a resolution of the constant <a href="/wiki/Sheaf_(mathematics)" title="Sheaf (mathematics)">sheaf</a> R, which in turn implies a form of de Rham's theorem: de Rham cohomology computes the <a href="/wiki/Sheaf_cohomology" title="Sheaf cohomology">sheaf cohomology</a> of R. <div class="mw-heading mw-heading2"><h2 id="Pullback">Pullback</h2>[<a href="/w/index.php?title=Differential_form&action=edit&section=13" title="Edit section: Pullback">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Pullback_(differential_geometry)" title="Pullback (differential geometry)">Pullback (differential geometry)</a></div> Suppose that f : M → N is smooth. The differential of f is a smooth map df : TM → TN between the tangent bundles of M and N. This map is also denoted f∗ and called the pushforward. For any point p ∈ M and any tangent vector v ∈ TpM, there is a well-defined pushforward vector f∗(v) in Tf(p)N. However, the same is not true of a vector field. If f is not injective, say because q ∈ N has two or more preimages, then the vector field may determine two or more distinct vectors in TqN. If f is not surjective, then there will be a point q ∈ N at which f∗ does not determine any tangent vector at all. Since a vector field on N determines, by definition, a unique tangent vector at every point of N, the pushforward of a vector field does not always exist. By contrast, it is always possible to pull back a differential form. A differential form on N may be viewed as a linear functional on each tangent space. Precomposing this functional with the differential df : TM → TN defines a linear functional on each tangent space of M and therefore a differential form on M. The existence of pullbacks is one of the key features of the theory of differential forms. It leads to the existence of pullback maps in other situations, such as pullback homomorphisms in de Rham cohomology. Formally, let f : M → N be smooth, and let ω be a smooth k-form on N. Then there is a differential form f∗ω on M, called the pullback of ω, which captures the behavior of ω as seen relative to f. To define the pullback, fix a point p of M and tangent vectors v1, ..., vk to M at p. The pullback of ω is defined by the formula <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f^{*}\omega )_{p}(v_{1},\ldots ,v_{k})=\omega _{f(p)}(f_{*}v_{1},\ldots ,f_{*}v_{k}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>ω</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f^{*}\omega )_{p}(v_{1},\ldots ,v_{k})=\omega _{f(p)}(f_{*}v_{1},\ldots ,f_{*}v_{k}).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cf28d38d6cec6fcb9916b65efe7435ceb359754" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:42.281ex; height:3.176ex;" alt="{\displaystyle (f^{*}\omega )_{p}(v_{1},\ldots ,v_{k})=\omega _{f(p)}(f_{*}v_{1},\ldots ,f_{*}v_{k}).}"></dd></dl> There are several more abstract ways to view this definition. If ω is a 1-form on N, then it may be viewed as a section of the cotangent bundle T∗N of N. Using ∗ to denote a dual map, the dual to the differential of f is (df)∗ : T∗N → T∗M. The pullback of ω may be defined to be the composite <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\ {\stackrel {f}{\to }}\ N\ {\stackrel {\omega }{\to }}\ T^{*}N\ {\stackrel {(df)^{*}}{\longrightarrow }}\ T^{*}M.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">→</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <mi>N</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">→</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>N</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">⟶</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>d</mi> <mi>f</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>M</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M\ {\stackrel {f}{\to }}\ N\ {\stackrel {\omega }{\to }}\ T^{*}N\ {\stackrel {(df)^{*}}{\longrightarrow }}\ T^{*}M.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f908d3dc04751cee07ca6cd1fbb8c0a23db4c66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:27.213ex; height:4.009ex;" alt="{\displaystyle M\ {\stackrel {f}{\to }}\ N\ {\stackrel {\omega }{\to }}\ T^{*}N\ {\stackrel {(df)^{*}}{\longrightarrow }}\ T^{*}M.}"></dd></dl> This is a section of the cotangent bundle of M and hence a differential 1-form on M. In full generality, let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \bigwedge ^{k}(df)^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mover> <mo>⋀</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </mover> <mo stretchy="false">(</mo> <mi>d</mi> <mi>f</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \bigwedge ^{k}(df)^{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4433bb78dabfd29e79d715d6707ae68c4f11a0b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.383ex; height:3.343ex;" alt="{\textstyle \bigwedge ^{k}(df)^{*}}"> denote the kth exterior power of the dual map to the differential. Then the pullback of a k-form ω is the composite <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\ {\stackrel {f}{\to }}\ N\ {\stackrel {\omega }{\to }}\ {\textstyle \bigwedge }^{k}T^{*}N\ {\stackrel {{\bigwedge }^{k}(df)^{*}}{\longrightarrow }}\ {\textstyle \bigwedge }^{k}T^{*}M.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">→</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <mi>N</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">→</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>⋀</mo> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>N</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">⟶</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo>⋀</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>d</mi> <mi>f</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>⋀</mo> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>M</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M\ {\stackrel {f}{\to }}\ N\ {\stackrel {\omega }{\to }}\ {\textstyle \bigwedge }^{k}T^{*}N\ {\stackrel {{\bigwedge }^{k}(df)^{*}}{\longrightarrow }}\ {\textstyle \bigwedge }^{k}T^{*}M.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0478733a69d46c69ce13ac590eb15bd3e4ed35e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.491ex; height:4.843ex;" alt="{\displaystyle M\ {\stackrel {f}{\to }}\ N\ {\stackrel {\omega }{\to }}\ {\textstyle \bigwedge }^{k}T^{*}N\ {\stackrel {{\bigwedge }^{k}(df)^{*}}{\longrightarrow }}\ {\textstyle \bigwedge }^{k}T^{*}M.}"></dd></dl> Another abstract way to view the pullback comes from viewing a k-form ω as a linear functional on tangent spaces. From this point of view, ω is a morphism of <a href="/wiki/Vector_bundle" title="Vector bundle">vector bundles</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textstyle \bigwedge }^{k}TN\ {\stackrel {\omega }{\to }}\ N\times \mathbf {R} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>⋀</mo> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mi>T</mi> <mi>N</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">→</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <mi>N</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\textstyle \bigwedge }^{k}TN\ {\stackrel {\omega }{\to }}\ N\times \mathbf {R} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97303ff95a376eeaecd80022ab7a82195be1ff15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.764ex; height:3.676ex;" alt="{\displaystyle {\textstyle \bigwedge }^{k}TN\ {\stackrel {\omega }{\to }}\ N\times \mathbf {R} ,}"></dd></dl> where N × R is the trivial rank one bundle on N. The composite map <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textstyle \bigwedge }^{k}TM\ {\stackrel {{\bigwedge }^{k}df}{\longrightarrow }}\ {\textstyle \bigwedge }^{k}TN\ {\stackrel {\omega }{\to }}\ N\times \mathbf {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>⋀</mo> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mi>T</mi> <mi>M</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">⟶</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo>⋀</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mi>d</mi> <mi>f</mi> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>⋀</mo> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mi>T</mi> <mi>N</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">→</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <mi>N</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\textstyle \bigwedge }^{k}TM\ {\stackrel {{\bigwedge }^{k}df}{\longrightarrow }}\ {\textstyle \bigwedge }^{k}TN\ {\stackrel {\omega }{\to }}\ N\times \mathbf {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b6dba872c8d2ca87d10c0be21f5a2452a9627eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.373ex; height:4.843ex;" alt="{\displaystyle {\textstyle \bigwedge }^{k}TM\ {\stackrel {{\bigwedge }^{k}df}{\longrightarrow }}\ {\textstyle \bigwedge }^{k}TN\ {\stackrel {\omega }{\to }}\ N\times \mathbf {R} }"></dd></dl> defines a linear functional on each tangent space of M, and therefore it factors through the trivial bundle M × R. The vector bundle morphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\textstyle \bigwedge }^{k}TM\to M\times \mathbf {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>⋀</mo> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mi>T</mi> <mi>M</mi> <mo stretchy="false">→</mo> <mi>M</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\textstyle \bigwedge }^{k}TM\to M\times \mathbf {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8cdf3faf24dbd8ae93a7865be9a22c06d95f534" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.003ex; height:3.343ex;" alt="{\textstyle {\textstyle \bigwedge }^{k}TM\to M\times \mathbf {R} }"> defined in this way is f∗ω. Pullback respects all of the basic operations on forms. If ω and η are forms and c is a real number, then <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}f^{*}(c\omega )&=c(f^{*}\omega ),\\f^{*}(\omega +\eta )&=f^{*}\omega +f^{*}\eta ,\\f^{*}(\omega \wedge \eta )&=f^{*}\omega \wedge f^{*}\eta ,\\f^{*}(d\omega )&=d(f^{*}\omega ).\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> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>c</mi> <mi>ω</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>c</mi> <mo stretchy="false">(</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>ω</mi> <mo>+</mo> <mi>η</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>ω</mi> <mo>+</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>η</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>ω</mi> <mo>∧</mo> <mi>η</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>ω</mi> <mo>∧</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>η</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>d</mi> <mi>ω</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>d</mi> <mo stretchy="false">(</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}f^{*}(c\omega )&=c(f^{*}\omega ),\\f^{*}(\omega +\eta )&=f^{*}\omega +f^{*}\eta ,\\f^{*}(\omega \wedge \eta )&=f^{*}\omega \wedge f^{*}\eta ,\\f^{*}(d\omega )&=d(f^{*}\omega ).\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec885091abaf1eb731bb87e67f138adff71e6676" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:24.342ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}f^{*}(c\omega )&=c(f^{*}\omega ),\\f^{*}(\omega +\eta )&=f^{*}\omega +f^{*}\eta ,\\f^{*}(\omega \wedge \eta )&=f^{*}\omega \wedge f^{*}\eta ,\\f^{*}(d\omega )&=d(f^{*}\omega ).\end{aligned}}}"></dd></dl> The pullback of a form can also be written in coordinates. Assume that x1, ..., xm are coordinates on M, that y1, ..., yn are coordinates on N, and that these coordinate systems are related by the formulas yi = fi(x1, ..., xm) for all i. Locally on N, ω can be written as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega =\sum _{i_{1}<\cdots <i_{k}}\omega _{i_{1}\cdots i_{k}}\,dy^{i_{1}}\wedge \cdots \wedge dy^{i_{k}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo><</mo> <mo>⋯</mo> <mo><</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </munder> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <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"> <mi>k</mi> </mrow> </msub> </mrow> </msub> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> <mo>∧</mo> <mo>⋯</mo> <mo>∧</mo> <mi>d</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega =\sum _{i_{1}<\cdots <i_{k}}\omega _{i_{1}\cdots i_{k}}\,dy^{i_{1}}\wedge \cdots \wedge dy^{i_{k}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e99161659367c4ce201c7417c25f06f1d33d87e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:35.635ex; height:5.843ex;" alt="{\displaystyle \omega =\sum _{i_{1}<\cdots <i_{k}}\omega _{i_{1}\cdots i_{k}}\,dy^{i_{1}}\wedge \cdots \wedge dy^{i_{k}},}"></dd></dl> where, for each choice of i1, ..., ik, ωi1⋅⋅⋅ik is a real-valued function of y1, ..., yn. Using the linearity of pullback and its compatibility with exterior product, the pullback of ω has the formula <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{*}\omega =\sum _{i_{1}<\cdots <i_{k}}(\omega _{i_{1}\cdots i_{k}}\circ f)\,df_{i_{1}}\wedge \cdots \wedge df_{i_{k}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>ω</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo><</mo> <mo>⋯</mo> <mo><</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <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"> <mi>k</mi> </mrow> </msub> </mrow> </msub> <mo>∘</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo>∧</mo> <mo>⋯</mo> <mo>∧</mo> <mi>d</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{*}\omega =\sum _{i_{1}<\cdots <i_{k}}(\omega _{i_{1}\cdots i_{k}}\circ f)\,df_{i_{1}}\wedge \cdots \wedge df_{i_{k}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83acbb0138869851c2549cce3bbbdcc5dfd71e98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:42.863ex; height:5.843ex;" alt="{\displaystyle f^{*}\omega =\sum _{i_{1}<\cdots <i_{k}}(\omega _{i_{1}\cdots i_{k}}\circ f)\,df_{i_{1}}\wedge \cdots \wedge df_{i_{k}}.}"></dd></dl> Each exterior derivative dfi can be expanded in terms of dx1, ..., dxm. The resulting k-form can be written using <a href="/wiki/Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian</a> matrices: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{*}\omega =\sum _{i_{1}<\cdots <i_{k}}\sum _{j_{1}<\cdots <j_{k}}(\omega _{i_{1}\cdots i_{k}}\circ f){\frac {\partial (f_{i_{1}},\ldots ,f_{i_{k}})}{\partial (x^{j_{1}},\ldots ,x^{j_{k}})}}\,dx^{j_{1}}\wedge \cdots \wedge dx^{j_{k}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>ω</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo><</mo> <mo>⋯</mo> <mo><</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </munder> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo><</mo> <mo>⋯</mo> <mo><</mo> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <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"> <mi>k</mi> </mrow> </msub> </mrow> </msub> <mo>∘</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> <mo>∧</mo> <mo>⋯</mo> <mo>∧</mo> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{*}\omega =\sum _{i_{1}<\cdots <i_{k}}\sum _{j_{1}<\cdots <j_{k}}(\omega _{i_{1}\cdots i_{k}}\circ f){\frac {\partial (f_{i_{1}},\ldots ,f_{i_{k}})}{\partial (x^{j_{1}},\ldots ,x^{j_{k}})}}\,dx^{j_{1}}\wedge \cdots \wedge dx^{j_{k}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3dcacea72b68e686df76df13540e74131b6d491a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:66.691ex; height:7.509ex;" alt="{\displaystyle f^{*}\omega =\sum _{i_{1}<\cdots <i_{k}}\sum _{j_{1}<\cdots <j_{k}}(\omega _{i_{1}\cdots i_{k}}\circ f){\frac {\partial (f_{i_{1}},\ldots ,f_{i_{k}})}{\partial (x^{j_{1}},\ldots ,x^{j_{k}})}}\,dx^{j_{1}}\wedge \cdots \wedge dx^{j_{k}}.}"></dd></dl> Here, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {\partial (f_{i_{1}},\ldots ,f_{i_{k}})}{\partial (x^{j_{1}},\ldots ,x^{j_{k}})}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {\partial (f_{i_{1}},\ldots ,f_{i_{k}})}{\partial (x^{j_{1}},\ldots ,x^{j_{k}})}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74924dabd955bc36b5a11a8f11d7c4f894d9c9e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.826ex; height:5.343ex;" alt="{\textstyle {\frac {\partial (f_{i_{1}},\ldots ,f_{i_{k}})}{\partial (x^{j_{1}},\ldots ,x^{j_{k}})}}}"> denotes the determinant of the matrix whose entries are <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {\partial f_{i_{m}}}{\partial x^{j_{n}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {\partial f_{i_{m}}}{\partial x^{j_{n}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef2ccbf9e1b6fa3e8410cbd4baca8cfe896acaf7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:4.503ex; height:4.509ex;" alt="{\textstyle {\frac {\partial f_{i_{m}}}{\partial x^{j_{n}}}}}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq m,n\leq k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>≤</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>≤</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\leq m,n\leq k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59174e4c31372263dfde35534c3ad664365666ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.04ex; height:2.509ex;" alt="{\displaystyle 1\leq m,n\leq k}">. <div class="mw-heading mw-heading2"><h2 id="Integration">Integration</h2>[<a href="/w/index.php?title=Differential_form&action=edit&section=14" title="Edit section: Integration">edit</a>]</div> A differential k-form can be integrated over an oriented k-dimensional manifold. When the k-form is defined on an n-dimensional manifold with n > k, then the k-form can be integrated over oriented k-dimensional submanifolds. If k = 0, integration over oriented 0-dimensional submanifolds is just the summation of the integrand evaluated at points, according to the orientation of those points. Other values of k = 1, 2, 3, ... correspond to line integrals, surface integrals, volume integrals, and so on. There are several equivalent ways to formally define the integral of a differential form, all of which depend on reducing to the case of Euclidean space. <div class="mw-heading mw-heading3"><h3 id="Integration_on_Euclidean_space">Integration on Euclidean space</h3>[<a href="/w/index.php?title=Differential_form&action=edit&section=15" title="Edit section: Integration on Euclidean space">edit</a>]</div> Let U be an open subset of Rn. Give Rn its standard orientation and U the restriction of that orientation. Every smooth n-form ω on U has the form <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega =f(x)\,dx^{1}\wedge \cdots \wedge dx^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>∧</mo> <mo>⋯</mo> <mo>∧</mo> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega =f(x)\,dx^{1}\wedge \cdots \wedge dx^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b184cf564269eb1fb65b9ae639be4961328608a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.601ex; height:3.176ex;" alt="{\displaystyle \omega =f(x)\,dx^{1}\wedge \cdots \wedge dx^{n}}"></dd></dl> for some smooth function f : Rn → R. Such a function has an integral in the usual Riemann or Lebesgue sense. This allows us to define the integral of ω to be the integral of f: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{U}\omega \ {\stackrel {\text{def}}{=}}\int _{U}f(x)\,dx^{1}\cdots dx^{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msub> <mi>ω</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>def</mtext> </mrow> </mover> </mrow> </mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>⋯</mo> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{U}\omega \ {\stackrel {\text{def}}{=}}\int _{U}f(x)\,dx^{1}\cdots dx^{n}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a96708ff183338b6b7eabb7c09ff6ef0ca1a5c50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.218ex; height:5.676ex;" alt="{\displaystyle \int _{U}\omega \ {\stackrel {\text{def}}{=}}\int _{U}f(x)\,dx^{1}\cdots dx^{n}.}"></dd></dl> Fixing an orientation is necessary for this to be well-defined. The skew-symmetry of differential forms means that the integral of, say, dx1 ∧ dx2 must be the negative of the integral of dx2 ∧ dx1. Riemann and Lebesgue integrals cannot see this dependence on the ordering of the coordinates, so they leave the sign of the integral undetermined. The orientation resolves this ambiguity. <div class="mw-heading mw-heading3"><h3 id="Integration_over_chains">Integration over chains</h3>[<a href="/w/index.php?title=Differential_form&action=edit&section=16" title="Edit section: Integration over chains">edit</a>]</div> Let M be an n-manifold and ω an n-form on M. First, assume that there is a parametrization of M by an open subset of Euclidean space. That is, assume that there exists a diffeomorphism <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \colon D\to M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>:</mo> <mi>D</mi> <mo stretchy="false">→</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi \colon D\to M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef380f8a39f96f217d37d90546daca30b8be31e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.535ex; height:2.676ex;" alt="{\displaystyle \varphi \colon D\to M}"></dd></dl> where D ⊆ Rn. Give M the orientation induced by φ. Then (<a href="#CITEREFRudin1976">Rudin 1976</a>) defines the integral of ω over M to be the integral of φ∗ω over D. In coordinates, this has the following expression. Fix an embedding of M in RI with coordinates x1, ..., xI. Then <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega =\sum _{i_{1}<\cdots <i_{n}}a_{i_{1},\ldots ,i_{n}}({\mathbf {x} })\,dx^{i_{1}}\wedge \cdots \wedge dx^{i_{n}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo><</mo> <mo>⋯</mo> <mo><</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> <mo>∧</mo> <mo>⋯</mo> <mo>∧</mo> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega =\sum _{i_{1}<\cdots <i_{n}}a_{i_{1},\ldots ,i_{n}}({\mathbf {x} })\,dx^{i_{1}}\wedge \cdots \wedge dx^{i_{n}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bb35a352bec19f14341b8100b6fdf11d07cf051" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:40.209ex; height:5.843ex;" alt="{\displaystyle \omega =\sum _{i_{1}<\cdots <i_{n}}a_{i_{1},\ldots ,i_{n}}({\mathbf {x} })\,dx^{i_{1}}\wedge \cdots \wedge dx^{i_{n}}.}"></dd></dl> Suppose that φ is defined by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi ({\mathbf {u} })=(x^{1}({\mathbf {u} }),\ldots ,x^{I}({\mathbf {u} })).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi ({\mathbf {u} })=(x^{1}({\mathbf {u} }),\ldots ,x^{I}({\mathbf {u} })).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbb64f82b6901794ee167d71d7be31dd7ecf5d93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.911ex; height:3.176ex;" alt="{\displaystyle \varphi ({\mathbf {u} })=(x^{1}({\mathbf {u} }),\ldots ,x^{I}({\mathbf {u} })).}"></dd></dl> Then the integral may be written in coordinates as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{M}\omega =\int _{D}\sum _{i_{1}<\cdots <i_{n}}a_{i_{1},\ldots ,i_{n}}(\varphi ({\mathbf {u} })){\frac {\partial (x^{i_{1}},\ldots ,x^{i_{n}})}{\partial (u^{1},\dots ,u^{n})}}\,du^{1}\cdots du^{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mi>ω</mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo><</mo> <mo>⋯</mo> <mo><</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>⋯</mo> <mi>d</mi> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{M}\omega =\int _{D}\sum _{i_{1}<\cdots <i_{n}}a_{i_{1},\ldots ,i_{n}}(\varphi ({\mathbf {u} })){\frac {\partial (x^{i_{1}},\ldots ,x^{i_{n}})}{\partial (u^{1},\dots ,u^{n})}}\,du^{1}\cdots du^{n},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b89e7893f7df02c405aaf31e10cffa8658ecc06a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:60.207ex; height:7.343ex;" alt="{\displaystyle \int _{M}\omega =\int _{D}\sum _{i_{1}<\cdots <i_{n}}a_{i_{1},\ldots ,i_{n}}(\varphi ({\mathbf {u} })){\frac {\partial (x^{i_{1}},\ldots ,x^{i_{n}})}{\partial (u^{1},\dots ,u^{n})}}\,du^{1}\cdots du^{n},}"></dd></dl> where <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial (x^{i_{1}},\ldots ,x^{i_{n}})}{\partial (u^{1},\ldots ,u^{n})}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial (x^{i_{1}},\ldots ,x^{i_{n}})}{\partial (u^{1},\ldots ,u^{n})}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e4ce3afe8ab62c2889014c38e7aebc661b7c880" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:15.197ex; height:6.676ex;" alt="{\displaystyle {\frac {\partial (x^{i_{1}},\ldots ,x^{i_{n}})}{\partial (u^{1},\ldots ,u^{n})}}}"></dd></dl> is the determinant of the <a href="/wiki/Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian</a>. The Jacobian exists because φ is differentiable. In general, an n-manifold cannot be parametrized by an open subset of Rn. But such a parametrization is always possible locally, so it is possible to define integrals over arbitrary manifolds by defining them as sums of integrals over collections of local parametrizations. Moreover, it is also possible to define parametrizations of k-dimensional subsets for k < n, and this makes it possible to define integrals of k-forms. To make this precise, it is convenient to fix a standard domain D in Rk, usually a cube or a simplex. A k-chain is a formal sum of smooth embeddings D → M. That is, it is a collection of smooth embeddings, each of which is assigned an integer multiplicity. Each smooth embedding determines a k-dimensional submanifold of M. If the chain is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c=\sum _{i=1}^{r}m_{i}\varphi _{i},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </munderover> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c=\sum _{i=1}^{r}m_{i}\varphi _{i},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dad7b70eb8df5f2ad4f5fa2c732c2a80fb7b4ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:13.654ex; height:6.843ex;" alt="{\displaystyle c=\sum _{i=1}^{r}m_{i}\varphi _{i},}"></dd></dl> then the integral of a k-form ω over c is defined to be the sum of the integrals over the terms of c: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{c}\omega =\sum _{i=1}^{r}m_{i}\int _{D}\varphi _{i}^{*}\omega .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mi>ω</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </munderover> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> <msubsup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mi>ω</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{c}\omega =\sum _{i=1}^{r}m_{i}\int _{D}\varphi _{i}^{*}\omega .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a891894a09dcc5a6b037f48f06678c6c5a473562" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:22.077ex; height:6.843ex;" alt="{\displaystyle \int _{c}\omega =\sum _{i=1}^{r}m_{i}\int _{D}\varphi _{i}^{*}\omega .}"></dd></dl> This approach to defining integration does not assign a direct meaning to integration over the whole manifold M. However, it is still possible to assign such a meaning indirectly because every smooth manifold may be smoothly <a href="/wiki/Triangulation_(topology)" title="Triangulation (topology)">triangulated</a> in an essentially unique way, and the integral over M may be defined to be the integral over the chain determined by a triangulation. <div class="mw-heading mw-heading3"><h3 id="Integration_using_partitions_of_unity">Integration using partitions of unity</h3>[<a href="/w/index.php?title=Differential_form&action=edit&section=17" title="Edit section: Integration using partitions of unity">edit</a>]</div> There is another approach, expounded in (<a href="#CITEREFDieudonné1972">Dieudonné 1972</a>), which does directly assign a meaning to integration over M, but this approach requires fixing an orientation of M. The integral of an n-form ω on an n-dimensional manifold is defined by working in charts. Suppose first that ω is supported on a single positively oriented chart. On this chart, it may be pulled back to an n-form on an open subset of Rn. Here, the form has a well-defined Riemann or Lebesgue integral as before. The change of variables formula and the assumption that the chart is positively oriented together ensure that the integral of ω is independent of the chosen chart. In the general case, use a partition of unity to write ω as a sum of n-forms, each of which is supported in a single positively oriented chart, and define the integral of ω to be the sum of the integrals of each term in the partition of unity. It is also possible to integrate k-forms on oriented k-dimensional submanifolds using this more intrinsic approach. The form is pulled back to the submanifold, where the integral is defined using charts as before. For example, given a path γ(t) : [0, 1] → R2, integrating a 1-form on the path is simply pulling back the form to a form f(t) dt on [0, 1], and this integral is the integral of the function f(t) on the interval. <div class="mw-heading mw-heading3"><h3 id="Integration_along_fibers">Integration along fibers</h3>[<a href="/w/index.php?title=Differential_form&action=edit&section=18" title="Edit section: Integration along fibers">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Integration_along_fibers" title="Integration along fibers">Integration along fibers</a></div> <a href="/wiki/Fubini%27s_theorem" title="Fubini's theorem">Fubini's theorem</a> states that the integral over a set that is a product may be computed as an iterated integral over the two factors in the product. This suggests that the integral of a differential form over a product ought to be computable as an iterated integral as well. The geometric flexibility of differential forms ensures that this is possible not just for products, but in more general situations as well. Under some hypotheses, it is possible to integrate along the fibers of a smooth map, and the analog of Fubini's theorem is the case where this map is the projection from a product to one of its factors. Because integrating a differential form over a submanifold requires fixing an orientation, a prerequisite to integration along fibers is the existence of a well-defined orientation on those fibers. Let M and N be two orientable manifolds of pure dimensions m and n, respectively. Suppose that f : M → N is a surjective submersion. This implies that each fiber f−1(y) is (m − n)-dimensional and that, around each point of M, there is a chart on which f looks like the projection from a product onto one of its factors. Fix x ∈ M and set y = f(x). Suppose that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\omega _{x}&\in {\textstyle \bigwedge }^{m}T_{x}^{*}M,\\\eta _{y}&\in {\textstyle \bigwedge }^{n}T_{y}^{*}N,\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> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>⋀</mo> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mi>M</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>⋀</mo> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mi>N</mi> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\omega _{x}&\in {\textstyle \bigwedge }^{m}T_{x}^{*}M,\\\eta _{y}&\in {\textstyle \bigwedge }^{n}T_{y}^{*}N,\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c1ed4145eef3190c02ff45a7a120c3572c3374e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.685ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}\omega _{x}&\in {\textstyle \bigwedge }^{m}T_{x}^{*}M,\\\eta _{y}&\in {\textstyle \bigwedge }^{n}T_{y}^{*}N,\end{aligned}}}"></dd></dl> and that ηy does not vanish. Following (<a href="#CITEREFDieudonné1972">Dieudonné 1972</a>), there is a unique <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{x}\in {\textstyle \bigwedge }^{m-n}T_{x}^{*}(f^{-1}(y))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>⋀</mo> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mi>n</mi> </mrow> </msup> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{x}\in {\textstyle \bigwedge }^{m-n}T_{x}^{*}(f^{-1}(y))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d9a726da6d6fe18e549c89ee39ce636f8176154" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.418ex; height:3.176ex;" alt="{\displaystyle \sigma _{x}\in {\textstyle \bigwedge }^{m-n}T_{x}^{*}(f^{-1}(y))}"></dd></dl> which may be thought of as the fibral part of ωx with respect to ηy. More precisely, define j : f−1(y) → M to be the inclusion. Then σx is defined by the property that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{x}=(f^{*}\eta _{y})_{x}\wedge \sigma '_{x}\in {\textstyle \bigwedge }^{m}T_{x}^{*}M,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>∧</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mo>′</mo> </msubsup> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>⋀</mo> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mi>M</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{x}=(f^{*}\eta _{y})_{x}\wedge \sigma '_{x}\in {\textstyle \bigwedge }^{m}T_{x}^{*}M,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ffa9a1dc0c5d09bc8ec91c6e2b778279b8a44d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:30.675ex; height:3.176ex;" alt="{\displaystyle \omega _{x}=(f^{*}\eta _{y})_{x}\wedge \sigma '_{x}\in {\textstyle \bigwedge }^{m}T_{x}^{*}M,}"></dd></dl> where <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma '_{x}\in {\textstyle \bigwedge }^{m-n}T_{x}^{*}M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mo>′</mo> </msubsup> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>⋀</mo> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mi>n</mi> </mrow> </msup> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma '_{x}\in {\textstyle \bigwedge }^{m-n}T_{x}^{*}M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab4c77882c126eac2da5212677d9153bc26639cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.433ex; height:3.176ex;" alt="{\displaystyle \sigma '_{x}\in {\textstyle \bigwedge }^{m-n}T_{x}^{*}M}"></dd></dl> is any (m − n)-covector for which <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{x}=j^{*}\sigma '_{x}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mo>′</mo> </msubsup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{x}=j^{*}\sigma '_{x}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/883569585dd8ec22863ee0bd69df18bbab06e016" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.757ex; height:2.676ex;" alt="{\displaystyle \sigma _{x}=j^{*}\sigma '_{x}.}"></dd></dl> The form σx may also be notated ωx / ηy. Moreover, for fixed y, σx varies smoothly with respect to x. That is, suppose that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega \colon f^{-1}(y)\to T^{*}M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>:</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega \colon f^{-1}(y)\to T^{*}M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6af0b4055031723afd2d066e5b1f6795fe0d95e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.928ex; height:3.176ex;" alt="{\displaystyle \omega \colon f^{-1}(y)\to T^{*}M}"></dd></dl> is a smooth section of the projection map; we say that ω is a smooth differential m-form on M along f−1(y). Then there is a smooth differential (m − n)-form σ on f−1(y) such that, at each x ∈ f−1(y), <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{x}=\omega _{x}/\eta _{y}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{x}=\omega _{x}/\eta _{y}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c12f91a8d44fbb7ccb2415851f45432aa799f523" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.231ex; height:3.009ex;" alt="{\displaystyle \sigma _{x}=\omega _{x}/\eta _{y}.}"></dd></dl> This form is denoted ω / ηy. The same construction works if ω is an m-form in a neighborhood of the fiber, and the same notation is used. A consequence is that each fiber f−1(y) is orientable. In particular, a choice of orientation forms on M and N defines an orientation of every fiber of f. The analog of Fubini's theorem is as follows. As before, M and N are two orientable manifolds of pure dimensions m and n, and f : M → N is a surjective submersion. Fix orientations of M and N, and give each fiber of f the induced orientation. Let ω be an m-form on M, and let η be an n-form on N that is almost everywhere positive with respect to the orientation of N. Then, for almost every y ∈ N, the form ω / ηy is a well-defined integrable m − n form on f−1(y). Moreover, there is an integrable n-form on N defined by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\mapsto {\bigg (}\int _{f^{-1}(y)}\omega /\eta _{y}{\bigg )}\,\eta _{y}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\mapsto {\bigg (}\int _{f^{-1}(y)}\omega /\eta _{y}{\bigg )}\,\eta _{y}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7247d0883e04e85ae30c5e797168913173c0251c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:23.441ex; height:6.343ex;" alt="{\displaystyle y\mapsto {\bigg (}\int _{f^{-1}(y)}\omega /\eta _{y}{\bigg )}\,\eta _{y}.}"></dd></dl> Denote this form by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bigg (}\int _{f^{-1}(y)}\omega /\eta {\bigg )}\,\eta .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mi>η</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bigg (}\int _{f^{-1}(y)}\omega /\eta {\bigg )}\,\eta .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f34f74b45503b77f471339833999d914220e428" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:16.601ex; height:6.343ex;" alt="{\displaystyle {\bigg (}\int _{f^{-1}(y)}\omega /\eta {\bigg )}\,\eta .}"></dd></dl> Then (<a href="#CITEREFDieudonné1972">Dieudonné 1972</a>) proves the generalized Fubini formula <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{M}\omega =\int _{N}{\bigg (}\int _{f^{-1}(y)}\omega /\eta {\bigg )}\,\eta .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mi>ω</mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mi>η</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{M}\omega =\int _{N}{\bigg (}\int _{f^{-1}(y)}\omega /\eta {\bigg )}\,\eta .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75d61c0bd0d63bc4a58a37c07cf5a37644e8f111" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:28.155ex; height:6.343ex;" alt="{\displaystyle \int _{M}\omega =\int _{N}{\bigg (}\int _{f^{-1}(y)}\omega /\eta {\bigg )}\,\eta .}"></dd></dl> It is also possible to integrate forms of other degrees along the fibers of a submersion. Assume the same hypotheses as before, and let α be a compactly supported (m − n + k)-form on M. Then there is a k-form γ on N which is the result of integrating α along the fibers of f. The form α is defined by specifying, at each y ∈ N, how γ pairs with each k-vector v at y, and the value of that pairing is an integral over f−1(y) that depends only on α, v, and the orientations of M and N. More precisely, at each y ∈ N, there is an isomorphism <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textstyle \bigwedge }^{k}T_{y}N\to {\textstyle \bigwedge }^{n-k}T_{y}^{*}N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>⋀</mo> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mi>N</mi> <mo stretchy="false">→</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>⋀</mo> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msup> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\textstyle \bigwedge }^{k}T_{y}N\to {\textstyle \bigwedge }^{n-k}T_{y}^{*}N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a671aea6f2940650f9965dc2095ae97cfe536e16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.236ex; height:3.509ex;" alt="{\displaystyle {\textstyle \bigwedge }^{k}T_{y}N\to {\textstyle \bigwedge }^{n-k}T_{y}^{*}N}"></dd></dl> defined by the interior product <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} \mapsto \mathbf {v} \,\lrcorner \,\zeta _{y},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mspace width="thinmathspace" /> <mo>⌟</mo> <mspace width="thinmathspace" /> <msub> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} \mapsto \mathbf {v} \,\lrcorner \,\zeta _{y},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae32d81c58eca6f7f828641609d5d9fbf5572ccf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.087ex; height:2.843ex;" alt="{\displaystyle \mathbf {v} \mapsto \mathbf {v} \,\lrcorner \,\zeta _{y},}"></dd></dl> for any choice of volume form ζ in the orientation of N. If x ∈ f−1(y), then a k-vector v at y determines an (n − k)-covector at x by pullback: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{*}(\mathbf {v} \,\lrcorner \,\zeta _{y})\in {\textstyle \bigwedge }^{n-k}T_{x}^{*}M.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mspace width="thinmathspace" /> <mo>⌟</mo> <mspace width="thinmathspace" /> <msub> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>⋀</mo> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msup> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mi>M</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{*}(\mathbf {v} \,\lrcorner \,\zeta _{y})\in {\textstyle \bigwedge }^{n-k}T_{x}^{*}M.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61d486403fe38f6ae083035a299126e6cec86803" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.593ex; height:3.509ex;" alt="{\displaystyle f^{*}(\mathbf {v} \,\lrcorner \,\zeta _{y})\in {\textstyle \bigwedge }^{n-k}T_{x}^{*}M.}"></dd></dl> Each of these covectors has an exterior product against α, so there is an (m − n)-form βv on M along f−1(y) defined by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\beta _{\mathbf {v} })_{x}=\left(\alpha _{x}\wedge f^{*}(\mathbf {v} \,\lrcorner \,\zeta _{y})\right){\big /}\zeta _{y}\in {\textstyle \bigwedge }^{m-n}T_{x}^{*}M.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>∧</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mspace width="thinmathspace" /> <mo>⌟</mo> <mspace width="thinmathspace" /> <msub> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo fence="true" stretchy="true" symmetric="true" maxsize="1.2em" minsize="1.2em">/</mo> </mrow> </mrow> <msub> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>⋀</mo> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mi>n</mi> </mrow> </msup> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mi>M</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\beta _{\mathbf {v} })_{x}=\left(\alpha _{x}\wedge f^{*}(\mathbf {v} \,\lrcorner \,\zeta _{y})\right){\big /}\zeta _{y}\in {\textstyle \bigwedge }^{m-n}T_{x}^{*}M.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca5341ffa14cab2dc8c1a91d96d8c8c96e71c40e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:43.655ex; height:3.343ex;" alt="{\displaystyle (\beta _{\mathbf {v} })_{x}=\left(\alpha _{x}\wedge f^{*}(\mathbf {v} \,\lrcorner \,\zeta _{y})\right){\big /}\zeta _{y}\in {\textstyle \bigwedge }^{m-n}T_{x}^{*}M.}"></dd></dl> This form depends on the orientation of N but not the choice of ζ. Then the k-form γ is uniquely defined by the property <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \gamma _{y},\mathbf {v} \rangle =\int _{f^{-1}(y)}\beta _{\mathbf {v} },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </msub> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \gamma _{y},\mathbf {v} \rangle =\int _{f^{-1}(y)}\beta _{\mathbf {v} },}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df2366a985b4c1547a632b346217b1397e85822e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:19.61ex; height:6.009ex;" alt="{\displaystyle \langle \gamma _{y},\mathbf {v} \rangle =\int _{f^{-1}(y)}\beta _{\mathbf {v} },}"></dd></dl> and γ is smooth (<a href="#CITEREFDieudonné1972">Dieudonné 1972</a>). This form also denoted α♭ and called the integral of α along the fibers of f. Integration along fibers is important for the construction of Gysin maps in de Rham cohomology. Integration along fibers satisfies the projection formula (<a href="#CITEREFDieudonné1972">Dieudonné 1972</a>). If λ is any ℓ-form on N, then <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ^{\flat }\wedge \lambda =(\alpha \wedge f^{*}\lambda )^{\flat }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">♭</mi> </mrow> </msup> <mo>∧</mo> <mi>λ</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>∧</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>λ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">♭</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ^{\flat }\wedge \lambda =(\alpha \wedge f^{*}\lambda )^{\flat }.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4803e954aed2442ef1d95cbb742b1476abb69a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.524ex; height:3.343ex;" alt="{\displaystyle \alpha ^{\flat }\wedge \lambda =(\alpha \wedge f^{*}\lambda )^{\flat }.}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Stokes's_theorem">Stokes's theorem</h3>[<a href="/w/index.php?title=Differential_form&action=edit&section=19" title="Edit section: Stokes's theorem">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Stokes%27s_theorem" class="mw-redirect" title="Stokes's theorem">Stokes's theorem</a></div> The fundamental relationship between the exterior derivative and integration is given by the <a href="/wiki/Stokes%27_theorem" title="Stokes' theorem">Stokes' theorem</a>: If ω is an (n − 1)-form with compact support on M and ∂M denotes the <a href="/wiki/Manifold#Manifold_with_boundary" title="Manifold">boundary</a> of M with its induced <a href="/wiki/Orientation_(mathematics)" class="mw-redirect" title="Orientation (mathematics)">orientation</a>, then <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{M}d\omega =\int _{\partial M}\omega .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mi>d</mi> <mi>ω</mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> <mi>M</mi> </mrow> </msub> <mi>ω</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{M}d\omega =\int _{\partial M}\omega .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/307e671f2b4e1a6d46dffed122e05bb90316ba2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.062ex; height:5.676ex;" alt="{\displaystyle \int _{M}d\omega =\int _{\partial M}\omega .}"></dd></dl> A key consequence of this is that "the integral of a closed form over homologous chains is equal": If ω is a closed k-form and M and N are k-chains that are homologous (such that M − N is the boundary of a (k + 1)-chain W), then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\int _{M}\omega =\int _{N}\omega }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mi>ω</mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mi>ω</mi> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\int _{M}\omega =\int _{N}\omega }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/928684b8d81d26cdb4c283cf58388c9b560c2d82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.61ex; height:3.176ex;" alt="{\displaystyle \textstyle {\int _{M}\omega =\int _{N}\omega }}">, since the difference is the integral <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \int _{W}d\omega =\int _{W}0=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>W</mi> </mrow> </msub> <mi>d</mi> <mi>ω</mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>W</mi> </mrow> </msub> <mn>0</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \int _{W}d\omega =\int _{W}0=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/414259701903c92b05693b056255f236910be584" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.061ex; height:3.176ex;" alt="{\displaystyle \textstyle \int _{W}d\omega =\int _{W}0=0}">. For example, if ω = df is the derivative of a potential function on the plane or Rn, then the integral of ω over a path from a to b does not depend on the choice of path (the integral is f(b) − f(a)), since different paths with given endpoints are <a href="/wiki/Homotopic" class="mw-redirect" title="Homotopic">homotopic</a>, hence homologous (a weaker condition). This case is called the <a href="/wiki/Gradient_theorem" title="Gradient theorem">gradient theorem</a>, and generalizes the <a href="/wiki/Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus">fundamental theorem of calculus</a>. This path independence is very useful in <a href="/wiki/Contour_integration" title="Contour integration">contour integration</a>. This theorem also underlies the duality between <a href="/wiki/De_Rham_cohomology" title="De Rham cohomology">de Rham cohomology</a> and the <a href="/wiki/Homology_(mathematics)" title="Homology (mathematics)">homology</a> of chains. <div class="mw-heading mw-heading3"><h3 id="Relation_with_measures">Relation with measures</h3>[<a href="/w/index.php?title=Differential_form&action=edit&section=20" title="Edit section: Relation with measures">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Density_on_a_manifold" title="Density on a manifold">Density on a manifold</a></div> On a general differentiable manifold (without additional structure), differential forms cannot be integrated over subsets of the manifold; this distinction is key to the distinction between differential forms, which are integrated over chains or oriented submanifolds, and measures, which are integrated over subsets. The simplest example is attempting to integrate the 1-form dx over the interval [0, 1]. Assuming the usual distance (and thus measure) on the real line, this integral is either 1 or −1, depending on orientation: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\int _{0}^{1}dx=1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mn>1</mn> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\int _{0}^{1}dx=1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9ec5b018e73810a8fb81105b9572012c554fe17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.762ex; height:3.676ex;" alt="{\displaystyle \textstyle {\int _{0}^{1}dx=1}}">, while <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\int _{1}^{0}dx=-\int _{0}^{1}dx=-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msubsup> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mo>−</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\int _{1}^{0}dx=-\int _{0}^{1}dx=-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e0f4b74aad2f95866c70d69f7d16271c133c7a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.365ex; height:3.676ex;" alt="{\displaystyle \textstyle {\int _{1}^{0}dx=-\int _{0}^{1}dx=-1}}">. By contrast, the integral of the measure |dx| on the interval is unambiguously 1 (i.e. the integral of the constant function 1 with respect to this measure is 1). Similarly, under a change of coordinates a differential n-form changes by the <a href="/wiki/Jacobian_determinant" class="mw-redirect" title="Jacobian determinant">Jacobian determinant</a> J, while a measure changes by the absolute value of the Jacobian determinant, |J|, which further reflects the issue of orientation. For example, under the map x ↦ −x on the line, the differential form dx pulls back to −dx; orientation has reversed; while the <a href="/wiki/Lebesgue_measure" title="Lebesgue measure">Lebesgue measure</a>, which here we denote |dx|, pulls back to |dx|; it does not change. In the presence of the additional data of an orientation, it is possible to integrate n-forms (top-dimensional forms) over the entire manifold or over compact subsets; integration over the entire manifold corresponds to integrating the form over the <a href="/wiki/Fundamental_class" title="Fundamental class">fundamental class</a> of the manifold, [M]. Formally, in the presence of an orientation, one may identify n-forms with <a href="/wiki/Densities_on_a_manifold" class="mw-redirect" title="Densities on a manifold">densities on a manifold</a>; densities in turn define a measure, and thus can be integrated (<a href="#CITEREFFolland1999">Folland 1999</a>, Section 11.4, pp. 361–362). On an orientable but not oriented manifold, there are two choices of orientation; either choice allows one to integrate n-forms over compact subsets, with the two choices differing by a sign. On a non-orientable manifold, n-forms and densities cannot be identified —notably, any top-dimensional form must vanish somewhere (there are no <a href="/wiki/Volume_form" title="Volume form">volume forms</a> on non-orientable manifolds), but there are nowhere-vanishing densities— thus while one can integrate densities over compact subsets, one cannot integrate n-forms. One can instead identify densities with top-dimensional <a href="/wiki/Volume_form#Relation_to_measures" title="Volume form">pseudoforms</a>. Even in the presence of an orientation, there is in general no meaningful way to integrate k-forms over subsets for k < n because there is no consistent way to use the ambient orientation to orient k-dimensional subsets. Geometrically, a k-dimensional subset can be turned around in place, yielding the same subset with the opposite orientation; for example, the horizontal axis in a plane can be rotated by 180 degrees. Compare the <a href="/wiki/Gram_determinant" class="mw-redirect" title="Gram determinant">Gram determinant</a> of a set of k vectors in an n-dimensional space, which, unlike the determinant of n vectors, is always positive, corresponding to a squared number. An orientation of a k-submanifold is therefore extra data not derivable from the ambient manifold. On a Riemannian manifold, one may define a k-dimensional <a href="/wiki/Hausdorff_measure" title="Hausdorff measure">Hausdorff measure</a> for any k (integer or real), which may be integrated over k-dimensional subsets of the manifold. A function times this Hausdorff measure can then be integrated over k-dimensional subsets, providing a measure-theoretic analog to integration of k-forms. The n-dimensional Hausdorff measure yields a density, as above. <div class="mw-heading mw-heading3"><h3 id="Currents">Currents</h3>[<a href="/w/index.php?title=Differential_form&action=edit&section=21" title="Edit section: Currents">edit</a>]</div> The differential form analog of a <a href="/wiki/Distribution_(mathematics)" title="Distribution (mathematics)">distribution</a> or generalized function is called a <a href="/wiki/Current_(mathematics)" title="Current (mathematics)">current</a>. The space of k-currents on M is the dual space to an appropriate space of differential k-forms. Currents play the role of generalized domains of integration, similar to but even more flexible than chains. <div class="mw-heading mw-heading2"><h2 id="Applications_in_physics">Applications in physics</h2>[<a href="/w/index.php?title=Differential_form&action=edit&section=22" title="Edit section: Applications in physics">edit</a>]</div> Differential forms arise in some important physical contexts. For example, in Maxwell's theory of <a href="/wiki/Electromagnetism" title="Electromagnetism">electromagnetism</a>, the Faraday 2-form, or <a href="/wiki/Electromagnetic_field_strength" class="mw-redirect" title="Electromagnetic field strength">electromagnetic field strength</a>, is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textbf {F}}={\frac {1}{2}}f_{ab}\,dx^{a}\wedge dx^{b}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">F</mtext> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> <mo>∧</mo> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\textbf {F}}={\frac {1}{2}}f_{ab}\,dx^{a}\wedge dx^{b}\,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aae9e7c27e275481a657d13cf845132f0eda97a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.86ex; height:5.176ex;" alt="{\displaystyle {\textbf {F}}={\frac {1}{2}}f_{ab}\,dx^{a}\wedge dx^{b}\,,}"></dd></dl> where the fab are formed from the electromagnetic fields <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {E}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>E</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {E}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bc18ae485a72f148e85ccbeff2b3dcdd4f5f3f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.843ex;" alt="{\displaystyle {\vec {E}}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {B}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83ae7d80cab55b606de217162280b2279142bbb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.843ex;" alt="{\displaystyle {\vec {B}}}">; e.g., f12 = Ez/c, f23 = −Bz, or equivalent definitions. This form is a special case of the <a href="/wiki/Curvature_form" title="Curvature form">curvature form</a> on the <a href="/wiki/U(1)" class="mw-redirect" title="U(1)">U(1)</a> <a href="/wiki/Principal_bundle" title="Principal bundle">principal bundle</a> on which both electromagnetism and general <a href="/wiki/Gauge_theories" class="mw-redirect" title="Gauge theories">gauge theories</a> may be described. The <a href="/wiki/Connection_form" title="Connection form">connection form</a> for the principal bundle is the vector potential, typically denoted by A, when represented in some gauge. One then has <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textbf {F}}=d{\textbf {A}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">F</mtext> </mrow> </mrow> <mo>=</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">A</mtext> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\textbf {F}}=d{\textbf {A}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42ab747c2036116ccb15e5d0dd5fa8022faef248" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.663ex; height:2.176ex;" alt="{\displaystyle {\textbf {F}}=d{\textbf {A}}.}"></dd></dl> The current 3-form is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textbf {J}}={\frac {1}{6}}j^{a}\,\varepsilon _{abcd}\,dx^{b}\wedge dx^{c}\wedge dx^{d}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">J</mtext> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <msup> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> <mspace width="thinmathspace" /> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> <mi>c</mi> <mi>d</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msup> <mo>∧</mo> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> <mo>∧</mo> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\textbf {J}}={\frac {1}{6}}j^{a}\,\varepsilon _{abcd}\,dx^{b}\wedge dx^{c}\wedge dx^{d}\,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02422160dec6693437eed137f7980233e78fe4c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:30.584ex; height:5.176ex;" alt="{\displaystyle {\textbf {J}}={\frac {1}{6}}j^{a}\,\varepsilon _{abcd}\,dx^{b}\wedge dx^{c}\wedge dx^{d}\,,}"></dd></dl> where ja are the four components of the current density. (Here it is a matter of convention to write Fab instead of fab, i.e. to use capital letters, and to write Ja instead of ja. However, the vector rsp. tensor components and the above-mentioned forms have different physical dimensions. Moreover, by decision of an international commission of the <a href="/wiki/International_Union_of_Pure_and_Applied_Physics" title="International Union of Pure and Applied Physics">International Union of Pure and Applied Physics</a>, the magnetic polarization vector has been called <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {J}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>J</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {J}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f64ab02528d2b80e1df79bc2a8762489a986afa8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.788ex; height:2.843ex;" alt="{\displaystyle {\vec {J}}}"> for several decades, and by some publishers J; i.e., the same name is used for different quantities.) Using the above-mentioned definitions, <a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's equations</a> can be written very compactly in <a href="/wiki/Geometrized_units" class="mw-redirect" title="Geometrized units">geometrized units</a> as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}d{\textbf {F}}&={\textbf {0}}\\d{\star {\textbf {F}}}&={\textbf {J}},\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> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">F</mtext> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">0</mtext> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⋆</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">F</mtext> </mrow> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">J</mtext> </mrow> </mrow> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}d{\textbf {F}}&={\textbf {0}}\\d{\star {\textbf {F}}}&={\textbf {J}},\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9b00cabd1fc7c6de64ff5fd55b9ce19c6abe3a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:9.939ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}d{\textbf {F}}&={\textbf {0}}\\d{\star {\textbf {F}}}&={\textbf {J}},\end{aligned}}}"></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \star }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋆</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \star }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd316a21eeb5079a850f223b1d096a06bfa788c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.035ex; margin-bottom: -0.206ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle \star }"> denotes the <a href="/wiki/Hodge_star" class="mw-redirect" title="Hodge star">Hodge star</a> operator. Similar considerations describe the geometry of gauge theories in general. The 2-form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\star }\mathbf {F} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo>⋆</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\star }\mathbf {F} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9e4891297b68a5ff7b039ce33e4ff5c7f14c76e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.845ex; height:2.176ex;" alt="{\displaystyle {\star }\mathbf {F} }">, which is <a href="/wiki/Duality_(mathematics)" title="Duality (mathematics)">dual</a> to the Faraday form, is also called Maxwell 2-form. Electromagnetism is an example of a <a href="/wiki/U(1)" class="mw-redirect" title="U(1)">U(1)</a> <a href="/wiki/Gauge_theory" title="Gauge theory">gauge theory</a>. Here the <a href="/wiki/Lie_group" title="Lie group">Lie group</a> is U(1), the one-dimensional <a href="/wiki/Unitary_group" title="Unitary group">unitary group</a>, which is in particular <a href="/wiki/Abelian_group" title="Abelian group">abelian</a>. There are gauge theories, such as <a href="/wiki/Yang%E2%80%93Mills_theory" title="Yang–Mills theory">Yang–Mills theory</a>, in which the Lie group is not abelian. In that case, one gets relations which are similar to those described here. The analog of the field F in such theories is the curvature form of the connection, which is represented in a gauge by a <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebra</a>-valued one-form A. The Yang–Mills field F is then defined by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} =d\mathbf {A} +\mathbf {A} \wedge \mathbf {A} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>=</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} =d\mathbf {A} +\mathbf {A} \wedge \mathbf {A} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14a0a39307e25aca9a9f8d7e168b86e04a733363" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.125ex; height:2.343ex;" alt="{\displaystyle \mathbf {F} =d\mathbf {A} +\mathbf {A} \wedge \mathbf {A} .}"></dd></dl> In the abelian case, such as electromagnetism, A ∧ A = 0, but this does not hold in general. Likewise the field equations are modified by additional terms involving exterior products of A and F, owing to the <a href="/wiki/Maurer%E2%80%93Cartan_form" title="Maurer–Cartan form">structure equations</a> of the gauge group. <div class="mw-heading mw-heading2"><h2 id="Applications_in_geometric_measure_theory">Applications in geometric measure theory</h2>[<a href="/w/index.php?title=Differential_form&action=edit&section=23" title="Edit section: Applications in geometric measure theory">edit</a>]</div> Numerous minimality results for complex analytic manifolds are based on the <a href="/wiki/Wirtinger_inequality_(2-forms)" title="Wirtinger inequality (2-forms)">Wirtinger inequality for 2-forms</a>. A succinct proof may be found in <a href="/wiki/Herbert_Federer" title="Herbert Federer">Herbert Federer</a>'s classic text Geometric Measure Theory. The Wirtinger inequality is also a key ingredient in <a href="/wiki/Gromov%27s_inequality_for_complex_projective_space" title="Gromov's inequality for complex projective space">Gromov's inequality for complex projective space</a> in <a href="/wiki/Systolic_geometry" title="Systolic geometry">systolic geometry</a>. <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Differential_form&action=edit&section=24" title="Edit section: See also">edit</a>]</div> <ul><li><a href="/wiki/Closed_and_exact_differential_forms" title="Closed and exact differential forms">Closed and exact differential forms</a></li> <li><a href="/wiki/Complex_differential_form" title="Complex differential form">Complex differential form</a></li> <li><a href="/wiki/Vector-valued_differential_form" title="Vector-valued differential form">Vector-valued differential form</a></li> <li><a href="/wiki/Equivariant_differential_form" title="Equivariant differential form">Equivariant differential form</a></li> <li><a href="/wiki/Calculus_on_Manifolds_(book)" title="Calculus on Manifolds (book)">Calculus on Manifolds</a></li> <li><a href="/wiki/Multilinear_form" title="Multilinear form">Multilinear form</a></li> <li><a href="/wiki/Polynomial_differential_form" title="Polynomial differential form">Polynomial differential form</a></li> <li><a href="/wiki/Presymplectic_form" title="Presymplectic form">Presymplectic form</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Differential_form&action=edit&section=25" title="Edit section: Notes">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"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</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 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rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTu,_Loring_W.2011" class="citation book cs1">Tu, Loring W. (2011). An introduction to manifolds (2nd ed.). New York: Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781441974006" title="Special:BookSources/9781441974006"><bdi>9781441974006</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/682907530">682907530</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+introduction+to+manifolds&rft.place=New+York&rft.edition=2nd&rft.pub=Springer&rft.date=2011&rft_id=info%3Aoclcnum%2F682907530&rft.isbn=9781441974006&rft.au=Tu%2C+Loring+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifferential+form" 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 class="citation web cs1"><a rel="nofollow" class="external text" href="https://mathoverflow.net/q/68033">"Linear algebra – "Natural" pairings between exterior powers of a vector space and its dual"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Linear+algebra+%E2%80%93+%22Natural%22+pairings+between+exterior+powers+of+a+vector+space+and+its+dual&rft_id=https%3A%2F%2Fmathoverflow.net%2Fq%2F68033&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifferential+form" class="Z3988"> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Differential_form&action=edit&section=26" title="Edit section: References">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBachman2006" class="citation cs2">Bachman, David (2006), A Geometric Approach to Differential Forms, Birkhäuser, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8176-4499-4" title="Special:BookSources/978-0-8176-4499-4"><bdi>978-0-8176-4499-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Geometric+Approach+to+Differential+Forms&rft.pub=Birkh%C3%A4user&rft.date=2006&rft.isbn=978-0-8176-4499-4&rft.aulast=Bachman&rft.aufirst=David&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifferential+form" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBachman2003" class="citation cs2">Bachman, David (2003), A Geometric Approach to Differential Forms, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0306194v1">math/0306194v1</a>, <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/2003math......6194B">2003math......6194B</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Geometric+Approach+to+Differential+Forms&rft.date=2003&rft_id=info%3Aarxiv%2Fmath%2F0306194v1&rft_id=info%3Abibcode%2F2003math......6194B&rft.aulast=Bachman&rft.aufirst=David&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifferential+form" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCartan2006" class="citation cs2"><a href="/wiki/Henri_Cartan" title="Henri Cartan">Cartan, Henri</a> (2006), Differential Forms, Dover, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-45010-4" title="Special:BookSources/0-486-45010-4"><bdi>0-486-45010-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Differential+Forms&rft.pub=Dover&rft.date=2006&rft.isbn=0-486-45010-4&rft.aulast=Cartan&rft.aufirst=Henri&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifferential+form" class="Z3988">—Translation of Formes différentielles (1967)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDieudonné1972" class="citation cs2"><a href="/wiki/Jean_Dieudonn%C3%A9" title="Jean Dieudonné">Dieudonné, Jean</a> (1972), Treatise on Analysis, vol. 3, New York-London: Academic Press, Inc., <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=0350769">0350769</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Treatise+on+Analysis&rft.place=New+York-London&rft.pub=Academic+Press%2C+Inc.&rft.date=1972&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0350769%23id-name%3DMR&rft.aulast=Dieudonn%C3%A9&rft.aufirst=Jean&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifferential+form" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEdwards1994" class="citation cs2"><a href="/wiki/Harold_M._Edwards" class="mw-redirect" title="Harold M. Edwards">Edwards, Harold M.</a> (1994), Advanced Calculus; A Differential Forms Approach, Modern Birkhäuser Classics, Boston, Basel, Berlin: Birkhäuser, <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%2F978-0-8176-8412-9">10.1007/978-0-8176-8412-9</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8176-8411-2" title="Special:BookSources/978-0-8176-8411-2"><bdi>978-0-8176-8411-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Advanced+Calculus%3B+A+Differential+Forms+Approach&rft.place=Boston%2C+Basel%2C+Berlin&rft.series=Modern+Birkh%C3%A4user+Classics&rft.pub=Birkh%C3%A4user&rft.date=1994&rft_id=info%3Adoi%2F10.1007%2F978-0-8176-8412-9&rft.isbn=978-0-8176-8411-2&rft.aulast=Edwards&rft.aufirst=Harold+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifferential+form" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFolland1999" class="citation cs2"><a href="/wiki/Gerald_Folland" title="Gerald Folland">Folland, Gerald B.</a> (1999), Real Analysis: Modern Techniques and Their Applications (Second ed.), <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-31716-6" title="Special:BookSources/978-0-471-31716-6"><bdi>978-0-471-31716-6</bdi></a>,</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Real+Analysis%3A+Modern+Techniques+and+Their+Applications&rft.edition=Second&rft.date=1999&rft.isbn=978-0-471-31716-6&rft.aulast=Folland&rft.aufirst=Gerald+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifferential+form" class="Z3988"> provides a brief discussion of integration on manifolds from the point of view of measure theory in the last section.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFlanders1989" class="citation cs2"><a href="/wiki/Harley_Flanders" title="Harley Flanders">Flanders, Harley</a> (1989) [1964], Differential forms with applications to the physical sciences, Mineola, New York: Dover Publications, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-66169-5" title="Special:BookSources/0-486-66169-5"><bdi>0-486-66169-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Differential+forms+with+applications+to+the+physical+sciences&rft.place=Mineola%2C+New+York&rft.pub=Dover+Publications&rft.date=1989&rft.isbn=0-486-66169-5&rft.aulast=Flanders&rft.aufirst=Harley&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifferential+form" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFleming1965" class="citation cs2"><a href="/wiki/Wendell_Fleming" title="Wendell Fleming">Fleming, Wendell H.</a> (1965), "Chapter 6: Exterior algebra and differential calculus", Functions of Several Variables, Addison-Wesley, pp. 205–238.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+6%3A+Exterior+algebra+and+differential+calculus&rft.btitle=Functions+of+Several+Variables&rft.pages=205-238&rft.pub=Addison-Wesley&rft.date=1965&rft.aulast=Fleming&rft.aufirst=Wendell+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifferential+form" class="Z3988"> This textbook in <a href="/wiki/Multivariate_calculus" class="mw-redirect" title="Multivariate calculus">multivariate calculus</a> introduces the exterior algebra of differential forms at the college calculus level.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMorita2001" class="citation cs2">Morita, Shigeyuki (2001), <a rel="nofollow" class="external text" href="https://archive.org/details/geometryofdiffer00mori">Geometry of Differential Forms</a>, AMS, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8218-1045-6" title="Special:BookSources/0-8218-1045-6"><bdi>0-8218-1045-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geometry+of+Differential+Forms&rft.pub=AMS&rft.date=2001&rft.isbn=0-8218-1045-6&rft.aulast=Morita&rft.aufirst=Shigeyuki&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fgeometryofdiffer00mori&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifferential+form" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRudin1976" class="citation cs2"><a href="/wiki/Walter_Rudin" title="Walter Rudin">Rudin, Walter</a> (1976), Principles of Mathematical Analysis, New York: McGraw-Hill, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-07-054235-X" title="Special:BookSources/0-07-054235-X"><bdi>0-07-054235-X</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Principles+of+Mathematical+Analysis&rft.place=New+York&rft.pub=McGraw-Hill&rft.date=1976&rft.isbn=0-07-054235-X&rft.aulast=Rudin&rft.aufirst=Walter&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifferential+form" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSpivak1965" class="citation cs2"><a href="/wiki/Michael_Spivak" title="Michael Spivak">Spivak, Michael</a> (1965), <a rel="nofollow" class="external text" href="https://archive.org/details/SpivakM.CalculusOnManifolds_201703">Calculus on Manifolds</a>, Menlo Park, California: W. A. Benjamin, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8053-9021-9" title="Special:BookSources/0-8053-9021-9"><bdi>0-8053-9021-9</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+on+Manifolds&rft.place=Menlo+Park%2C+California&rft.pub=W.+A.+Benjamin&rft.date=1965&rft.isbn=0-8053-9021-9&rft.aulast=Spivak&rft.aufirst=Michael&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2FSpivakM.CalculusOnManifolds_201703&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifferential+form" class="Z3988"> standard introductory text.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTu2008" class="citation cs2"><a href="/wiki/Loring_W._Tu" title="Loring W. Tu">Tu, Loring W.</a> (2008), An Introduction to Manifolds, Universitext, Springer, <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%2F978-1-4419-7400-6">10.1007/978-1-4419-7400-6</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-48098-5" title="Special:BookSources/978-0-387-48098-5"><bdi>978-0-387-48098-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+Manifolds&rft.series=Universitext&rft.pub=Springer&rft.date=2008&rft_id=info%3Adoi%2F10.1007%2F978-1-4419-7400-6&rft.isbn=978-0-387-48098-5&rft.aulast=Tu&rft.aufirst=Loring+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifferential+form" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZorich2004" class="citation cs2"><a href="/wiki/Vladimir_A._Zorich" title="Vladimir A. Zorich">Zorich, Vladimir A.</a> (2004), Mathematical Analysis II, Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-40633-6" title="Special:BookSources/3-540-40633-6"><bdi>3-540-40633-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Analysis+II&rft.pub=Springer&rft.date=2004&rft.isbn=3-540-40633-6&rft.aulast=Zorich&rft.aufirst=Vladimir+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifferential+form" class="Z3988"></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=Differential_form&action=edit&section=27" title="Edit section: External links">edit</a>]</div> <ul><li><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/Differentialk-Form.html">"Differential form"</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=Differential+form&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FDifferentialk-Form.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifferential+form" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSjamaar2006" class="citation cs2">Sjamaar, Reyer (2006), <a rel="nofollow" class="external text" href="http://pi.math.cornell.edu/~sjamaar/manifolds/manifold.pdf">Manifolds and differential forms lecture notes</a> (PDF)</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Manifolds+and+differential+forms+lecture+notes&rft.date=2006&rft.aulast=Sjamaar&rft.aufirst=Reyer&rft_id=http%3A%2F%2Fpi.math.cornell.edu%2F~sjamaar%2Fmanifolds%2Fmanifold.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifferential+form" class="Z3988">, a course taught at <a href="/wiki/Cornell_University" title="Cornell University">Cornell University</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBachman2003" class="citation cs2">Bachman, David (2003), A Geometric Approach to Differential Forms, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0306194">math/0306194</a>, <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/2003math......6194B">2003math......6194B</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Geometric+Approach+to+Differential+Forms&rft.date=2003&rft_id=info%3Aarxiv%2Fmath%2F0306194&rft_id=info%3Abibcode%2F2003math......6194B&rft.aulast=Bachman&rft.aufirst=David&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifferential+form" class="Z3988">, an undergraduate text.</li> <li><a href="/wiki/Tristan_Needham" title="Tristan Needham">Needham, Tristan</a>. <a rel="nofollow" class="external text" href="https://www.vdgf.space/table-of-contents">Visual differential geometry and forms: a mathematical drama in five acts</a>. Princeton University Press, 2021.</li></ul> <div class="navbox-styles"><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 ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist 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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 .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Manifolds" title="Template:Manifolds"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Manifolds" title="Template talk:Manifolds"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Manifolds" title="Special:EditPage/Template:Manifolds"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Manifolds_(Glossary)" style="font-size:114%;margin:0 4em"><a href="/wiki/Manifold" title="Manifold">Manifolds</a> (<a href="/wiki/Glossary_of_differential_geometry_and_topology" title="Glossary of differential geometry and topology">Glossary</a>)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Topological_manifold" title="Topological manifold">Topological manifold</a> <ul><li><a href="/wiki/Atlas_(topology)" title="Atlas (topology)">Atlas</a></li></ul></li> <li><a href="/wiki/Differentiable_manifold" title="Differentiable manifold">Differentiable/Smooth manifold</a> <ul><li><a href="/wiki/Differential_structure" title="Differential structure">Differential structure</a></li> <li><a href="/wiki/Smooth_structure" title="Smooth structure">Smooth atlas</a></li></ul></li> <li><a href="/wiki/Submanifold" title="Submanifold">Submanifold</a></li> <li><a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a></li> <li><a href="/wiki/Smoothness" title="Smoothness">Smooth map</a></li> <li><a href="/wiki/Submersion_(mathematics)" title="Submersion (mathematics)">Submersion</a></li> <li><a href="/wiki/Pushforward_(differential)" title="Pushforward (differential)">Pushforward</a></li> <li><a href="/wiki/Tangent_space" title="Tangent space">Tangent space</a></li> <li><a class="mw-selflink selflink">Differential form</a></li> <li><a href="/wiki/Vector_field" title="Vector field">Vector field</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Main results <a href="/wiki/Category:Theorems_in_differential_geometry" title="Category:Theorems in differential geometry">(list)</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/Atiyah%E2%80%93Singer_index_theorem" title="Atiyah–Singer index theorem">Atiyah–Singer index</a></li> <li><a href="/wiki/Darboux%27s_theorem" title="Darboux's theorem">Darboux's</a></li> <li><a href="/wiki/De_Rham_cohomology#De_Rham's_theorem" title="De Rham cohomology">De Rham's</a></li> <li><a href="/wiki/Frobenius_theorem_(differential_topology)" title="Frobenius theorem (differential topology)">Frobenius</a></li> <li><a href="/wiki/Generalized_Stokes_theorem" title="Generalized Stokes theorem">Generalized Stokes</a></li> <li><a href="/wiki/Hopf%E2%80%93Rinow_theorem" title="Hopf–Rinow theorem">Hopf–Rinow</a></li> <li><a href="/wiki/Noether%27s_theorem" title="Noether's theorem">Noether's</a></li> <li><a href="/wiki/Sard%27s_theorem" title="Sard's theorem">Sard's</a></li> <li><a href="/wiki/Whitney_embedding_theorem" title="Whitney embedding theorem">Whitney embedding</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Smoothness" title="Smoothness">Maps</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Differentiable_curve" title="Differentiable curve">Curve</a></li> <li><a href="/wiki/Diffeomorphism" title="Diffeomorphism">Diffeomorphism</a> <ul><li><a href="/wiki/Local_diffeomorphism" title="Local diffeomorphism">Local</a></li></ul></li> <li><a href="/wiki/Geodesic" title="Geodesic">Geodesic</a></li> <li><a href="/wiki/Exponential_map_(Riemannian_geometry)" title="Exponential map (Riemannian geometry)">Exponential map</a> <ul><li><a href="/wiki/Exponential_map_(Lie_theory)" title="Exponential map (Lie theory)">in Lie theory</a></li></ul></li> <li><a href="/wiki/Foliation" title="Foliation">Foliation</a></li> <li><a href="/wiki/Immersion_(mathematics)" title="Immersion (mathematics)">Immersion</a></li> <li><a href="/wiki/Integral_curve" title="Integral curve">Integral curve</a></li> <li><a href="/wiki/Lie_derivative" title="Lie derivative">Lie derivative</a></li> <li><a href="/wiki/Section_(fiber_bundle)" title="Section (fiber bundle)">Section</a></li> <li><a href="/wiki/Submersion_(mathematics)" title="Submersion (mathematics)">Submersion</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of manifolds</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/Closed_manifold" title="Closed manifold">Closed</a></li> <li>(<a href="/wiki/Almost_complex_manifold" title="Almost complex manifold">Almost</a>) <a href="/wiki/Complex_manifold" title="Complex manifold">Complex</a></li> <li>(<a href="/wiki/Almost-contact_manifold" title="Almost-contact manifold">Almost</a>) <a href="/wiki/Contact_manifold" class="mw-redirect" title="Contact manifold">Contact</a></li> <li><a href="/wiki/Fibered_manifold" title="Fibered manifold">Fibered</a></li> <li><a href="/wiki/Finsler_manifold" title="Finsler manifold">Finsler</a></li> <li><a href="/wiki/Flat_manifold" title="Flat manifold">Flat</a></li> <li><a href="/wiki/G-structure_on_a_manifold" title="G-structure on a manifold">G-structure</a></li> <li><a href="/wiki/Hadamard_manifold" title="Hadamard manifold">Hadamard</a></li> <li><a href="/wiki/Hermitian_manifold" title="Hermitian manifold">Hermitian</a></li> <li><a href="/wiki/Hyperbolic_manifold" title="Hyperbolic manifold">Hyperbolic</a></li> <li><a href="/wiki/K%C3%A4hler_manifold" title="Kähler manifold">Kähler</a></li> <li><a href="/wiki/Kenmotsu_manifold" title="Kenmotsu manifold">Kenmotsu</a></li> <li><a href="/wiki/Lie_group" title="Lie group">Lie group</a> <ul><li><a href="/wiki/Lie_group%E2%80%93Lie_algebra_correspondence" title="Lie group–Lie algebra correspondence">Lie algebra</a></li></ul></li> <li><a href="/wiki/Manifold_with_boundary" class="mw-redirect" title="Manifold with boundary">Manifold with boundary</a></li> <li><a href="/wiki/Orientability" title="Orientability">Oriented</a></li> <li><a href="/wiki/Parallelizable_manifold" title="Parallelizable manifold">Parallelizable</a></li> <li><a href="/wiki/Poisson_manifold" title="Poisson manifold">Poisson</a></li> <li><a href="/wiki/Prime_manifold" title="Prime manifold">Prime</a></li> <li><a href="/wiki/Quaternionic_manifold" title="Quaternionic manifold">Quaternionic</a></li> <li><a href="/wiki/Hypercomplex_manifold" title="Hypercomplex manifold">Hypercomplex</a></li> <li>(<a href="/wiki/Pseudo-Riemannian_manifold" title="Pseudo-Riemannian manifold">Pseudo−</a>, <a href="/wiki/Sub-Riemannian_manifold" title="Sub-Riemannian manifold">Sub−</a>) <a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian</a></li> <li><a href="/wiki/Rizza_manifold" title="Rizza manifold">Rizza</a></li> <li>(<a href="/wiki/Almost_symplectic_manifold" title="Almost symplectic manifold">Almost</a>) <a href="/wiki/Symplectic_manifold" title="Symplectic manifold">Symplectic</a></li> <li><a href="/wiki/Tame_manifold" title="Tame manifold">Tame</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Tensor" title="Tensor">Tensors</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Vectors</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Distribution_(differential_geometry)" title="Distribution (differential geometry)">Distribution</a></li> <li><a href="/wiki/Lie_bracket_of_vector_fields" title="Lie bracket of vector fields">Lie bracket</a></li> <li><a href="/wiki/Pushforward_(differential)" title="Pushforward (differential)">Pushforward</a></li> <li><a href="/wiki/Tangent_space" title="Tangent space">Tangent space</a> <ul><li><a href="/wiki/Tangent_bundle" title="Tangent bundle">bundle</a></li></ul></li> <li><a href="/wiki/Torsion_tensor" title="Torsion tensor">Torsion</a></li> <li><a href="/wiki/Vector_field" title="Vector field">Vector field</a></li> <li><a href="/wiki/Vector_flow" title="Vector flow">Vector flow</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Covectors</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/Closed_and_exact_differential_forms" title="Closed and exact differential forms">Closed/Exact</a></li> <li><a href="/wiki/Covariant_derivative" title="Covariant derivative">Covariant derivative</a></li> <li><a href="/wiki/Cotangent_space" title="Cotangent space">Cotangent space</a> <ul><li><a href="/wiki/Cotangent_bundle" title="Cotangent bundle">bundle</a></li></ul></li> <li><a href="/wiki/De_Rham_cohomology" title="De Rham cohomology">De Rham cohomology</a></li> <li><a class="mw-selflink selflink">Differential form</a> <ul><li><a href="/wiki/Vector-valued_differential_form" title="Vector-valued differential form">Vector-valued</a></li></ul></li> <li><a href="/wiki/Exterior_derivative" title="Exterior derivative">Exterior derivative</a></li> <li><a href="/wiki/Interior_product" title="Interior product">Interior product</a></li> <li><a href="/wiki/Pullback_(differential_geometry)" title="Pullback (differential geometry)">Pullback</a></li> <li><a href="/wiki/Ricci_curvature" title="Ricci curvature">Ricci curvature</a> <ul><li><a href="/wiki/Ricci_flow" title="Ricci flow">flow</a></li></ul></li> <li><a href="/wiki/Riemann_curvature_tensor" title="Riemann curvature tensor">Riemann curvature tensor</a></li> <li><a href="/wiki/Tensor_field" title="Tensor field">Tensor field</a> <ul><li><a href="/wiki/Tensor_density" title="Tensor density">density</a></li></ul></li> <li><a href="/wiki/Volume_form" title="Volume form">Volume form</a></li> <li><a href="/wiki/Wedge_product" class="mw-redirect" title="Wedge product">Wedge product</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Fiber_bundle" title="Fiber bundle">Bundles</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Adjoint_bundle" title="Adjoint bundle">Adjoint</a></li> <li><a href="/wiki/Affine_bundle" title="Affine bundle">Affine</a></li> <li><a href="/wiki/Associated_bundle" title="Associated bundle">Associated</a></li> <li><a href="/wiki/Cotangent_bundle" title="Cotangent bundle">Cotangent</a></li> <li><a href="/wiki/Dual_bundle" title="Dual bundle">Dual</a></li> <li><a href="/wiki/Fiber_bundle" title="Fiber bundle">Fiber</a></li> <li>(<a href="/wiki/Cofibration" title="Cofibration">Co</a>) <a href="/wiki/Fibration" title="Fibration">Fibration</a></li> <li><a href="/wiki/Jet_bundle" title="Jet bundle">Jet</a></li> <li><a href="/wiki/Lie_algebra_bundle" title="Lie algebra bundle">Lie algebra</a></li> <li>(<a href="/wiki/Stable_normal_bundle" title="Stable normal bundle">Stable</a>) <a href="/wiki/Normal_bundle" title="Normal bundle">Normal</a></li> <li><a href="/wiki/Principal_bundle" title="Principal bundle">Principal</a></li> <li><a href="/wiki/Spinor_bundle" title="Spinor bundle">Spinor</a></li> <li><a href="/wiki/Subbundle" title="Subbundle">Subbundle</a></li> <li><a href="/wiki/Tangent_bundle" title="Tangent bundle">Tangent</a></li> <li><a href="/wiki/Tensor_bundle" title="Tensor bundle">Tensor</a></li> <li><a href="/wiki/Vector_bundle" title="Vector bundle">Vector</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Connection_(mathematics)" title="Connection (mathematics)">Connections</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/Affine_connection" title="Affine connection">Affine</a></li> <li><a href="/wiki/Cartan_connection" title="Cartan connection">Cartan</a></li> <li><a href="/wiki/Ehresmann_connection" title="Ehresmann connection">Ehresmann</a></li> <li><a href="/wiki/Connection_form" title="Connection form">Form</a></li> <li><a href="/wiki/Connection_(fibred_manifold)" title="Connection (fibred manifold)">Generalized</a></li> <li><a href="/wiki/Koszul_connection" class="mw-redirect" title="Koszul connection">Koszul</a></li> <li><a href="/wiki/Levi-Civita_connection" title="Levi-Civita connection">Levi-Civita</a></li> <li><a href="/wiki/Connection_(principal_bundle)" title="Connection (principal bundle)">Principal</a></li> <li><a href="/wiki/Connection_(vector_bundle)" title="Connection (vector bundle)">Vector</a></li> <li><a href="/wiki/Parallel_transport" title="Parallel transport">Parallel transport</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Classification_of_manifolds" title="Classification of manifolds">Classification of manifolds</a></li> <li><a href="/wiki/Gauge_theory_(mathematics)" title="Gauge theory (mathematics)">Gauge theory</a></li> <li><a href="/wiki/History_of_manifolds_and_varieties" title="History of manifolds and varieties">History</a></li> <li><a href="/wiki/Morse_theory" title="Morse theory">Morse theory</a></li> <li><a href="/wiki/Moving_frame" title="Moving frame">Moving frame</a></li> <li><a href="/wiki/Singularity_theory" title="Singularity theory">Singularity theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Generalizations</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/Banach_manifold" title="Banach manifold">Banach manifold</a></li> <li><a href="/wiki/Diffeology" title="Diffeology">Diffeology</a></li> <li><a href="/wiki/Diffiety" title="Diffiety">Diffiety</a></li> <li><a href="/wiki/Fr%C3%A9chet_manifold" title="Fréchet manifold">Fréchet manifold</a></li> <li><a href="/wiki/K-theory" title="K-theory">K-theory</a></li> <li><a href="/wiki/Orbifold" title="Orbifold">Orbifold</a></li> <li><a href="/wiki/Secondary_calculus_and_cohomological_physics" title="Secondary calculus and cohomological physics">Secondary calculus</a> <ul><li><a href="/wiki/Differential_calculus_over_commutative_algebras" title="Differential calculus over commutative algebras">over commutative algebras</a></li></ul></li> <li><a href="/wiki/Sheaf_(mathematics)" title="Sheaf (mathematics)">Sheaf</a></li> <li><a href="/wiki/Stratifold" title="Stratifold">Stratifold</a></li> <li><a href="/wiki/Supermanifold" title="Supermanifold">Supermanifold</a></li> <li><a href="/wiki/Stratified_space" title="Stratified space">Stratified space</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link 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href="/wiki/Special:EditPage/Template:Tensors" title="Special:EditPage/Template:Tensors"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Tensors" style="font-size:114%;margin:0 4em"><a href="/wiki/Tensor" title="Tensor">Tensors</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div><a href="/wiki/Glossary_of_tensor_theory" title="Glossary of tensor theory">Glossary of tensor theory</a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Scope</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Mathematics" title="Mathematics">Mathematics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Coordinate_system" title="Coordinate system">Coordinate system</a></li> <li><a href="/wiki/Differential_geometry" title="Differential geometry">Differential geometry</a></li> <li><a href="/wiki/Dyadics" title="Dyadics">Dyadic algebra</a></li> <li><a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a></li> <li><a href="/wiki/Exterior_calculus" class="mw-redirect" title="Exterior calculus">Exterior calculus</a></li> <li><a href="/wiki/Multilinear_algebra" title="Multilinear algebra">Multilinear algebra</a></li> <li><a href="/wiki/Tensor_algebra" title="Tensor algebra">Tensor algebra</a></li> <li><a href="/wiki/Tensor_calculus" class="mw-redirect" title="Tensor calculus">Tensor calculus</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><div class="hlist"><ul><li><a href="/wiki/Physics" title="Physics">Physics</a></li><li><a href="/wiki/Engineering" title="Engineering">Engineering</a></li></ul></div></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/Computer_vision" title="Computer vision">Computer vision</a></li> <li><a href="/wiki/Continuum_mechanics" title="Continuum mechanics">Continuum mechanics</a></li> <li><a href="/wiki/Electromagnetism" title="Electromagnetism">Electromagnetism</a></li> <li><a href="/wiki/General_relativity" title="General relativity">General relativity</a></li> <li><a href="/wiki/Transport_phenomena" title="Transport phenomena">Transport phenomena</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Notation</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abstract_index_notation" title="Abstract index notation">Abstract index notation</a></li> <li><a href="/wiki/Einstein_notation" title="Einstein notation">Einstein notation</a></li> <li><a href="/wiki/Index_notation" title="Index notation">Index notation</a></li> <li><a href="/wiki/Multi-index_notation" title="Multi-index notation">Multi-index notation</a></li> <li><a href="/wiki/Penrose_graphical_notation" title="Penrose graphical notation">Penrose graphical notation</a></li> <li><a href="/wiki/Ricci_calculus" title="Ricci calculus">Ricci calculus</a></li> <li><a href="/wiki/Tetrad_(index_notation)" class="mw-redirect" title="Tetrad (index notation)">Tetrad (index notation)</a></li> <li><a href="/wiki/Van_der_Waerden_notation" title="Van der Waerden notation">Van der Waerden notation</a></li> <li><a href="/wiki/Voigt_notation" title="Voigt notation">Voigt notation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Tensor definitions</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/Tensor_(intrinsic_definition)" title="Tensor (intrinsic definition)">Tensor (intrinsic definition)</a></li> <li><a href="/wiki/Tensor_field" title="Tensor field">Tensor field</a></li> <li><a href="/wiki/Tensor_density" title="Tensor density">Tensor density</a></li> <li><a href="/wiki/Tensors_in_curvilinear_coordinates" title="Tensors in curvilinear coordinates">Tensors in curvilinear coordinates</a></li> <li><a href="/wiki/Mixed_tensor" title="Mixed tensor">Mixed tensor</a></li> <li><a href="/wiki/Antisymmetric_tensor" title="Antisymmetric tensor">Antisymmetric tensor</a></li> <li><a href="/wiki/Symmetric_tensor" title="Symmetric tensor">Symmetric tensor</a></li> <li><a href="/wiki/Tensor_operator" title="Tensor operator">Tensor operator</a></li> <li><a href="/wiki/Tensor_bundle" title="Tensor bundle">Tensor bundle</a></li> <li><a href="/wiki/Two-point_tensor" title="Two-point tensor">Two-point tensor</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">Operations</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Covariant_derivative" title="Covariant derivative">Covariant derivative</a></li> <li><a href="/wiki/Exterior_covariant_derivative" title="Exterior covariant derivative">Exterior covariant derivative</a></li> <li><a href="/wiki/Exterior_derivative" title="Exterior derivative">Exterior derivative</a></li> <li><a href="/wiki/Exterior_product" class="mw-redirect" title="Exterior product">Exterior product</a></li> <li><a href="/wiki/Hodge_star_operator" title="Hodge star operator">Hodge star operator</a></li> <li><a href="/wiki/Lie_derivative" title="Lie derivative">Lie derivative</a></li> <li><a href="/wiki/Raising_and_lowering_indices" title="Raising and lowering indices">Raising and lowering indices</a></li> <li><a href="/wiki/Symmetrization" title="Symmetrization">Symmetrization</a></li> <li><a href="/wiki/Tensor_contraction" title="Tensor contraction">Tensor contraction</a></li> <li><a href="/wiki/Tensor_product" title="Tensor product">Tensor product</a></li> <li><a href="/wiki/Transpose" title="Transpose">Transpose</a> (2nd-order tensors)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related abstractions</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/Affine_connection" title="Affine connection">Affine connection</a></li> <li><a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">Basis</a></li> <li><a href="/wiki/Cartan_formalism_(physics)" class="mw-redirect" title="Cartan formalism (physics)">Cartan formalism (physics)</a></li> <li><a href="/wiki/Connection_form" title="Connection form">Connection form</a></li> <li><a href="/wiki/Covariance_and_contravariance_of_vectors" title="Covariance and contravariance of vectors">Covariance and contravariance of vectors</a></li> <li><a class="mw-selflink selflink">Differential form</a></li> <li><a href="/wiki/Dimension" title="Dimension">Dimension</a></li> <li><a href="/wiki/Exterior_form" class="mw-redirect" title="Exterior form">Exterior form</a></li> <li><a href="/wiki/Fiber_bundle" title="Fiber bundle">Fiber bundle</a></li> <li><a href="/wiki/Geodesic" title="Geodesic">Geodesic</a></li> <li><a href="/wiki/Levi-Civita_connection" title="Levi-Civita connection">Levi-Civita connection</a></li> <li><a href="/wiki/Linear_map" title="Linear map">Linear map</a></li> <li><a href="/wiki/Manifold" title="Manifold">Manifold</a></li> <li><a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">Matrix</a></li> <li><a href="/wiki/Multivector" title="Multivector">Multivector</a></li> <li><a href="/wiki/Pseudotensor" title="Pseudotensor">Pseudotensor</a></li> <li><a href="/wiki/Spinor" title="Spinor">Spinor</a></li> <li><a href="/wiki/Vector_(mathematics_and_physics)" title="Vector (mathematics and physics)">Vector</a></li> <li><a href="/wiki/Vector_space" title="Vector space">Vector space</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Notable tensors</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Mathematics</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Kronecker_delta" title="Kronecker delta">Kronecker delta</a></li> <li><a href="/wiki/Levi-Civita_symbol" title="Levi-Civita symbol">Levi-Civita symbol</a></li> <li><a href="/wiki/Metric_tensor" title="Metric tensor">Metric tensor</a></li> <li><a href="/wiki/Nonmetricity_tensor" title="Nonmetricity tensor">Nonmetricity tensor</a></li> <li><a href="/wiki/Ricci_curvature" title="Ricci curvature">Ricci curvature</a></li> <li><a href="/wiki/Riemann_curvature_tensor" title="Riemann curvature tensor">Riemann curvature tensor</a></li> <li><a href="/wiki/Torsion_tensor" title="Torsion tensor">Torsion tensor</a></li> <li><a href="/wiki/Weyl_tensor" title="Weyl tensor">Weyl tensor</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Physics</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/Moment_of_inertia#Inertia_tensor" title="Moment of inertia">Moment of inertia</a></li> <li><a href="/wiki/Angular_momentum#Angular_momentum_in_relativistic_mechanics" title="Angular momentum">Angular momentum tensor</a></li> <li><a href="/wiki/Spin_tensor" title="Spin tensor">Spin tensor</a></li> <li><a href="/wiki/Cauchy_stress_tensor" title="Cauchy stress tensor">Cauchy stress tensor</a></li> <li><a href="/wiki/Stress%E2%80%93energy_tensor" title="Stress–energy tensor">stress–energy tensor</a></li> <li><a href="/wiki/Einstein_tensor" title="Einstein tensor">Einstein tensor</a></li> <li><a href="/wiki/Electromagnetic_tensor" title="Electromagnetic tensor">EM tensor</a></li> <li><a href="/wiki/Gluon_field_strength_tensor" title="Gluon field strength tensor">Gluon field strength tensor</a></li> <li><a href="/wiki/Metric_tensor_(general_relativity)" title="Metric tensor (general relativity)">Metric tensor (GR)</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Mathematician" title="Mathematician">Mathematicians</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%C3%89lie_Cartan" title="Élie Cartan">Élie Cartan</a></li> <li><a href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Augustin-Louis Cauchy</a></li> <li><a href="/wiki/Elwin_Bruno_Christoffel" title="Elwin Bruno Christoffel">Elwin Bruno Christoffel</a></li> <li><a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a></li> <li><a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a></li> <li><a href="/wiki/Hermann_Grassmann" title="Hermann Grassmann">Hermann Grassmann</a></li> <li><a href="/wiki/Tullio_Levi-Civita" title="Tullio Levi-Civita">Tullio Levi-Civita</a></li> <li><a href="/wiki/Gregorio_Ricci-Curbastro" title="Gregorio Ricci-Curbastro">Gregorio Ricci-Curbastro</a></li> <li><a href="/wiki/Bernhard_Riemann" title="Bernhard Riemann">Bernhard Riemann</a></li> <li><a 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Differential form - Wikipedia