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data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Definition" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definition"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Definition</span> </div> </a> <button aria-controls="toc-Definition-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Definition subsection</span> </button> <ul id="toc-Definition-sublist" class="vector-toc-list"> <li id="toc-Vector_fields_on_subsets_of_Euclidean_space" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Vector_fields_on_subsets_of_Euclidean_space"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Vector fields on subsets of Euclidean space</span> </div> </a> <ul id="toc-Vector_fields_on_subsets_of_Euclidean_space-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Coordinate_transformation_law" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Coordinate_transformation_law"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Coordinate transformation law</span> </div> </a> <ul id="toc-Coordinate_transformation_law-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vector_fields_on_manifolds" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Vector_fields_on_manifolds"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Vector fields on manifolds</span> </div> </a> <ul id="toc-Vector_fields_on_manifolds-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Examples</span> </div> </a> <button aria-controls="toc-Examples-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Examples subsection</span> </button> <ul id="toc-Examples-sublist" class="vector-toc-list"> <li id="toc-Gradient_field_in_Euclidean_spaces" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Gradient_field_in_Euclidean_spaces"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Gradient field in Euclidean spaces</span> </div> </a> <ul id="toc-Gradient_field_in_Euclidean_spaces-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Central_field_in_euclidean_spaces" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Central_field_in_euclidean_spaces"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Central field in euclidean spaces</span> </div> </a> <ul id="toc-Central_field_in_euclidean_spaces-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Operations_on_vector_fields" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Operations_on_vector_fields"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Operations on vector fields</span> </div> </a> <button aria-controls="toc-Operations_on_vector_fields-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Operations on vector fields subsection</span> </button> <ul id="toc-Operations_on_vector_fields-sublist" class="vector-toc-list"> <li id="toc-Line_integral" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Line_integral"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Line integral</span> </div> </a> <ul id="toc-Line_integral-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Divergence" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Divergence"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Divergence</span> </div> </a> <ul id="toc-Divergence-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Curl_in_three_dimensions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Curl_in_three_dimensions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Curl in three dimensions</span> </div> </a> <ul id="toc-Curl_in_three_dimensions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Index_of_a_vector_field" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Index_of_a_vector_field"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Index of a vector field</span> </div> </a> <ul id="toc-Index_of_a_vector_field-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Physical_intuition" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Physical_intuition"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Physical intuition</span> </div> </a> <ul id="toc-Physical_intuition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Flow_curves" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Flow_curves"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Flow curves</span> </div> </a> <button aria-controls="toc-Flow_curves-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Flow curves subsection</span> </button> <ul id="toc-Flow_curves-sublist" class="vector-toc-list"> <li id="toc-Complete_vector_fields" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Complete_vector_fields"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Complete vector fields</span> </div> </a> <ul id="toc-Complete_vector_fields-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_Lie_bracket" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_Lie_bracket"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>The Lie bracket</span> </div> </a> <ul id="toc-The_Lie_bracket-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-f-relatedness" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#f-relatedness"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span><i>f</i>-relatedness</span> </div> </a> <ul id="toc-f-relatedness-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generalizations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Generalizations"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Generalizations</span> </div> </a> <ul id="toc-Generalizations-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"> <span class="vector-toc-numb">8</span> <span>See also</span> </div> </a> <ul id="toc-See_also-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"> <span class="vector-toc-numb">9</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliography" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliography"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Bibliography</span> </div> </a> <ul id="toc-Bibliography-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"> <span class="vector-toc-numb">11</span> <span>External links</span> </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 vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Vector field</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 54 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-54" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">54 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Vektorveld" title="Vektorveld – Afrikaans" lang="af" hreflang="af" data-title="Vektorveld" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AD%D9%82%D9%84_%D9%85%D8%AA%D8%AC%D9%87%D8%A7%D8%AA" title="حقل متجهات – Arabic" lang="ar" hreflang="ar" data-title="حقل متجهات" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%92%D0%B5%D0%BA%D1%82%D0%B0%D1%80%D0%BD%D0%B0%D0%B5_%D0%BF%D0%BE%D0%BB%D0%B5" title="Вектарнае поле – Belarusian" lang="be" hreflang="be" data-title="Вектарнае поле" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80%D0%BD%D0%BE_%D0%BF%D0%BE%D0%BB%D0%B5" title="Векторно поле – Bulgarian" lang="bg" hreflang="bg" data-title="Векторно поле" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Vektorsko_polje" title="Vektorsko polje – Bosnian" lang="bs" hreflang="bs" data-title="Vektorsko polje" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Camp_vectorial" title="Camp vectorial – Catalan" lang="ca" hreflang="ca" data-title="Camp vectorial" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80%D0%BB%C4%83_%D1%83%D0%B9" title="Векторлă уй – Chuvash" lang="cv" hreflang="cv" data-title="Векторлă уй" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Vektorov%C3%A9_pole" title="Vektorové pole – Czech" lang="cs" hreflang="cs" data-title="Vektorové pole" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Vektorfeld" title="Vektorfeld – German" lang="de" hreflang="de" data-title="Vektorfeld" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Vektorv%C3%A4li" title="Vektorväli – Estonian" lang="et" hreflang="et" data-title="Vektorväli" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Campo_vectorial" title="Campo vectorial – Spanish" lang="es" hreflang="es" data-title="Campo vectorial" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Vektora_kampo" title="Vektora kampo – Esperanto" lang="eo" hreflang="eo" data-title="Vektora kampo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Bektore-eremua" title="Bektore-eremua – Basque" lang="eu" hreflang="eu" data-title="Bektore-eremua" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%DB%8C%D8%AF%D8%A7%D9%86_%D8%A8%D8%B1%D8%AF%D8%A7%D8%B1%DB%8C" title="میدان برداری – Persian" lang="fa" hreflang="fa" data-title="میدان برداری" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Champ_de_vecteurs" title="Champ de vecteurs – French" lang="fr" hreflang="fr" data-title="Champ de vecteurs" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Campo_vectorial" title="Campo vectorial – Galician" lang="gl" hreflang="gl" data-title="Campo vectorial" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B2%A1%ED%84%B0%EC%9E%A5" title="벡터장 – Korean" lang="ko" hreflang="ko" data-title="벡터장" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%8E%D5%A5%D5%AF%D5%BF%D5%B8%D6%80%D5%A1%D5%AF%D5%A1%D5%B6_%D5%A4%D5%A1%D5%B7%D5%BF" title="Վեկտորական դաշտ – Armenian" lang="hy" hreflang="hy" data-title="Վեկտորական դաշտ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B8%E0%A4%A6%E0%A4%BF%E0%A4%B6_%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A5%87%E0%A4%A4%E0%A5%8D%E0%A4%B0" title="सदिश क्षेत्र – Hindi" lang="hi" hreflang="hi" data-title="सदिश क्षेत्र" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Vektorsko_polje" title="Vektorsko polje – Croatian" lang="hr" hreflang="hr" data-title="Vektorsko polje" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Vektorofeldo" title="Vektorofeldo – Ido" lang="io" hreflang="io" data-title="Vektorofeldo" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Vigursvi%C3%B0" title="Vigursvið – Icelandic" lang="is" hreflang="is" data-title="Vigursvið" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Campo_vettoriale" title="Campo vettoriale – Italian" lang="it" hreflang="it" data-title="Campo vettoriale" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A9%D7%93%D7%94_%D7%95%D7%A7%D7%98%D7%95%D7%A8%D7%99" title="שדה וקטורי – Hebrew" lang="he" hreflang="he" data-title="שדה וקטורי" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80%D0%BB%D1%8B%D2%9B_%D3%A9%D1%80%D1%96%D1%81" title="Векторлық өріс – Kazakh" lang="kk" hreflang="kk" data-title="Векторлық өріс" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Chan_vektory%C3%A8l" title="Chan vektoryèl – Haitian Creole" lang="ht" hreflang="ht" data-title="Chan vektoryèl" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80%D0%B4%D1%83%D0%BA_%D1%82%D0%B0%D0%BB%D0%B0%D0%B0" title="Вектордук талаа – Kyrgyz" lang="ky" hreflang="ky" data-title="Вектордук талаа" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Vektorinis_laukas" title="Vektorinis laukas – Lithuanian" lang="lt" hreflang="lt" data-title="Vektorinis laukas" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Vektormez%C5%91" title="Vektormező – Hungarian" lang="hu" hreflang="hu" data-title="Vektormező" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80_%D0%BE%D1%80%D0%BE%D0%BD" title="Вектор орон – Mongolian" lang="mn" hreflang="mn" data-title="Вектор орон" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Vectorveld" title="Vectorveld – Dutch" lang="nl" hreflang="nl" data-title="Vectorveld" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%99%E3%82%AF%E3%83%88%E3%83%AB%E5%A0%B4" title="ベクトル場 – Japanese" lang="ja" hreflang="ja" data-title="ベクトル場" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Vektorfelt" title="Vektorfelt – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Vektorfelt" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Vektorfelt" title="Vektorfelt – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Vektorfelt" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Vektor_maydon" title="Vektor maydon – Uzbek" lang="uz" hreflang="uz" data-title="Vektor maydon" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / 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searchaux" style="display:none">Assignment of a vector to each point in a subset of Euclidean space</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:VectorField.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b9/VectorField.svg/250px-VectorField.svg.png" decoding="async" width="250" height="250" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b9/VectorField.svg/375px-VectorField.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b9/VectorField.svg/500px-VectorField.svg.png 2x" data-file-width="600" data-file-height="600" /></a><figcaption>A portion of the vector field (sin <i>y</i>, sin <i>x</i>)</figcaption></figure> <p>In <a href="/wiki/Vector_calculus" title="Vector calculus">vector calculus</a> and <a href="/wiki/Physics" title="Physics">physics</a>, a <b>vector field</b> is an assignment of a <a href="/wiki/Vector_(mathematics_and_physics)" title="Vector (mathematics and physics)">vector</a> to each point in a <a href="/wiki/Space_(mathematics)" title="Space (mathematics)">space</a>, most commonly <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span>.<sup id="cite_ref-Galbis-2012-p12_1-0" class="reference"><a href="#cite_note-Galbis-2012-p12-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> A vector field on a <a href="/wiki/Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">plane</a> can be visualized as a collection of arrows with given magnitudes and directions, each attached to a point on the plane. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout <a href="/wiki/Three_dimensional_space" class="mw-redirect" title="Three dimensional space">three dimensional space</a>, such as the <a href="/wiki/Wind" title="Wind">wind</a>, or the strength and direction of some <a href="/wiki/Force" title="Force">force</a>, such as the <a href="/wiki/Magnetic_field" title="Magnetic field">magnetic</a> or <a href="/wiki/Gravity" title="Gravity">gravitational</a> force, as it changes from one point to another point. </p><p>The elements of <a href="/wiki/Differential_and_integral_calculus" class="mw-redirect" title="Differential and integral calculus">differential and integral calculus</a> extend naturally to vector fields. When a vector field represents <a href="/wiki/Force" title="Force">force</a>, the <a href="/wiki/Line_integral" title="Line integral">line integral</a> of a vector field represents the <a href="/wiki/Work_(physics)" title="Work (physics)">work</a> done by a force moving along a path, and under this interpretation <a href="/wiki/Conservation_of_energy" title="Conservation of energy">conservation of energy</a> is exhibited as a special case of the <a href="/wiki/Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus">fundamental theorem of calculus</a>. Vector fields can usefully be thought of as representing the velocity of a moving flow in space, and this physical intuition leads to notions such as the <a href="/wiki/Divergence" title="Divergence">divergence</a> (which represents the rate of change of <a href="/wiki/Volume" title="Volume">volume</a> of a flow) and <a href="/wiki/Curl_(mathematics)" title="Curl (mathematics)">curl</a> (which represents the rotation of a flow). </p><p>A vector field is a special case of a <i><a href="/wiki/Vector-valued_function" title="Vector-valued function">vector-valued function</a></i>, whose domain's dimension has no relation to the dimension of its range; for example, the <a href="/wiki/Position_vector" class="mw-redirect" title="Position vector">position vector</a> of a <a href="/wiki/Space_curve" class="mw-redirect" title="Space curve">space curve</a> is defined only for smaller subset of the ambient space. Likewise, n <a href="/wiki/Coordinate_system" title="Coordinate system">coordinates</a>, a vector field on a domain in <i>n</i>-dimensional Euclidean space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span> can be represented as a vector-valued function that associates an <i>n</i>-tuple of real numbers to each point of the domain. This representation of a vector field depends on the coordinate system, and there is a well-defined transformation law (<i><a href="/wiki/Covariance_and_contravariance_of_vectors" title="Covariance and contravariance of vectors">covariance and contravariance of vectors</a></i>) in passing from one coordinate system to the other. </p><p>Vector fields are often discussed on <a href="/wiki/Open_set" title="Open set">open subsets</a> of Euclidean space, but also make sense on other subsets such as <a href="/wiki/Surface_(topology)" title="Surface (topology)">surfaces</a>, where they associate an arrow tangent to the surface at each point (a <a href="/wiki/Differential_geometry_of_curves" class="mw-redirect" title="Differential geometry of curves">tangent vector</a>). More generally, vector fields are defined on <a href="/wiki/Differentiable_manifold" title="Differentiable manifold">differentiable manifolds</a>, which are spaces that look like Euclidean space on small scales, but may have more complicated structure on larger scales. In this setting, a vector field gives a tangent vector at each point of the manifold (that is, a <a href="/wiki/Section_(fiber_bundle)" title="Section (fiber bundle)">section</a> of the <a href="/wiki/Tangent_bundle" title="Tangent bundle">tangent bundle</a> to the manifold). Vector fields are one kind of <a href="/wiki/Tensor_field" title="Tensor field">tensor field</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vector_field&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Vector_fields_on_subsets_of_Euclidean_space">Vector fields on subsets of Euclidean space</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vector_field&action=edit&section=2" title="Edit section: Vector fields on subsets of Euclidean space"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1237032888/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:288px;max-width:288px"><div class="trow"><div class="tsingle" style="width:142px;max-width:142px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Radial_vector_field_sparse.svg" class="mw-file-description"><img alt="Sparse vector field representation" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b1/Radial_vector_field_sparse.svg/140px-Radial_vector_field_sparse.svg.png" decoding="async" width="140" height="140" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b1/Radial_vector_field_sparse.svg/210px-Radial_vector_field_sparse.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b1/Radial_vector_field_sparse.svg/280px-Radial_vector_field_sparse.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span></div></div><div class="tsingle" style="width:142px;max-width:142px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Radial_vector_field_dense.svg" class="mw-file-description"><img alt="Dense vector field representation." src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Radial_vector_field_dense.svg/140px-Radial_vector_field_dense.svg.png" decoding="async" width="140" height="140" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Radial_vector_field_dense.svg/210px-Radial_vector_field_dense.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Radial_vector_field_dense.svg/280px-Radial_vector_field_dense.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span></div></div></div><div class="trow" style="display:flex"><div class="thumbcaption">Two representations of the same vector field: <span class="nowrap"><b>v</b>(<i>x</i>, <i>y</i>) = −<b>r</b></span>. The arrows depict the field at discrete points, however, the field exists everywhere.</div></div></div></div> <p>Given a subset <span class="texhtml"><i>S</i></span> of <span class="texhtml"><b>R</b><sup><i>n</i></sup></span>, a <b>vector field</b> is represented by a <a href="/wiki/Vector-valued_function" title="Vector-valued function">vector-valued function</a> <span class="texhtml"><i>V</i>: <i>S</i> → <b>R</b><sup><i>n</i></sup></span> in standard <a href="/wiki/Cartesian_coordinates" class="mw-redirect" title="Cartesian coordinates">Cartesian coordinates</a> <span class="texhtml">(<i>x</i><sub>1</sub>, …, <i>x</i><sub><i>n</i></sub>)</span>. If each component of <span class="texhtml"><i>V</i></span> is continuous, then <span class="texhtml"><i>V</i></span> is a continuous vector field. It is common to focus on <b>smooth</b> vector fields, meaning that each component is a <a href="/wiki/Smooth_function" class="mw-redirect" title="Smooth function">smooth function</a> (differentiable any number of times). A vector field can be visualized as assigning a vector to individual points within an <i>n</i>-dimensional space.<sup id="cite_ref-Galbis-2012-p12_1-1" class="reference"><a href="#cite_note-Galbis-2012-p12-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>One standard notation is to write <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial }{\partial x_{1}}},\ldots ,{\frac {\partial }{\partial x_{n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial }{\partial x_{1}}},\ldots ,{\frac {\partial }{\partial x_{n}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7e09a94b38e8dd975fb95b2efb34d22bf955daa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.419ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial }{\partial x_{1}}},\ldots ,{\frac {\partial }{\partial x_{n}}}}"></span> for the unit vectors in the coordinate directions. In these terms, every smooth vector field <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> on an open subset <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathbf {R} }^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbf {R} }^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f96fef157cdfd6a7bb01b79ea8467bcbb6c3d78b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.222ex; height:2.343ex;" alt="{\displaystyle {\mathbf {R} }^{n}}"></span> can be written as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=1}^{n}V_{i}(x_{1},\ldots ,x_{n}){\frac {\partial }{\partial x_{i}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{n}V_{i}(x_{1},\ldots ,x_{n}){\frac {\partial }{\partial x_{i}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37efd61cf76b090b52c08b0b75b7d33cb4cccab3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:22.1ex; height:6.843ex;" alt="{\displaystyle \sum _{i=1}^{n}V_{i}(x_{1},\ldots ,x_{n}){\frac {\partial }{\partial x_{i}}}}"></span></dd></dl> <p>for some smooth functions <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{1},\ldots ,V_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{1},\ldots ,V_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/056aea51dd62880a078cf84fe69ff7d09afb20b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.161ex; height:2.509ex;" alt="{\displaystyle V_{1},\ldots ,V_{n}}"></span> on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span>.<sup id="cite_ref-Tu-2010-p149_2-0" class="reference"><a href="#cite_note-Tu-2010-p149-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The reason for this notation is that a vector field determines a <a href="/wiki/Linear_map" title="Linear map">linear map</a> from the space of smooth functions to itself, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\colon C^{\infty }(S)\to C^{\infty }(S)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>:</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V\colon C^{\infty }(S)\to C^{\infty }(S)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d108dba0bf92b0744b9dbf56f67d4aef4789139" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.399ex; height:2.843ex;" alt="{\displaystyle V\colon C^{\infty }(S)\to C^{\infty }(S)}"></span>, given by differentiating in the direction of the vector field. </p><p><b>Example</b>: The vector field <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -x_{2}{\frac {\partial }{\partial x_{1}}}+x_{1}{\frac {\partial }{\partial x_{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -x_{2}{\frac {\partial }{\partial x_{1}}}+x_{1}{\frac {\partial }{\partial x_{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0083b374f3494a2aacbdd9cf4c5d1b3dbc6f45f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.493ex; height:5.843ex;" alt="{\displaystyle -x_{2}{\frac {\partial }{\partial x_{1}}}+x_{1}{\frac {\partial }{\partial x_{2}}}}"></span> describes a counterclockwise rotation around the origin in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/feb97520ae70482ae41b49980ec140d871cb8243" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.057ex; height:2.676ex;" alt="{\displaystyle \mathbf {R} ^{2}}"></span>. To show that the function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}^{2}+x_{2}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}^{2}+x_{2}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21689e64baca0c6322c9e6d12c3d73647c7cd7a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.608ex; height:3.176ex;" alt="{\displaystyle x_{1}^{2}+x_{2}^{2}}"></span> is rotationally invariant, compute: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bigg (}-x_{2}{\frac {\partial }{\partial x_{1}}}+x_{1}{\frac {\partial }{\partial x_{2}}}{\bigg )}(x_{1}^{2}+x_{2}^{2})=-x_{2}(2x_{1})+x_{1}(2x_{2})=0.}"> <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> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mo stretchy="false">(</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>2</mn> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>2</mn> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bigg (}-x_{2}{\frac {\partial }{\partial x_{1}}}+x_{1}{\frac {\partial }{\partial x_{2}}}{\bigg )}(x_{1}^{2}+x_{2}^{2})=-x_{2}(2x_{1})+x_{1}(2x_{2})=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b5787dbf72f8811212742f48fe57a9d1a014b00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:60.498ex; height:6.176ex;" alt="{\displaystyle {\bigg (}-x_{2}{\frac {\partial }{\partial x_{1}}}+x_{1}{\frac {\partial }{\partial x_{2}}}{\bigg )}(x_{1}^{2}+x_{2}^{2})=-x_{2}(2x_{1})+x_{1}(2x_{2})=0.}"></span></dd></dl> <p>Given vector fields <span class="texhtml"><i>V</i></span>, <span class="texhtml"><i>W</i></span> defined on <span class="texhtml"><i>S</i></span> and a smooth function <span class="texhtml mvar" style="font-style:italic;">f</span> defined on <span class="texhtml"><i>S</i></span>, the operations of scalar multiplication and vector addition, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (fV)(p):=f(p)V(p)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mi>V</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (fV)(p):=f(p)V(p)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7429dfffc675d7a99350684b346b6e91603f0108" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.622ex; height:2.843ex;" alt="{\displaystyle (fV)(p):=f(p)V(p)}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (V+W)(p):=V(p)+W(p),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>V</mi> <mo>+</mo> <mi>W</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>W</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (V+W)(p):=V(p)+W(p),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5061da271a1d99d730ead8af5df98ae337ba65a7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.263ex; height:2.843ex;" alt="{\displaystyle (V+W)(p):=V(p)+W(p),}"></span> make the smooth vector fields into a <a href="/wiki/Module_(mathematics)" title="Module (mathematics)">module</a> over the <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a> of smooth functions, where multiplication of functions is defined pointwise. </p> <div class="mw-heading mw-heading3"><h3 id="Coordinate_transformation_law">Coordinate transformation law</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vector_field&action=edit&section=3" title="Edit section: Coordinate transformation law"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In physics, a <a href="/wiki/Euclidean_vector" title="Euclidean vector">vector</a> is additionally distinguished by how its coordinates change when one measures the same vector with respect to a different background coordinate system. The <a href="/wiki/Euclidean_vector#Vectors,_pseudovectors,_and_transformations" title="Euclidean vector">transformation properties of vectors</a> distinguish a vector as a geometrically distinct entity from a simple list of scalars, or from a <a href="/wiki/Covector" class="mw-redirect" title="Covector">covector</a>. </p><p>Thus, suppose that <span class="texhtml">(<i>x</i><sub>1</sub>, ..., <i>x</i><sub><i>n</i></sub>)</span> is a choice of Cartesian coordinates, in terms of which the components of the vector <span class="texhtml mvar" style="font-style:italic;">V</span> are <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{x}=(V_{1,x},\dots ,V_{n,x})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mi>x</mi> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>x</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{x}=(V_{1,x},\dots ,V_{n,x})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a68b37373763bbaf4536732a4f13724e83559e3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.392ex; height:3.009ex;" alt="{\displaystyle V_{x}=(V_{1,x},\dots ,V_{n,x})}"></span> and suppose that (<i>y</i><sub>1</sub>,...,<i>y</i><sub><i>n</i></sub>) are <i>n</i> functions of the <i>x</i><sub><i>i</i></sub> defining a different coordinate system. Then the components of the vector <i>V</i> in the new coordinates are required to satisfy the transformation law </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{i,y}=\sum _{j=1}^{n}{\frac {\partial y_{i}}{\partial x_{j}}}V_{j,x}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>y</mi> </mrow> </msub> <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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>,</mo> <mi>x</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{i,y}=\sum _{j=1}^{n}{\frac {\partial y_{i}}{\partial x_{j}}}V_{j,x}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ba34807a0a319437f1251ed665782f7cc9f223a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:18.973ex; height:7.176ex;" alt="{\displaystyle V_{i,y}=\sum _{j=1}^{n}{\frac {\partial y_{i}}{\partial x_{j}}}V_{j,x}.}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_1" class="reference nourlexpansion" style="font-weight:bold;">1</span>)</b></td></tr></tbody></table> <p>Such a transformation law is called <a href="/wiki/Covariance_and_contravariance_of_vectors" title="Covariance and contravariance of vectors">contravariant</a>. A similar transformation law characterizes vector fields in physics: specifically, a vector field is a specification of <i>n</i> functions in each coordinate system subject to the transformation law (<b><a href="#math_1">1</a></b>) relating the different coordinate systems. </p><p>Vector fields are thus contrasted with <a href="/wiki/Scalar_field" title="Scalar field">scalar fields</a>, which associate a number or <i>scalar</i> to every point in space, and are also contrasted with simple lists of scalar fields, which do not transform under coordinate changes. </p> <div class="mw-heading mw-heading3"><h3 id="Vector_fields_on_manifolds">Vector fields on manifolds</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vector_field&action=edit&section=4" title="Edit section: Vector fields on manifolds"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Vector_field_E.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/dd/Vector_field_E.png/200px-Vector_field_E.png" decoding="async" width="200" height="218" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/dd/Vector_field_E.png/300px-Vector_field_E.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/dd/Vector_field_E.png/400px-Vector_field_E.png 2x" data-file-width="480" data-file-height="522" /></a><figcaption>A vector field on a <a href="/wiki/Sphere" title="Sphere">sphere</a></figcaption></figure> <p>Given a <a href="/wiki/Differentiable_manifold" title="Differentiable manifold">differentiable manifold</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span>, a <b>vector field</b> on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> is an assignment of a <a href="/wiki/Tangent_space" title="Tangent space">tangent vector</a> to each point in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span>.<sup id="cite_ref-Tu-2010-p149_2-1" class="reference"><a href="#cite_note-Tu-2010-p149-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> More precisely, a vector field <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> is a <a href="/wiki/Map_(mathematics)" title="Map (mathematics)">mapping</a> from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> into the <a href="/wiki/Tangent_bundle" title="Tangent bundle">tangent bundle</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle TM}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle TM}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea000afb5769206ddd5fd43f458430d04422ddeb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.078ex; height:2.176ex;" alt="{\displaystyle TM}"></span> so that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\circ F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>∘</mo> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\circ F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/180b247af1ad933f8c9cf4ebd34b3b7dcdcdb6a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.194ex; height:2.509ex;" alt="{\displaystyle p\circ F}"></span> is the identity mapping where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> denotes the projection from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle TM}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle TM}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea000afb5769206ddd5fd43f458430d04422ddeb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.078ex; height:2.176ex;" alt="{\displaystyle TM}"></span> to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span>. In other words, a vector field is a <a href="/wiki/Section_(fiber_bundle)" title="Section (fiber bundle)">section</a> of the <a href="/wiki/Tangent_bundle" title="Tangent bundle">tangent bundle</a>. </p><p>An alternative definition: A smooth vector field <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> on a manifold <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> is a linear map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X:C^{\infty }(M)\to C^{\infty }(M)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>:</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X:C^{\infty }(M)\to C^{\infty }(M)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/724987ca5c95e72eb850ccb86258c629e08eda4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.381ex; height:2.843ex;" alt="{\displaystyle X:C^{\infty }(M)\to C^{\infty }(M)}"></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> is a <a href="/wiki/Derivation_(differential_algebra)" title="Derivation (differential algebra)">derivation</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X(fg)=fX(g)+X(f)g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mi>g</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mi>X</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>X</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X(fg)=fX(g)+X(f)g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddd3afb2f70deae169b942efa3fa54f3791f89e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.491ex; height:2.843ex;" alt="{\displaystyle X(fg)=fX(g)+X(f)g}"></span> for all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f,g\in C^{\infty }(M)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo>∈</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f,g\in C^{\infty }(M)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee0c3fbb7f70cd0f1da1e09cc81f21ec9e1606e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.194ex; height:2.843ex;" alt="{\displaystyle f,g\in C^{\infty }(M)}"></span>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>If the manifold <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> is smooth or <a href="/wiki/Analytic_function" title="Analytic function">analytic</a>—that is, the change of coordinates is smooth (analytic)—then one can make sense of the notion of smooth (analytic) vector fields. The collection of all smooth vector fields on a smooth manifold <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> is often denoted by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma (TM)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mi>M</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma (TM)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2a0815a341f6a1b87b432c1ba9756425361a729" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.341ex; height:2.843ex;" alt="{\displaystyle \Gamma (TM)}"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{\infty }(M,TM)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>M</mi> <mo>,</mo> <mi>T</mi> <mi>M</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{\infty }(M,TM)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08b76d6a0543adf307b1ff2cea798b22f5002088" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.037ex; height:2.843ex;" alt="{\displaystyle C^{\infty }(M,TM)}"></span> (especially when thinking of vector fields as <a href="/wiki/Section_(fiber_bundle)" title="Section (fiber bundle)">sections</a>); the collection of all smooth vector fields is also denoted by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\mathfrak {X}}(M)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">X</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\mathfrak {X}}(M)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ff83454e3b326c6a7b492267c12330fd639294c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.923ex; height:2.843ex;" alt="{\textstyle {\mathfrak {X}}(M)}"></span> (a <a href="/wiki/Fraktur_(typeface_sub-classification)" class="mw-redirect" title="Fraktur (typeface sub-classification)">fraktur</a> "X"). </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vector_field&action=edit&section=5" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Cessna_182_model-wingtip-vortex.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Cessna_182_model-wingtip-vortex.jpg/250px-Cessna_182_model-wingtip-vortex.jpg" decoding="async" width="250" height="145" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Cessna_182_model-wingtip-vortex.jpg/375px-Cessna_182_model-wingtip-vortex.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Cessna_182_model-wingtip-vortex.jpg/500px-Cessna_182_model-wingtip-vortex.jpg 2x" data-file-width="2271" data-file-height="1313" /></a><figcaption>The flow field around an airplane is a vector field in <b>R</b><sup>3</sup>, here visualized by bubbles that follow the <a href="/wiki/Streamlines,_streaklines,_and_pathlines" title="Streamlines, streaklines, and pathlines">streamlines</a> showing a <a href="/wiki/Wingtip_vortex" class="mw-redirect" title="Wingtip vortex">wingtip vortex</a>.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Bezier_curves_composition_ray-traced_in_3D.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Bezier_curves_composition_ray-traced_in_3D.png/220px-Bezier_curves_composition_ray-traced_in_3D.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Bezier_curves_composition_ray-traced_in_3D.png/330px-Bezier_curves_composition_ray-traced_in_3D.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Bezier_curves_composition_ray-traced_in_3D.png/440px-Bezier_curves_composition_ray-traced_in_3D.png 2x" data-file-width="1936" data-file-height="1936" /></a><figcaption>Vector fields are commonly used to create patterns in <a href="/wiki/Computer_graphics" title="Computer graphics">computer graphics</a>. Here: abstract composition of curves following a vector field generated with <a href="/wiki/OpenSimplex_noise" title="OpenSimplex noise">OpenSimplex noise</a>.</figcaption></figure> <ul><li>A vector field for the movement of air on Earth will associate for every point on the surface of the Earth a vector with the wind speed and direction for that point. This can be drawn using arrows to represent the wind; the length (<a href="/wiki/Magnitude_(mathematics)" title="Magnitude (mathematics)">magnitude</a>) of the arrow will be an indication of the wind speed. A "high" on the usual <a href="/wiki/Barometric_pressure" class="mw-redirect" title="Barometric pressure">barometric pressure</a> map would then act as a source (arrows pointing away), and a "low" would be a sink (arrows pointing towards), since air tends to move from high pressure areas to low pressure areas.</li> <li><a href="/wiki/Velocity" title="Velocity">Velocity</a> field of a moving <a href="/wiki/Fluid" title="Fluid">fluid</a>. In this case, a <a href="/wiki/Velocity" title="Velocity">velocity</a> vector is associated to each point in the fluid.</li> <li><a href="/wiki/Streamlines,_Streaklines_and_Pathlines" class="mw-redirect" title="Streamlines, Streaklines and Pathlines">Streamlines, streaklines and pathlines</a> are 3 types of lines that can be made from (time-dependent) vector fields. They are: <ul><li>streaklines: the line produced by particles passing through a specific fixed point over various times</li> <li>pathlines: showing the path that a given particle (of zero mass) would follow.</li> <li>streamlines (or fieldlines): the path of a particle influenced by the instantaneous field (i.e., the path of a particle if the field is held fixed).</li></ul></li> <li><a href="/wiki/Magnetic_field" title="Magnetic field">Magnetic fields</a>. The fieldlines can be revealed using small <a href="/wiki/Iron" title="Iron">iron</a> filings.</li> <li><a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's equations</a> allow us to use a given set of initial and boundary conditions to deduce, for every point in <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>, a magnitude and direction for the <a href="/wiki/Force" title="Force">force</a> experienced by a charged test particle at that point; the resulting vector field is the <a href="/wiki/Electric_field" title="Electric field">electric field</a>.</li> <li>A <a href="/wiki/Gravitational_field" title="Gravitational field">gravitational field</a> generated by any massive object is also a vector field. For example, the gravitational field vectors for a spherically symmetric body would all point towards the sphere's center with the magnitude of the vectors reducing as radial distance from the body increases.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Gradient_field_in_Euclidean_spaces">Gradient field in Euclidean spaces</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vector_field&action=edit&section=6" title="Edit section: Gradient field in Euclidean spaces"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Irrotationalfield.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Irrotationalfield.svg/300px-Irrotationalfield.svg.png" decoding="async" width="300" height="300" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Irrotationalfield.svg/450px-Irrotationalfield.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Irrotationalfield.svg/600px-Irrotationalfield.svg.png 2x" data-file-width="512" data-file-height="512" /></a><figcaption>A vector field that has circulation about a point cannot be written as the gradient of a function.</figcaption></figure> <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">Further information: <a href="/wiki/Gradient" title="Gradient">Gradient</a></div> <p>Vector fields can be constructed out of <a href="/wiki/Scalar_field" title="Scalar field">scalar fields</a> using the <a href="/wiki/Gradient" title="Gradient">gradient</a> operator (denoted by the <a href="/wiki/Del" title="Del">del</a>: ∇).<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>A vector field <i>V</i> defined on an open set <i>S</i> is called a <b>gradient field</b> or a <b><a href="/wiki/Conservative_field" class="mw-redirect" title="Conservative field">conservative field</a></b> if there exists a real-valued function (a scalar field) <i>f</i> on <i>S</i> such that <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=\nabla f=\left({\frac {\partial f}{\partial x_{1}}},{\frac {\partial f}{\partial x_{2}}},{\frac {\partial f}{\partial x_{3}}},\dots ,{\frac {\partial f}{\partial x_{n}}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mi>f</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=\nabla f=\left({\frac {\partial f}{\partial x_{1}}},{\frac {\partial f}{\partial x_{2}}},{\frac {\partial f}{\partial x_{3}}},\dots ,{\frac {\partial f}{\partial x_{n}}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9072db2637fb193e4921e3516a4f8e75dd023b9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:41.217ex; height:6.176ex;" alt="{\displaystyle V=\nabla f=\left({\frac {\partial f}{\partial x_{1}}},{\frac {\partial f}{\partial x_{2}}},{\frac {\partial f}{\partial x_{3}}},\dots ,{\frac {\partial f}{\partial x_{n}}}\right).}"></span> </p><p>The associated <a href="/wiki/Flow_(mathematics)" title="Flow (mathematics)">flow</a> is called the <b><style data-mw-deduplicate="TemplateStyles:r1238216509">.mw-parser-output .vanchor>:target~.vanchor-text{background-color:#b1d2ff}@media screen{html.skin-theme-clientpref-night .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}</style><span class="vanchor"><span id="gradient_flow"></span><span class="vanchor-text">gradient flow</span></span></b>, and is used in the method of <a href="/wiki/Gradient_descent" title="Gradient descent">gradient descent</a>. </p><p>The <a href="/wiki/Line_integral" title="Line integral">path integral</a> along any <a href="/wiki/Closed_curve" class="mw-redirect" title="Closed curve">closed curve</a> <i>γ</i> (<i>γ</i>(0) = <i>γ</i>(1)) in a conservative field is zero: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oint _{\gamma }V(\mathbf {x} )\cdot \mathrm {d} \mathbf {x} =\oint _{\gamma }\nabla f(\mathbf {x} )\cdot \mathrm {d} \mathbf {x} =f(\gamma (1))-f(\gamma (0)).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msub> <mi>V</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>=</mo> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msub> <mi mathvariant="normal">∇</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>γ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>γ</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint _{\gamma }V(\mathbf {x} )\cdot \mathrm {d} \mathbf {x} =\oint _{\gamma }\nabla f(\mathbf {x} )\cdot \mathrm {d} \mathbf {x} =f(\gamma (1))-f(\gamma (0)).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/318b853b664638637eedebd6f7196d1bb371d857" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:50.144ex; height:6.009ex;" alt="{\displaystyle \oint _{\gamma }V(\mathbf {x} )\cdot \mathrm {d} \mathbf {x} =\oint _{\gamma }\nabla f(\mathbf {x} )\cdot \mathrm {d} \mathbf {x} =f(\gamma (1))-f(\gamma (0)).}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Central_field_in_euclidean_spaces">Central field in euclidean spaces</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vector_field&action=edit&section=7" title="Edit section: Central field in euclidean spaces"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <span class="texhtml"><i>C</i><sup>∞</sup></span>-vector field over <span class="texhtml"><b>R</b><sup><i>n</i></sup> \ {0}</span> is called a <b>central field</b> if <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(T(p))=T(V(p))\qquad (T\in \mathrm {O} (n,\mathbb {R} ))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="2em" /> <mo stretchy="false">(</mo> <mi>T</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V(T(p))=T(V(p))\qquad (T\in \mathrm {O} (n,\mathbb {R} ))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d62f18e5f584162d54952c0caaa4a18954a5e759" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.177ex; height:2.843ex;" alt="{\displaystyle V(T(p))=T(V(p))\qquad (T\in \mathrm {O} (n,\mathbb {R} ))}"></span> where <span class="texhtml">O(<i>n</i>, <b>R</b>)</span> is the <a href="/wiki/Orthogonal_group" title="Orthogonal group">orthogonal group</a>. We say central fields are <a href="/wiki/Invariant_(mathematics)" title="Invariant (mathematics)">invariant</a> under <a href="/wiki/Orthogonal_matrix" title="Orthogonal matrix">orthogonal transformations</a> around 0. </p><p>The point 0 is called the <b>center</b> of the field. </p><p>Since orthogonal transformations are actually rotations and reflections, the invariance conditions mean that vectors of a central field are always directed towards, or away from, 0; this is an alternate (and simpler) definition. A central field is always a gradient field, since defining it on one semiaxis and integrating gives an antigradient. </p> <div class="mw-heading mw-heading2"><h2 id="Operations_on_vector_fields">Operations on vector fields</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vector_field&action=edit&section=8" title="Edit section: Operations on vector fields"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Line_integral">Line integral</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vector_field&action=edit&section=9" title="Edit section: Line integral"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Line_integral" title="Line integral">Line integral</a></div> <p>A common technique in physics is to integrate a vector field along a <a href="/wiki/Differential_geometry_of_curves" class="mw-redirect" title="Differential geometry of curves">curve</a>, also called determining its <a href="/wiki/Line_integral" title="Line integral">line integral</a>. Intuitively this is summing up all vector components in line with the tangents to the curve, expressed as their scalar products. For example, given a particle in a force field (e.g. gravitation), where each vector at some point in space represents the force acting there on the particle, the line integral along a certain path is the work done on the particle, when it travels along this path. Intuitively, it is the sum of the scalar products of the force vector and the small tangent vector in each point along the curve. </p><p>The line integral is constructed analogously to the <a href="/wiki/Riemann_integral" title="Riemann integral">Riemann integral</a> and it exists if the curve is rectifiable (has finite length) and the vector field is continuous. </p><p>Given a vector field <span class="texhtml mvar" style="font-style:italic;">V</span> and a curve <span class="texhtml mvar" style="font-style:italic;">γ</span>, <a href="/wiki/Parametric_equation" title="Parametric equation">parametrized</a> by <span class="texhtml mvar" style="font-style:italic;">t</span> in <span class="texhtml">[<i>a</i>, <i>b</i>]</span> (where <span class="texhtml mvar" style="font-style:italic;">a</span> and <span class="texhtml mvar" style="font-style:italic;">b</span> are <a href="/wiki/Real_number" title="Real number">real numbers</a>), the line integral is defined as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{\gamma }V(\mathbf {x} )\cdot \mathrm {d} \mathbf {x} =\int _{a}^{b}V(\gamma (t))\cdot {\dot {\gamma }}(t)\,\mathrm {d} t.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msub> <mi>V</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <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>V</mi> <mo stretchy="false">(</mo> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>γ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{\gamma }V(\mathbf {x} )\cdot \mathrm {d} \mathbf {x} =\int _{a}^{b}V(\gamma (t))\cdot {\dot {\gamma }}(t)\,\mathrm {d} t.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae806e0c23c74a4257467a26f1a81c2da365a9b9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:35.346ex; height:6.676ex;" alt="{\displaystyle \int _{\gamma }V(\mathbf {x} )\cdot \mathrm {d} \mathbf {x} =\int _{a}^{b}V(\gamma (t))\cdot {\dot {\gamma }}(t)\,\mathrm {d} t.}"></span> </p><p>To show vector field topology one can use <a href="/wiki/Line_integral_convolution" title="Line integral convolution">line integral convolution</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Divergence">Divergence</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vector_field&action=edit&section=10" title="Edit section: Divergence"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Divergence" title="Divergence">Divergence</a></div> <p>The <a href="/wiki/Divergence" title="Divergence">divergence</a> of a vector field on Euclidean space is a function (or scalar field). In three-dimensions, the divergence is defined by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {div} \mathbf {F} =\nabla \cdot \mathbf {F} ={\frac {\partial F_{1}}{\partial x}}+{\frac {\partial F_{2}}{\partial y}}+{\frac {\partial F_{3}}{\partial z}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>div</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </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"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </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"> <mn>3</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {div} \mathbf {F} =\nabla \cdot \mathbf {F} ={\frac {\partial F_{1}}{\partial x}}+{\frac {\partial F_{2}}{\partial y}}+{\frac {\partial F_{3}}{\partial z}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07be0b6db45b09236c2ca48b40a2d0e96e1839a6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:37.168ex; height:6.009ex;" alt="{\displaystyle \operatorname {div} \mathbf {F} =\nabla \cdot \mathbf {F} ={\frac {\partial F_{1}}{\partial x}}+{\frac {\partial F_{2}}{\partial y}}+{\frac {\partial F_{3}}{\partial z}},}"></span> </p><p>with the obvious generalization to arbitrary dimensions. The divergence at a point represents the degree to which a small volume around the point is a <a href="/wiki/Sources_and_sinks" title="Sources and sinks">source or a sink</a> for the vector flow, a result which is made precise by the <a href="/wiki/Divergence_theorem" title="Divergence theorem">divergence theorem</a>. </p><p>The divergence can also be defined on a <a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a>, that is, a manifold with a <a href="/wiki/Riemannian_metric" class="mw-redirect" title="Riemannian metric">Riemannian metric</a> that measures the length of vectors. </p> <div class="mw-heading mw-heading3"><h3 id="Curl_in_three_dimensions">Curl in three dimensions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vector_field&action=edit&section=11" title="Edit section: Curl in three dimensions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Curl_(mathematics)" title="Curl (mathematics)">Curl (mathematics)</a></div> <p>The <a href="/wiki/Curl_(mathematics)" title="Curl (mathematics)">curl</a> is an operation which takes a vector field and produces another vector field. The curl is defined only in three dimensions, but some properties of the curl can be captured in higher dimensions with the <a href="/wiki/Exterior_derivative" title="Exterior derivative">exterior derivative</a>. In three dimensions, it is defined by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {curl} \mathbf {F} =\nabla \times \mathbf {F} =\left({\frac {\partial F_{3}}{\partial y}}-{\frac {\partial F_{2}}{\partial z}}\right)\mathbf {e} _{1}-\left({\frac {\partial F_{3}}{\partial x}}-{\frac {\partial F_{1}}{\partial z}}\right)\mathbf {e} _{2}+\left({\frac {\partial F_{2}}{\partial x}}-{\frac {\partial F_{1}}{\partial y}}\right)\mathbf {e} _{3}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>curl</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </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"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </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"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </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"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {curl} \mathbf {F} =\nabla \times \mathbf {F} =\left({\frac {\partial F_{3}}{\partial y}}-{\frac {\partial F_{2}}{\partial z}}\right)\mathbf {e} _{1}-\left({\frac {\partial F_{3}}{\partial x}}-{\frac {\partial F_{1}}{\partial z}}\right)\mathbf {e} _{2}+\left({\frac {\partial F_{2}}{\partial x}}-{\frac {\partial F_{1}}{\partial y}}\right)\mathbf {e} _{3}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7817501699a78926548651742bf0424cdd4e283" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:79.939ex; height:6.176ex;" alt="{\displaystyle \operatorname {curl} \mathbf {F} =\nabla \times \mathbf {F} =\left({\frac {\partial F_{3}}{\partial y}}-{\frac {\partial F_{2}}{\partial z}}\right)\mathbf {e} _{1}-\left({\frac {\partial F_{3}}{\partial x}}-{\frac {\partial F_{1}}{\partial z}}\right)\mathbf {e} _{2}+\left({\frac {\partial F_{2}}{\partial x}}-{\frac {\partial F_{1}}{\partial y}}\right)\mathbf {e} _{3}.}"></span> </p><p>The curl measures the density of the <a href="/wiki/Angular_momentum" title="Angular momentum">angular momentum</a> of the vector flow at a point, that is, the amount to which the flow circulates around a fixed axis. This intuitive description is made precise by <a href="/wiki/Stokes%27_theorem" title="Stokes' theorem">Stokes' theorem</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Index_of_a_vector_field">Index of a vector field</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vector_field&action=edit&section=12" title="Edit section: Index of a vector field"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The index of a vector field is an integer that helps describe its behaviour around an isolated zero (i.e., an isolated singularity of the field). In the plane, the index takes the value −1 at a saddle singularity but +1 at a source or sink singularity. </p><p>Let <i>n be</i> the dimension of the manifold on which the vector field is defined. Take a closed surface (homeomorphic to the (n-1)-sphere) S around the zero, so that no other zeros lie in the interior of S. A map from this sphere to a unit sphere of dimension <i>n</i> − 1 can be constructed by dividing each vector on this sphere by its length to form a unit length vector, which is a point on the unit sphere S<sup><i>n</i>−1</sup>. This defines a continuous map from S to S<sup><i>n</i>−1</sup>. The index of the vector field at the point is the <a href="/wiki/Degree_of_a_continuous_mapping#Differential_topology" title="Degree of a continuous mapping">degree</a> of this map. It can be shown that this integer does not depend on the choice of S, and therefore depends only on the vector field itself. </p><p>The index is not defined at any non-singular point (i.e., a point where the vector is non-zero). It is equal to +1 around a source, and more generally equal to (−1)<sup><i>k</i></sup> around a saddle that has <i>k</i> contracting dimensions and <i>n</i>−<i>k</i> expanding dimensions. </p><p><b>The index of the vector field</b> as a whole is defined when it has just finitely many zeroes. In this case, all zeroes are isolated, and the index of the vector field is defined to be the sum of the indices at all zeroes. </p><p>For an ordinary (2-dimensional) sphere in three-dimensional space, it can be shown that the index of any vector field on the sphere must be 2. This shows that every such vector field must have a zero. This implies the <a href="/wiki/Hairy_ball_theorem" title="Hairy ball theorem">hairy ball theorem</a>. </p><p>For a vector field on a compact manifold with finitely many zeroes, the <a href="/wiki/Poincar%C3%A9-Hopf_theorem" class="mw-redirect" title="Poincaré-Hopf theorem">Poincaré-Hopf theorem</a> states that the vector field’s index is the manifold’s <a href="/wiki/Euler_characteristic" title="Euler characteristic">Euler characteristic</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Physical_intuition">Physical intuition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vector_field&action=edit&section=13" title="Edit section: Physical intuition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Magnet0873.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/Magnet0873.png/220px-Magnet0873.png" decoding="async" width="220" height="148" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/Magnet0873.png/330px-Magnet0873.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/57/Magnet0873.png/440px-Magnet0873.png 2x" data-file-width="444" data-file-height="298" /></a><figcaption><a href="/wiki/Magnetism" title="Magnetism">Magnetic</a> field lines of an iron bar (<a href="/wiki/Magnetic_dipole" title="Magnetic dipole">magnetic dipole</a>)</figcaption></figure> <p><a href="/wiki/Michael_Faraday" title="Michael Faraday">Michael Faraday</a>, in his concept of <i><a href="/wiki/Lines_of_force" class="mw-redirect" title="Lines of force">lines of force</a>,</i> emphasized that the field <i>itself</i> should be an object of study, which it has become throughout physics in the form of <a href="/wiki/Field_theory_(physics)" class="mw-redirect" title="Field theory (physics)">field theory</a>. </p><p>In addition to the magnetic field, other phenomena that were modeled by Faraday include the electrical field and <a href="/wiki/Light_field" title="Light field">light field</a>. </p><p>In recent decades many phenomenological formulations of irreversible dynamics and evolution equations in physics, from the mechanics of complex fluids and solids to chemical kinetics and quantum thermodynamics, have converged towards the geometric idea of "steepest entropy ascent" or "gradient flow" as a consistent universal modeling framework that guarantees compatibility with the second law of thermodynamics and extends well-known near-equilibrium results such as Onsager reciprocity to the far-nonequilibrium realm.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Flow_curves">Flow curves</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vector_field&action=edit&section=14" title="Edit section: Flow curves"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Integral_curve" title="Integral curve">Integral curve</a></div> <p>Consider the flow of a fluid through a region of space. At any given time, any point of the fluid has a particular velocity associated with it; thus there is a vector field associated to any flow. The converse is also true: it is possible to associate a flow to a vector field having that vector field as its velocity. </p><p>Given a vector field <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> defined on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span>, one defines curves <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma (t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma (t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54fa4a5d64e164410e4a18106677bebefe1a1f1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.911ex; height:2.843ex;" alt="{\displaystyle \gamma (t)}"></span> on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> such that for each <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span> in an interval <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma '(t)=V(\gamma (t))\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>γ</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma '(t)=V(\gamma (t))\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b9da6888a814d3a6e6f38d89dcc8899fa7ced76" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.254ex; height:3.009ex;" alt="{\displaystyle \gamma '(t)=V(\gamma (t))\,.}"></span> </p><p>By the <a href="/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem" title="Picard–Lindelöf theorem">Picard–Lindelöf theorem</a>, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> is <a href="/wiki/Lipschitz_continuity" title="Lipschitz continuity">Lipschitz continuous</a> there is a <i>unique</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd24bae0d7570018e828e19851902c09c618af91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.852ex; height:2.676ex;" alt="{\displaystyle C^{1}}"></span>-curve <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/caf2e6d6984dc3ef9963ec7e44329a8a2c755df5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.377ex; height:2.176ex;" alt="{\displaystyle \gamma _{x}}"></span> for each point <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> so that, for some <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e04ec3670b50384a3ce48aca42e7cc5131a06b12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.344ex; height:2.176ex;" alt="{\displaystyle \varepsilon >0}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\gamma _{x}(0)&=x\\\gamma '_{x}(t)&=V(\gamma _{x}(t))\qquad \forall t\in (-\varepsilon ,+\varepsilon )\subset \mathbb {R} .\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>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mo>′</mo> </msubsup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>V</mi> <mo stretchy="false">(</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="2em" /> <mi mathvariant="normal">∀</mi> <mi>t</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>ε</mi> <mo>,</mo> <mo>+</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mo>⊂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\gamma _{x}(0)&=x\\\gamma '_{x}(t)&=V(\gamma _{x}(t))\qquad \forall t\in (-\varepsilon ,+\varepsilon )\subset \mathbb {R} .\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fb897220ccc6a4e28bae83e4d35e2b7cad53c5e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:41.488ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}\gamma _{x}(0)&=x\\\gamma '_{x}(t)&=V(\gamma _{x}(t))\qquad \forall t\in (-\varepsilon ,+\varepsilon )\subset \mathbb {R} .\end{aligned}}}"></span> </p><p>The curves <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/caf2e6d6984dc3ef9963ec7e44329a8a2c755df5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.377ex; height:2.176ex;" alt="{\displaystyle \gamma _{x}}"></span> are called <b>integral curves</b> or <b>trajectories</b> (or less commonly, flow lines) of the vector field <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> and partition <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> into <a href="/wiki/Equivalence_class" title="Equivalence class">equivalence classes</a>. It is not always possible to extend the interval <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-\varepsilon ,+\varepsilon )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mi>ε</mi> <mo>,</mo> <mo>+</mo> <mi>ε</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-\varepsilon ,+\varepsilon )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a165d9d1ef3ef8f1a7b42d0065dbec019b82b491" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.626ex; height:2.843ex;" alt="{\displaystyle (-\varepsilon ,+\varepsilon )}"></span> to the whole <a href="/wiki/Real_number_line" class="mw-redirect" title="Real number line">real number line</a>. The flow may for example reach the edge of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> in a finite time. In two or three dimensions one can visualize the vector field as giving rise to a <a href="/wiki/Flow_(mathematics)" title="Flow (mathematics)">flow</a> on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span>. If we drop a particle into this flow at a point <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> it will move along the curve <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8f10308dc5371da421a037c991f1c3a3b5d3b72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.263ex; height:2.343ex;" alt="{\displaystyle \gamma _{p}}"></span> in the flow depending on the initial point <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>. If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> is a stationary point of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> (i.e., the vector field is equal to the zero vector at the point <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>), then the particle will remain at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>. </p><p>Typical applications are <a href="/wiki/Streamlines,_streaklines,_and_pathlines" title="Streamlines, streaklines, and pathlines">pathline</a> in <a href="/wiki/Fluid_flow" class="mw-redirect" title="Fluid flow">fluid</a>, <a href="/wiki/Geodesic_flow" class="mw-redirect" title="Geodesic flow">geodesic flow</a>, and <a href="/wiki/One-parameter_subgroup" class="mw-redirect" title="One-parameter subgroup">one-parameter subgroups</a> and the <a href="/wiki/Exponential_map_(Lie_theory)" title="Exponential map (Lie theory)">exponential map</a> in <a href="/wiki/Lie_group" title="Lie group">Lie groups</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Complete_vector_fields">Complete vector fields</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vector_field&action=edit&section=15" title="Edit section: Complete vector fields"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>By definition, a vector field on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> is called <b>complete</b> if each of its flow curves exists for all time.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> In particular, <a href="/wiki/Compact_support" class="mw-redirect" title="Compact support">compactly supported</a> vector fields on a manifold are complete. If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> is a complete vector field on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span>, then the <a href="/wiki/One-parameter_group" title="One-parameter group">one-parameter group</a> of <a href="/wiki/Diffeomorphism" title="Diffeomorphism">diffeomorphisms</a> generated by the flow along <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> exists for all time; it is described by a smooth mapping </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} \times M\to M.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>×</mo> <mi>M</mi> <mo stretchy="false">→</mo> <mi>M</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} \times M\to M.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbaf7d9f78e21fcbd0432fd8d7c7b38bc9756f2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.989ex; height:2.176ex;" alt="{\displaystyle \mathbf {R} \times M\to M.}"></span></dd></dl> <p>On a compact manifold without boundary, every smooth vector field is complete. An example of an <b>incomplete</b> vector field <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> on the real line <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span> is given by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(x)=x^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V(x)=x^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a39a6656cf7be1a5316a512378de02c38e43d3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.409ex; height:3.176ex;" alt="{\displaystyle V(x)=x^{2}}"></span>. For, the differential equation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x'(t)=x^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle x'(t)=x^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01649a7cdba43309bd732a878f8f1a0b6e399c27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.146ex; height:3.009ex;" alt="{\textstyle x'(t)=x^{2}}"></span>, with initial condition <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(0)=x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(0)=x_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20c92f17b65057cd0714e255b61557b6ba31f61b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.784ex; height:2.843ex;" alt="{\displaystyle x(0)=x_{0}}"></span>, has as its unique solution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x(t)={\frac {x_{0}}{1-tx_{0}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mn>1</mn> <mo>−</mo> <mi>t</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle x(t)={\frac {x_{0}}{1-tx_{0}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e114b1e8060d85276e2b269aedf8715cce0f244" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:12.379ex; height:3.843ex;" alt="{\textstyle x(t)={\frac {x_{0}}{1-tx_{0}}}}"></span> if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30c873d33292c8b8b4f57edbade8876f4894a5a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.645ex; height:2.676ex;" alt="{\displaystyle x_{0}\neq 0}"></span> (and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(t)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39d5a44241e71c6e25141d278126efa41a5de3dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.24ex; height:2.843ex;" alt="{\displaystyle x(t)=0}"></span> for all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/592bced0c39b10fc90e74c6a66223abfbfb029de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.358ex; height:2.176ex;" alt="{\displaystyle t\in \mathbb {R} }"></span> if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d18a96da37e1748deeb8d4c590dd4ad6629efef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.645ex; height:2.509ex;" alt="{\displaystyle x_{0}=0}"></span>). Hence for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30c873d33292c8b8b4f57edbade8876f4894a5a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.645ex; height:2.676ex;" alt="{\displaystyle x_{0}\neq 0}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d54c275db3a1e620737b58e143b0818107fa5f5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.979ex; height:2.843ex;" alt="{\displaystyle x(t)}"></span> is undefined at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle t={\frac {1}{x_{0}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle t={\frac {1}{x_{0}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f162bb98d148cdf8908b82acabeb79557f309794" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:6.546ex; height:3.676ex;" alt="{\textstyle t={\frac {1}{x_{0}}}}"></span> so cannot be defined for all values of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="The_Lie_bracket">The Lie bracket</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vector_field&action=edit&section=16" title="Edit section: The Lie bracket"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The flows associated to two vector fields need not <a href="/wiki/Commutative_property" title="Commutative property">commute</a> with each other. Their failure to commute is described by the <a href="/wiki/Lie_bracket_of_vector_fields" title="Lie bracket of vector fields">Lie bracket</a> of two vector fields, which is again a vector field. The Lie bracket has a simple definition in terms of the action of vector fields on smooth functions <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [X,Y](f):=X(Y(f))-Y(X(f)).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mi>X</mi> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>−</mo> <mi>Y</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [X,Y](f):=X(Y(f))-Y(X(f)).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6b7aff0e6f237662023631084e44441fab03d83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.702ex; height:2.843ex;" alt="{\displaystyle [X,Y](f):=X(Y(f))-Y(X(f)).}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="f-relatedness"><i>f</i>-relatedness</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vector_field&action=edit&section=17" title="Edit section: f-relatedness"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Given a <a href="/wiki/Smooth_function" class="mw-redirect" title="Smooth function">smooth function</a> between manifolds, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:M\to N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>M</mi> <mo stretchy="false">→</mo> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:M\to N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dbd50e2de9728ee14a7c232441137f588b109f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.336ex; height:2.509ex;" alt="{\displaystyle f:M\to N}"></span>, the <a href="/wiki/Derivative" title="Derivative">derivative</a> is an induced map on <a href="/wiki/Tangent_bundle" title="Tangent bundle">tangent bundles</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{*}:TM\to TN}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> <mo>:</mo> <mi>T</mi> <mi>M</mi> <mo stretchy="false">→</mo> <mi>T</mi> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{*}:TM\to TN}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e65328656819eb7d35be618b249d4e41a7d8819" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.523ex; height:2.509ex;" alt="{\displaystyle f_{*}:TM\to TN}"></span>. Given vector fields <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V:M\to TM}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>:</mo> <mi>M</mi> <mo stretchy="false">→</mo> <mi>T</mi> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V:M\to TM}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b106611267ae14b545c2e59f43b668852e4a8a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.859ex; height:2.176ex;" alt="{\displaystyle V:M\to TM}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W:N\to TN}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>:</mo> <mi>N</mi> <mo stretchy="false">→</mo> <mi>T</mi> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W:N\to TN}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ee2ab660c096451bfb970892f072c926a8279a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.75ex; height:2.176ex;" alt="{\displaystyle W:N\to TN}"></span>, we say that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>-related to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> if the equation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W\circ f=f_{*}\circ V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>∘</mo> <mi>f</mi> <mo>=</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> <mo>∘</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W\circ f=f_{*}\circ V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fe18aac870b908a34c871b0ca0570c21ecb9afe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.182ex; height:2.509ex;" alt="{\displaystyle W\circ f=f_{*}\circ V}"></span> holds. </p><p>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f300b83673e961a9d48f3862216b167f94e5668c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.155ex; height:2.509ex;" alt="{\displaystyle V_{i}}"></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>-related to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7301a4cfd04d4f5db4549fdf23746a0d2ce9f387" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.993ex; height:2.509ex;" alt="{\displaystyle W_{i}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=1,2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i=1,2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/608b3c5e448c465889913a88a105e38e7316fba7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.26ex; height:2.509ex;" alt="{\displaystyle i=1,2}"></span>, then the Lie bracket <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [V_{1},V_{2}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [V_{1},V_{2}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51dc17a2963bc6c7ab701aa8dd43720214abe766" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.147ex; height:2.843ex;" alt="{\displaystyle [V_{1},V_{2}]}"></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>-related to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [W_{1},W_{2}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [W_{1},W_{2}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/420dd448b9831f238dce99132b2aacd13122acae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.823ex; height:2.843ex;" alt="{\displaystyle [W_{1},W_{2}]}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Generalizations">Generalizations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vector_field&action=edit&section=18" title="Edit section: Generalizations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Replacing vectors by <a href="/wiki/P-vector" class="mw-redirect" title="P-vector"><i>p</i>-vectors</a> (<i>p</i>th exterior power of vectors) yields <i>p</i>-vector fields; taking the <a href="/wiki/Dual_space" title="Dual space">dual space</a> and exterior powers yields <a href="/wiki/Differential_form" title="Differential form">differential <i>k</i>-forms</a>, and combining these yields general <a href="/wiki/Tensor_field" title="Tensor field">tensor fields</a>. </p><p>Algebraically, vector fields can be characterized as <a href="/wiki/Derivation_(abstract_algebra)" class="mw-redirect" title="Derivation (abstract algebra)">derivations</a> of the algebra of smooth functions on the manifold, which leads to defining a vector field on a commutative algebra as a derivation on the algebra, which is developed in the theory of <a href="/wiki/Differential_calculus_over_commutative_algebras" title="Differential calculus over commutative algebras">differential calculus over commutative algebras</a>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vector_field&action=edit&section=19" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style 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dynamics">atmospheric dynamics</a></i></li> <li><a href="/wiki/Lie_derivative" title="Lie derivative">Lie derivative</a></li> <li><a href="/wiki/Scalar_field" title="Scalar field">Scalar field</a></li> <li><a href="/wiki/Time-dependent_vector_field" class="mw-redirect" title="Time-dependent vector field">Time-dependent vector field</a></li> <li><a href="/wiki/Vector_fields_in_cylindrical_and_spherical_coordinates" title="Vector fields in cylindrical and spherical coordinates">Vector fields in cylindrical and spherical coordinates</a></li> <li><a href="/wiki/Tensor_fields" class="mw-redirect" title="Tensor fields">Tensor fields</a></li> <li><a href="/wiki/Slope_field" title="Slope field">Slope field</a></li></ul></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vector_field&action=edit&section=20" title="Edit section: References"><span>edit</span></a><span 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src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Vector_field" title="Special:EditPage/Vector field">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Vector+field%22">"Vector field"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Vector+field%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Vector+field%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Vector+field%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Vector+field%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Vector+field%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">April 2012</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <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-Galbis-2012-p12-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-Galbis-2012-p12_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Galbis-2012-p12_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFGalbis,_AntonioMaestre,_Manuel2012" class="citation book cs1">Galbis, Antonio; Maestre, Manuel (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=tdF8uTn2cnMC&pg=PA12"><i>Vector Analysis Versus Vector Calculus</i></a>. Springer. p. 12. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4614-2199-3" title="Special:BookSources/978-1-4614-2199-3"><bdi>978-1-4614-2199-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Vector+Analysis+Versus+Vector+Calculus&rft.pages=12&rft.pub=Springer&rft.date=2012&rft.isbn=978-1-4614-2199-3&rft.au=Galbis%2C+Antonio&rft.au=Maestre%2C+Manuel&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DtdF8uTn2cnMC%26pg%3DPA12&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVector+field" class="Z3988"></span></span> </li> <li id="cite_note-Tu-2010-p149-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Tu-2010-p149_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Tu-2010-p149_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTu,_Loring_W.2010" class="citation book cs1">Tu, Loring W. (2010). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=PZ8Pvk7b6bUC&pg=PA149">"Vector fields"</a>. <i>An Introduction to Manifolds</i>. Springer. p. 149. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4419-7399-3" title="Special:BookSources/978-1-4419-7399-3"><bdi>978-1-4419-7399-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Vector+fields&rft.btitle=An+Introduction+to+Manifolds&rft.pages=149&rft.pub=Springer&rft.date=2010&rft.isbn=978-1-4419-7399-3&rft.au=Tu%2C+Loring+W.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DPZ8Pvk7b6bUC%26pg%3DPA149&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVector+field" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLerman2011" class="citation web cs1">Lerman, Eugene (August 19, 2011). <a rel="nofollow" class="external text" href="https://faculty.math.illinois.edu/~lerman/518/f11/8-19-11.pdf#page=18">"An Introduction to Differential Geometry"</a> <span class="cs1-format">(PDF)</span>. Definition 3.23.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=An+Introduction+to+Differential+Geometry&rft.pages=Definition+3.23&rft.date=2011-08-19&rft.aulast=Lerman&rft.aufirst=Eugene&rft_id=https%3A%2F%2Ffaculty.math.illinois.edu%2F~lerman%2F518%2Ff11%2F8-19-11.pdf%23page%3D18&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVector+field" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDawber,_P.G.1987" class="citation book cs1">Dawber, P.G. (1987). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=luBlL7oGgUIC&pg=PA29"><i>Vectors and Vector Operators</i></a>. CRC Press. p. 29. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-85274-585-4" title="Special:BookSources/978-0-85274-585-4"><bdi>978-0-85274-585-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Vectors+and+Vector+Operators&rft.pages=29&rft.pub=CRC+Press&rft.date=1987&rft.isbn=978-0-85274-585-4&rft.au=Dawber%2C+P.G.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DluBlL7oGgUIC%26pg%3DPA29&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVector+field" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBeretta2020" class="citation journal cs1">Beretta, Gian Paolo (2020-05-01). "The fourth law of thermodynamics: steepest entropy ascent". <i>Philosophical Transactions of the Royal Society A</i>. <b>378</b> (2170): 20190168. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1908.05768">1908.05768</a></span>. <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/2020RSPTA.37890168B">2020RSPTA.37890168B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frsta.2019.0168">10.1098/rsta.2019.0168</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1471-2962">1471-2962</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/32223406">32223406</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:201058607">201058607</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophical+Transactions+of+the+Royal+Society+A&rft.atitle=The+fourth+law+of+thermodynamics%3A+steepest+entropy+ascent&rft.volume=378&rft.issue=2170&rft.pages=20190168&rft.date=2020-05-01&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A201058607%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2020RSPTA.37890168B&rft_id=info%3Aarxiv%2F1908.05768&rft.issn=1471-2962&rft_id=info%3Adoi%2F10.1098%2Frsta.2019.0168&rft_id=info%3Apmid%2F32223406&rft.aulast=Beretta&rft.aufirst=Gian+Paolo&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVector+field" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSharpe1997" class="citation book cs1">Sharpe, R. (1997). <i>Differential geometry</i>. Springer-Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-94732-9" title="Special:BookSources/0-387-94732-9"><bdi>0-387-94732-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Differential+geometry&rft.pub=Springer-Verlag&rft.date=1997&rft.isbn=0-387-94732-9&rft.aulast=Sharpe&rft.aufirst=R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVector+field" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vector_field&action=edit&section=21" title="Edit section: Bibliography"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHubbardHubbard1999" class="citation book cs1"><a href="/wiki/John_H._Hubbard" title="John H. Hubbard">Hubbard, J. H.</a>; <a href="/wiki/Barbara_Burke_Hubbard" title="Barbara Burke Hubbard">Hubbard, B. B.</a> (1999). <i>Vector calculus, linear algebra, and differential forms. A unified approach</i>. Upper Saddle River, NJ: Prentice Hall. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-13-657446-7" title="Special:BookSources/0-13-657446-7"><bdi>0-13-657446-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Vector+calculus%2C+linear+algebra%2C+and+differential+forms.+A+unified+approach&rft.place=Upper+Saddle+River%2C+NJ&rft.pub=Prentice+Hall&rft.date=1999&rft.isbn=0-13-657446-7&rft.aulast=Hubbard&rft.aufirst=J.+H.&rft.au=Hubbard%2C+B.+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVector+field" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWarner1983" class="citation book cs1">Warner, Frank (1983) [1971]. <i>Foundations of differentiable manifolds and Lie groups</i>. New York-Berlin: Springer-Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-90894-3" title="Special:BookSources/0-387-90894-3"><bdi>0-387-90894-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Foundations+of+differentiable+manifolds+and+Lie+groups&rft.place=New+York-Berlin&rft.pub=Springer-Verlag&rft.date=1983&rft.isbn=0-387-90894-3&rft.aulast=Warner&rft.aufirst=Frank&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVector+field" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBoothby1986" class="citation book cs1"><a href="/wiki/William_M._Boothby" title="William M. Boothby">Boothby, William</a> (1986). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/introductiontodi0000boot"><i>An introduction to differentiable manifolds and Riemannian geometry</i></a></span>. Pure and Applied Mathematics, volume 120 (second ed.). Orlando, FL: Academic Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-12-116053-X" title="Special:BookSources/0-12-116053-X"><bdi>0-12-116053-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=An+introduction+to+differentiable+manifolds+and+Riemannian+geometry&rft.place=Orlando%2C+FL&rft.series=Pure+and+Applied+Mathematics%2C+volume+120&rft.edition=second&rft.pub=Academic+Press&rft.date=1986&rft.isbn=0-12-116053-X&rft.aulast=Boothby&rft.aufirst=William&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontodi0000boot&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVector+field" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vector_field&action=edit&section=22" 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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 href="/wiki/Differential_form" title="Differential form">Differential form</a></li> <li><a class="mw-selflink selflink">Vector field</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Main results <span style="font-size:85%;"><a href="/wiki/Category:Theorems_in_differential_geometry" title="Category:Theorems in differential geometry">(list)</a></span></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<br />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 class="mw-selflink selflink">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 href="/wiki/Differential_form" title="Differential form">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 rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"><style data-mw-deduplicate="TemplateStyles:r1038841319">.mw-parser-output .tooltip-dotted{border-bottom:1px dotted;cursor:help}</style></div><div role="navigation" class="navbox authority-control" aria-label="Navbox" style="padding:3px"><table class="nowraplinks hlist 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