Line integral - Wikipedia

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id="toc-Vector_calculus-sublist" class="vector-toc-list"> <li id="toc-Line_integral_of_a_scalar_field" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Line_integral_of_a_scalar_field"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Line integral of a scalar field</span> </div> </a> <ul id="toc-Line_integral_of_a_scalar_field-sublist" class="vector-toc-list"> <li id="toc-Definition" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Definition"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1.1</span> <span>Definition</span> </div> </a> <ul id="toc-Definition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Derivation" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Derivation"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1.2</span> <span>Derivation</span> </div> </a> <ul id="toc-Derivation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Line_integral_of_a_vector_field" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Line_integral_of_a_vector_field"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Line integral of a vector field</span> </div> </a> <ul id="toc-Line_integral_of_a_vector_field-sublist" class="vector-toc-list"> <li id="toc-Definition_2" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Definition_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2.1</span> <span>Definition</span> </div> </a> <ul id="toc-Definition_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Derivation_2" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Derivation_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2.2</span> <span>Derivation</span> </div> </a> <ul id="toc-Derivation_2-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Path_independence" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Path_independence"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Path independence</span> </div> </a> <ul id="toc-Path_independence-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Applications</span> </div> </a> <ul id="toc-Applications-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Flow_across_a_curve" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Flow_across_a_curve"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Flow across a curve</span> </div> </a> <ul id="toc-Flow_across_a_curve-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Complex_line_integral" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Complex_line_integral"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Complex line integral</span> </div> </a> <button aria-controls="toc-Complex_line_integral-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 Complex line integral subsection</span> </button> <ul id="toc-Complex_line_integral-sublist" class="vector-toc-list"> <li id="toc-Example" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Example"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Example</span> </div> </a> <ul id="toc-Example-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relation_of_complex_line_integral_and_line_integral_of_vector_field" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relation_of_complex_line_integral_and_line_integral_of_vector_field"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Relation of complex line integral and line integral of vector field</span> </div> </a> <ul id="toc-Relation_of_complex_line_integral_and_line_integral_of_vector_field-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Quantum_mechanics" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Quantum_mechanics"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Quantum mechanics</span> </div> </a> <ul id="toc-Quantum_mechanics-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">5</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">6</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</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 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</div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Line integral</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 34 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-34" 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">34 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D9%83%D8%A7%D9%85%D9%84_%D8%AE%D8%B7" title="تكامل خط – Arabic" lang="ar" hreflang="ar" data-title="تكامل خط" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9A%D1%80%D0%B8%D0%B2%D0%BE%D0%BB%D0%B8%D0%BD%D0%B5%D0%B5%D0%BD_%D0%B8%D0%BD%D1%82%D0%B5%D0%B3%D1%80%D0%B0%D0%BB" title="Криволинеен интеграл – Bulgarian" lang="bg" hreflang="bg" data-title="Криволинеен интеграл" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Integral_curvil%C3%ADnia" title="Integral curvilínia – Catalan" lang="ca" hreflang="ca" data-title="Integral curvilínia" 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%99%C4%95%D1%80_%D0%B8%D0%BD%D1%82%D0%B5%D0%B3%D1%80%D0%B0%D0%BB%C4%95" title="Йĕр интегралĕ – Chuvash" lang="cv" hreflang="cv" data-title="Йĕр интегралĕ" 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/K%C5%99ivkov%C3%BD_integr%C3%A1l" title="Křivkový integrál – Czech" lang="cs" hreflang="cs" data-title="Křivkový integrál" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Kurveintegral" title="Kurveintegral – Danish" lang="da" hreflang="da" data-title="Kurveintegral" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Kurvenintegral" title="Kurvenintegral – German" lang="de" hreflang="de" data-title="Kurvenintegral" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%95%CF%80%CE%B9%CE%BA%CE%B1%CE%BC%CF%80%CF%8D%CE%BB%CE%B9%CE%BF_%CE%BF%CE%BB%CE%BF%CE%BA%CE%BB%CE%AE%CF%81%CF%89%CE%BC%CE%B1" title="Επικαμπύλιο ολοκλήρωμα – Greek" lang="el" hreflang="el" data-title="Επικαμπύλιο ολοκλήρωμα" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Integral_de_l%C3%ADnea" title="Integral de línea – Spanish" lang="es" hreflang="es" data-title="Integral de línea" 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/Kurba_integralo" title="Kurba integralo – Esperanto" lang="eo" hreflang="eo" data-title="Kurba integralo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A7%D9%86%D8%AA%DA%AF%D8%B1%D8%A7%D9%84_%D8%AE%D8%B7%DB%8C" title="انتگرال خطی – Persian" lang="fa" hreflang="fa" data-title="انتگرال خطی" 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/Int%C3%A9grale_curviligne" title="Intégrale curviligne – French" lang="fr" hreflang="fr" data-title="Intégrale curviligne" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%84%A0%EC%A0%81%EB%B6%84" title="선적분 – Korean" lang="ko" hreflang="ko" data-title="선적분" 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/%D4%BF%D5%B8%D6%80%D5%A1%D5%A3%D5%AB%D5%AE_%D5%AB%D5%B6%D5%BF%D5%A5%D5%A3%D6%80%D5%A1%D5%AC" title="Կորագիծ ինտեգրալ – Armenian" lang="hy" hreflang="hy" data-title="Կորագիծ ինտեգրալ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Integral_garis" title="Integral garis – Indonesian" lang="id" hreflang="id" data-title="Integral garis" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Ferilheildi" title="Ferilheildi – Icelandic" lang="is" hreflang="is" data-title="Ferilheildi" 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/Integrale_di_linea" title="Integrale di linea – Italian" lang="it" hreflang="it" data-title="Integrale di linea" 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%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C_%D7%A7%D7%95%D7%95%D7%99" title="אינטגרל קווי – Hebrew" lang="he" hreflang="he" data-title="אינטגרל קווי" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9A%D1%80%D0%B8%D0%B2%D0%BE%D0%BB%D0%B8%D0%BD%D0%B8%D1%81%D0%BA%D0%B8_%D0%B8%D0%BD%D1%82%D0%B5%D0%B3%D1%80%D0%B0%D0%BB" title="Криволиниски интеграл – Macedonian" lang="mk" hreflang="mk" data-title="Криволиниски интеграл" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Lijnintegraal" title="Lijnintegraal – Dutch" lang="nl" hreflang="nl" data-title="Lijnintegraal" 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/%E7%B7%9A%E7%A9%8D%E5%88%86" title="線積分 – Japanese" lang="ja" hreflang="ja" data-title="線積分" 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-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%B2%E0%A8%BE%E0%A8%88%E0%A8%A8_%E0%A8%87%E0%A9%B0%E0%A8%9F%E0%A8%97%E0%A9%8D%E0%A8%B0%E0%A8%B2" title="ਲਾਈਨ ਇੰਟਗ੍ਰਲ – Punjabi" lang="pa" hreflang="pa" data-title="ਲਾਈਨ ਇੰਟਗ੍ਰਲ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Ca%C5%82ka_krzywoliniowa" title="Całka krzywoliniowa – Polish" lang="pl" hreflang="pl" data-title="Całka krzywoliniowa" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Integral_de_linha" title="Integral de linha – Portuguese" lang="pt" hreflang="pt" data-title="Integral de linha" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Integral%C4%83_curbilinie" title="Integrală curbilinie – Romanian" lang="ro" hreflang="ro" data-title="Integrală curbilinie" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9A%D1%80%D0%B8%D0%B2%D0%BE%D0%BB%D0%B8%D0%BD%D0%B5%D0%B9%D0%BD%D1%8B%D0%B9_%D0%B8%D0%BD%D1%82%D0%B5%D0%B3%D1%80%D0%B0%D0%BB" title="Криволинейный интеграл – Russian" lang="ru" hreflang="ru" data-title="Криволинейный интеграл" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Integrali_kurbolinear" title="Integrali kurbolinear – Albanian" lang="sq" hreflang="sq" data-title="Integrali kurbolinear" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9A%D1%80%D0%B8%D0%B2%D0%BE%D0%BB%D0%B8%D0%BD%D0%B8%D1%98%D1%81%D0%BA%D0%B8_%D0%B8%D0%BD%D1%82%D0%B5%D0%B3%D1%80%D0%B0%D0%BB" title="Криволинијски интеграл – Serbian" lang="sr" hreflang="sr" data-title="Криволинијски интеграл" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Viivaintegraali" title="Viivaintegraali – Finnish" lang="fi" hreflang="fi" data-title="Viivaintegraali" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Kurvintegral" title="Kurvintegral – Swedish" lang="sv" hreflang="sv" data-title="Kurvintegral" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/%C3%87izgi_integrali" title="Çizgi integrali – Turkish" lang="tr" hreflang="tr" data-title="Çizgi integrali" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9A%D1%80%D0%B8%D0%B2%D0%BE%D0%BB%D1%96%D0%BD%D1%96%D0%B9%D0%BD%D0%B8%D0%B9_%D1%96%D0%BD%D1%82%D0%B5%D0%B3%D1%80%D0%B0%D0%BB" title="Криволінійний інтеграл – Ukrainian" lang="uk" hreflang="uk" data-title="Криволінійний інтеграл" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/T%C3%ADch_ph%C3%A2n_%C4%91%C6%B0%E1%BB%9Dng" title="Tích phân đường – Vietnamese" lang="vi" hreflang="vi" data-title="Tích phân đường" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%9B%B2%E7%BA%BF%E7%A7%AF%E5%88%86" title="曲线积分 – Chinese" lang="zh" hreflang="zh" data-title="曲线积分" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q467699#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Line_integral" title="View the content 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rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><td class="sidebar-pretitle">Part of a series of articles about</td></tr><tr><th class="sidebar-title-with-pretitle" style="padding-bottom:0.25em;"><a href="/wiki/Calculus" title="Calculus">Calculus</a></th></tr><tr><td class="sidebar-image"><big><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 \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17d063dc86a53a2efb1fe86f4a5d47d498652766" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.228ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}"></span></big></td></tr><tr><td class="sidebar-above" style="padding:0.15em 0.25em 0.3em;font-weight:normal;"> <ul><li><a href="/wiki/Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus">Fundamental theorem</a></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Limit_of_a_function" title="Limit of a function">Limits</a></li> <li><a href="/wiki/Continuous_function" title="Continuous function">Continuity</a></li></ul> </div><div class="hlist"> <ul><li><a href="/wiki/Rolle%27s_theorem" title="Rolle's theorem">Rolle's theorem</a></li> <li><a href="/wiki/Mean_value_theorem" title="Mean value theorem">Mean value theorem</a></li> <li><a href="/wiki/Inverse_function_theorem" title="Inverse function theorem">Inverse function theorem</a></li></ul> </div></td></tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base);display:block;margin-top:0.65em;"><span style="font-size:120%"><a href="/wiki/Differential_calculus" title="Differential calculus">Differential</a></span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><th class="sidebar-heading"> Definitions</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Derivative" title="Derivative">Derivative</a> (<a href="/wiki/Generalizations_of_the_derivative" title="Generalizations of the derivative">generalizations</a>)</li> <li><a href="/wiki/Differential_(mathematics)" title="Differential (mathematics)">Differential</a> <ul><li><a href="/wiki/Differential_(infinitesimal)" class="mw-redirect" title="Differential (infinitesimal)">infinitesimal</a></li> <li><a href="/wiki/Differential_of_a_function" title="Differential of a function">of a function</a></li> <li><a href="/wiki/Differential_of_a_function#Differentials_in_several_variables" title="Differential of a function">total</a></li></ul></li></ul></td> </tr><tr><th class="sidebar-heading"> Concepts</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Notation_for_differentiation" title="Notation for differentiation">Differentiation notation</a></li> <li><a href="/wiki/Second_derivative" title="Second derivative">Second derivative</a></li> <li><a href="/wiki/Implicit_function" title="Implicit function">Implicit differentiation</a></li> <li><a href="/wiki/Logarithmic_differentiation" title="Logarithmic differentiation">Logarithmic differentiation</a></li> <li><a href="/wiki/Related_rates" title="Related rates">Related rates</a></li> <li><a href="/wiki/Taylor%27s_theorem" title="Taylor's theorem">Taylor's theorem</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Differentiation_rules" title="Differentiation rules">Rules and identities</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Sum_rule_in_differentiation" class="mw-redirect" title="Sum rule in differentiation">Sum</a></li> <li><a href="/wiki/Product_rule" title="Product rule">Product</a></li> <li><a href="/wiki/Chain_rule" title="Chain rule">Chain</a></li> <li><a href="/wiki/Power_rule" title="Power rule">Power</a></li> <li><a href="/wiki/Quotient_rule" title="Quotient rule">Quotient</a></li> <li><a href="/wiki/L%27H%C3%B4pital%27s_rule" title="L'Hôpital's rule">L'Hôpital's rule</a></li> <li><a href="/wiki/Inverse_function_rule" title="Inverse function rule">Inverse</a></li> <li><a href="/wiki/General_Leibniz_rule" title="General Leibniz rule">General Leibniz</a></li> <li><a href="/wiki/Fa%C3%A0_di_Bruno%27s_formula" title="Faà di Bruno's formula">Faà di Bruno's formula</a></li> <li><a href="/wiki/Reynolds_transport_theorem" title="Reynolds transport theorem">Reynolds</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><span style="font-size:120%"><a href="/wiki/Integral" title="Integral">Integral</a></span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Lists_of_integrals" title="Lists of integrals">Lists of integrals</a></li> <li><a href="/wiki/Integral_transform" title="Integral transform">Integral transform</a></li> <li><a href="/wiki/Leibniz_integral_rule" title="Leibniz integral rule">Leibniz integral rule</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Definitions</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Antiderivative" title="Antiderivative">Antiderivative</a></li> <li><a href="/wiki/Integral" title="Integral">Integral</a> (<a href="/wiki/Improper_integral" title="Improper integral">improper</a>)</li> <li><a href="/wiki/Riemann_integral" title="Riemann integral">Riemann integral</a></li> <li><a href="/wiki/Lebesgue_integration" class="mw-redirect" title="Lebesgue integration">Lebesgue integration</a></li> <li><a href="/wiki/Contour_integration" title="Contour integration">Contour integration</a></li> <li><a href="/wiki/Integral_of_inverse_functions" title="Integral of inverse functions">Integral of inverse functions</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Integration by</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Integration_by_parts" title="Integration by parts">Parts</a></li> <li><a href="/wiki/Disc_integration" title="Disc integration">Discs</a></li> <li><a href="/wiki/Shell_integration" title="Shell integration">Cylindrical shells</a></li> <li><a href="/wiki/Integration_by_substitution" title="Integration by substitution">Substitution</a> (<a href="/wiki/Trigonometric_substitution" title="Trigonometric substitution">trigonometric</a>, <a href="/wiki/Tangent_half-angle_substitution" title="Tangent half-angle substitution">tangent half-angle</a>, <a href="/wiki/Euler_substitution" title="Euler substitution">Euler</a>)</li> <li><a href="/wiki/Integration_using_Euler%27s_formula" title="Integration using Euler's formula">Euler's formula</a></li> <li><a href="/wiki/Partial_fractions_in_integration" class="mw-redirect" title="Partial fractions in integration">Partial fractions</a> (<a href="/wiki/Heaviside_cover-up_method" title="Heaviside cover-up method">Heaviside's method</a>)</li> <li><a href="/wiki/Order_of_integration_(calculus)" title="Order of integration (calculus)">Changing order</a></li> <li><a href="/wiki/Integration_by_reduction_formulae" title="Integration by reduction formulae">Reduction formulae</a></li> <li><a href="/wiki/Leibniz_integral_rule#Evaluating_definite_integrals" title="Leibniz integral rule">Differentiating under the integral sign</a></li> <li><a href="/wiki/Risch_algorithm" title="Risch algorithm">Risch algorithm</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><span style="font-size:120%"><a href="/wiki/Series_(mathematics)" title="Series (mathematics)">Series</a></span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Geometric_series" title="Geometric series">Geometric</a> (<a href="/wiki/Arithmetico%E2%80%93geometric_sequence" class="mw-redirect" title="Arithmetico–geometric sequence">arithmetico-geometric</a>)</li> <li><a href="/wiki/Harmonic_series_(mathematics)" title="Harmonic series (mathematics)">Harmonic</a></li> <li><a href="/wiki/Alternating_series" title="Alternating series">Alternating</a></li> <li><a href="/wiki/Power_series" title="Power series">Power</a></li> <li><a href="/wiki/Binomial_series" title="Binomial series">Binomial</a></li> <li><a href="/wiki/Taylor_series" title="Taylor series">Taylor</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Convergence_tests" title="Convergence tests">Convergence tests</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Term_test" class="mw-redirect" title="Term test">Summand limit (term test)</a></li> <li><a href="/wiki/Ratio_test" title="Ratio test">Ratio</a></li> <li><a href="/wiki/Root_test" title="Root test">Root</a></li> <li><a href="/wiki/Integral_test_for_convergence" title="Integral test for convergence">Integral</a></li> <li><a href="/wiki/Direct_comparison_test" title="Direct comparison test">Direct comparison</a></li> <li><br /><a href="/wiki/Limit_comparison_test" title="Limit comparison test">Limit comparison</a></li> <li><a href="/wiki/Alternating_series_test" title="Alternating series test">Alternating series</a></li> <li><a href="/wiki/Cauchy_condensation_test" title="Cauchy condensation test">Cauchy condensation</a></li> <li><a href="/wiki/Dirichlet%27s_test" title="Dirichlet's test">Dirichlet</a></li> <li><a href="/wiki/Abel%27s_test" title="Abel's test">Abel</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><span style="font-size:120%"><a href="/wiki/Vector_calculus" title="Vector calculus">Vector</a></span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Gradient" title="Gradient">Gradient</a></li> <li><a href="/wiki/Divergence" title="Divergence">Divergence</a></li> <li><a href="/wiki/Curl_(mathematics)" title="Curl (mathematics)">Curl</a></li> <li><a href="/wiki/Laplace_operator" title="Laplace operator">Laplacian</a></li> <li><a href="/wiki/Directional_derivative" title="Directional derivative">Directional derivative</a></li> <li><a href="/wiki/Vector_calculus_identities" title="Vector calculus identities">Identities</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Theorems</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Gradient_theorem" title="Gradient theorem">Gradient</a></li> <li><a href="/wiki/Green%27s_theorem" title="Green's theorem">Green's</a></li> <li><a href="/wiki/Stokes%27_theorem" title="Stokes' theorem">Stokes'</a></li> <li><a href="/wiki/Divergence_theorem" title="Divergence theorem">Divergence</a></li> <li><a href="/wiki/Generalized_Stokes_theorem" title="Generalized Stokes theorem">generalized Stokes</a></li> <li><a href="/wiki/Helmholtz_decomposition" title="Helmholtz decomposition">Helmholtz decomposition</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><span style="font-size:120%"><a href="/wiki/Multivariable_calculus" title="Multivariable calculus">Multivariable</a></span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><th class="sidebar-heading"> Formalisms</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Matrix_calculus" title="Matrix calculus">Matrix</a></li> <li><a href="/wiki/Tensor_calculus" class="mw-redirect" title="Tensor calculus">Tensor</a></li> <li><a href="/wiki/Exterior_derivative" title="Exterior derivative">Exterior</a></li> <li><a href="/wiki/Geometric_calculus" title="Geometric calculus">Geometric</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Definitions</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Partial_derivative" title="Partial derivative">Partial derivative</a></li> <li><a href="/wiki/Multiple_integral" title="Multiple integral">Multiple integral</a></li> <li><a class="mw-selflink selflink">Line integral</a></li> <li><a href="/wiki/Surface_integral" title="Surface integral">Surface integral</a></li> <li><a href="/wiki/Volume_integral" title="Volume integral">Volume integral</a></li> <li><a href="/wiki/Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian</a></li> <li><a href="/wiki/Hessian_matrix" title="Hessian matrix">Hessian</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><span style="font-size:120%">Advanced</span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Calculus_on_Euclidean_space" title="Calculus on Euclidean space">Calculus on Euclidean space</a></li> <li><a href="/wiki/Generalized_function" title="Generalized function">Generalized functions</a></li> <li><a href="/wiki/Limit_of_distributions" title="Limit of distributions">Limit of distributions</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><span style="font-size:120%">Specialized</span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Fractional_calculus" title="Fractional calculus">Fractional</a></li> <li><a href="/wiki/Malliavin_calculus" title="Malliavin calculus">Malliavin</a></li> <li><a href="/wiki/Stochastic_calculus" title="Stochastic calculus">Stochastic</a></li> <li><a href="/wiki/Calculus_of_variations" title="Calculus of variations">Variations</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><span style="font-size:120%">Miscellanea</span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Precalculus" title="Precalculus">Precalculus</a></li> <li><a href="/wiki/History_of_calculus" title="History of calculus">History</a></li> <li><a href="/wiki/Glossary_of_calculus" title="Glossary of calculus">Glossary</a></li> <li><a href="/wiki/List_of_calculus_topics" title="List of calculus topics">List of topics</a></li> <li><a href="/wiki/Integration_Bee" title="Integration Bee">Integration Bee</a></li> <li><a href="/wiki/Mathematical_analysis" title="Mathematical analysis">Mathematical analysis</a></li> <li><a href="/wiki/Nonstandard_analysis" title="Nonstandard analysis">Nonstandard analysis</a></li></ul></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Calculus" title="Template:Calculus"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Calculus" title="Template talk:Calculus"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Calculus" title="Special:EditPage/Template:Calculus"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a <b>line integral</b> is an <a href="/wiki/Integral" title="Integral">integral</a> where the <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> to be integrated is evaluated along a <a href="/wiki/Curve" title="Curve">curve</a>.<sup id="cite_ref-Tang2006_1-0" class="reference"><a href="#cite_note-Tang2006-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The terms <i><b>path integral</b></i>, <i><b>curve integral</b></i>, and <i><b>curvilinear integral</b></i> are also used; <i><a href="/wiki/Contour_integral" class="mw-redirect" title="Contour integral">contour integral</a></i> is used as well, although that is typically reserved for <a href="#Complex_line_integral">line integrals in the complex plane</a>. </p><p>The function to be integrated may be a <a href="/wiki/Scalar_field" title="Scalar field">scalar field</a> or a <a href="/wiki/Vector_field" title="Vector field">vector field</a>. The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly <a href="/wiki/Arc_length" title="Arc length">arc length</a> or, for a vector field, the <a href="/wiki/Dot_product" title="Dot product">scalar product</a> of the vector field with a <a href="/wiki/Differential_(infinitesimal)" class="mw-redirect" title="Differential (infinitesimal)">differential</a> vector in the curve). This weighting distinguishes the line integral from simpler integrals defined on <a href="/wiki/Interval_(mathematics)" title="Interval (mathematics)">intervals</a>. Many simple formulae in physics, such as the definition of <a href="/wiki/Work_(physics)" title="Work (physics)">work</a> as <span class="nowrap"><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=\mathbf {F} \cdot \mathbf {s} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">s</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=\mathbf {F} \cdot \mathbf {s} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f62e7b41bc9ba33e6f1d9c7e36d77c48d41a2149" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.951ex; height:2.176ex;" alt="{\displaystyle W=\mathbf {F} \cdot \mathbf {s} }"></span>,</span> have natural continuous analogues in terms of line integrals, in this case <span class="nowrap"><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 W=\int _{L}\mathbf {F} (\mathbf {s} )\cdot d\mathbf {s} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">s</mi> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">s</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle W=\int _{L}\mathbf {F} (\mathbf {s} )\cdot d\mathbf {s} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adcc983069f7d70133cc2efadc0560f8de7a93f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.868ex; height:3.176ex;" alt="{\textstyle W=\int _{L}\mathbf {F} (\mathbf {s} )\cdot d\mathbf {s} }"></span>,</span> which computes the <a href="/wiki/Mechanical_work" class="mw-redirect" title="Mechanical work">work</a> done on an object moving through an electric or gravitational field <span class="texhtml"><b>F</b></span> along a path <span class="nowrap"><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 L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span>.</span> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Vector_calculus">Vector calculus</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Line_integral&action=edit&section=1" title="Edit section: Vector calculus"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In qualitative terms, a line integral in vector calculus can be thought of as a measure of the total effect of a given <a href="/wiki/Tensor_field" title="Tensor field">tensor field</a> along a given curve. For example, the line integral over a scalar field (rank 0 tensor) can be interpreted as the area under the field carved out by a particular curve. This can be visualized as the surface created by <span class="texhtml"><i>z</i> = <i>f</i>(<i>x</i>,<i>y</i>)</span> and a curve <i>C</i> in the <i>xy</i> plane. The line integral of <i>f</i> would be the area of the "curtain" created—when the points of the surface that are directly over <i>C</i> are carved out. </p> <div class="mw-heading mw-heading3"><h3 id="Line_integral_of_a_scalar_field">Line integral of a scalar field</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Line_integral&action=edit&section=2" title="Edit section: Line integral of a scalar field"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Frame"><a href="/wiki/File:Line_integral_of_scalar_field.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/4/42/Line_integral_of_scalar_field.gif" decoding="async" width="400" height="300" class="mw-file-element" data-file-width="400" data-file-height="300" /></a><figcaption>The line integral over a scalar field <i>f</i> can be thought of as the area under the curve <i>C</i> along a surface <span class="texhtml"><i>z</i> = <i>f</i>(<i>x</i>,<i>y</i>)</span>, described by the field.</figcaption></figure> <div class="mw-heading mw-heading4"><h4 id="Definition">Definition</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Line_integral&action=edit&section=3" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For some <a href="/wiki/Scalar_field" title="Scalar field">scalar field</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\colon U\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>U</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon U\to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d71097bf5f00c4116453f42be6ed7f74e74b3be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.387ex; height:2.509ex;" alt="{\displaystyle f\colon U\to \mathbb {R} }"></span> 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 U\subseteq \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>⊆</mo> <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 U\subseteq \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a4ffabb41c43db92055a8fd5be74cd7633756a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.778ex; height:2.509ex;" alt="{\displaystyle U\subseteq \mathbb {R} ^{n}}"></span>, the line integral along a <a href="/wiki/Piecewise_smooth" class="mw-redirect" title="Piecewise smooth">piecewise smooth</a> <a href="/wiki/Curve" title="Curve">curve</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 {\mathcal {C}}\subset U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> <mo>⊂</mo> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}\subset U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37aa7703ab68113f80a4c75db47bd603a37fb3a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.12ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}\subset U}"></span> 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 _{\mathcal {C}}f(\mathbf {r} )\,ds=\int _{a}^{b}f\left(\mathbf {r} (t)\right)\left|\mathbf {r} '(t)\right|\,dt,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>s</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mrow> <mo>|</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{\mathcal {C}}f(\mathbf {r} )\,ds=\int _{a}^{b}f\left(\mathbf {r} (t)\right)\left|\mathbf {r} '(t)\right|\,dt,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69d84403ed856a17f677afc97c29498564afcf8b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:33.378ex; height:6.343ex;" alt="{\displaystyle \int _{\mathcal {C}}f(\mathbf {r} )\,ds=\int _{a}^{b}f\left(\mathbf {r} (t)\right)\left|\mathbf {r} '(t)\right|\,dt,}"></span> 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 \mathbf {r} \colon [a,b]\to {\mathcal {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>:</mo> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} \colon [a,b]\to {\mathcal {C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00e254c282fe91e1b451afd278d990ae34283d19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.544ex; height:2.843ex;" alt="{\displaystyle \mathbf {r} \colon [a,b]\to {\mathcal {C}}}"></span> is an arbitrary <a href="/wiki/Bijective" class="mw-redirect" title="Bijective">bijective</a> <a href="/wiki/Parametric_equation" title="Parametric equation">parametrization</a> of 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 {\mathcal {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7b3edab7022ca9e2976651bc59c489513ee9019" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.239ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}}"></span> such that <span class="texhtml"><b>r</b>(<i>a</i>)</span> and <span class="texhtml"><b>r</b>(<i>b</i>)</span> give the endpoints 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 {\mathcal {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7b3edab7022ca9e2976651bc59c489513ee9019" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.239ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}}"></span> and <span class="texhtml"><i>a</i> < <i>b</i></span>. Here, and in the rest of the article, the absolute value bars denote the <a href="/wiki/Euclidean_distance" title="Euclidean distance">standard (Euclidean) norm</a> of a vector. </p><p>The function <span class="texhtml mvar" style="font-style:italic;">f</span> is called the integrand, 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 {\mathcal {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7b3edab7022ca9e2976651bc59c489513ee9019" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.239ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}}"></span> is the domain of integration, and the symbol <span class="texhtml"><i>ds</i></span> may be intuitively interpreted as an elementary <a href="/wiki/Arc_length" title="Arc length">arc length</a> of 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 {\mathcal {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7b3edab7022ca9e2976651bc59c489513ee9019" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.239ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}}"></span> (i.e., a differential length 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 {\mathcal {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7b3edab7022ca9e2976651bc59c489513ee9019" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.239ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}}"></span>). Line integrals of scalar fields over a 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 {\mathcal {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7b3edab7022ca9e2976651bc59c489513ee9019" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.239ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}}"></span> do not depend on the chosen parametrization <span class="texhtml"><b>r</b></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 {\mathcal {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7b3edab7022ca9e2976651bc59c489513ee9019" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.239ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}}"></span>.<sup id="cite_ref-:1_2-0" class="reference"><a href="#cite_note-:1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Geometrically, when the scalar field <span class="texhtml mvar" style="font-style:italic;">f</span> is defined over a plane <span class="texhtml">(<i>n</i> = 2)</span>, its graph is a surface <span class="texhtml"><i>z</i> = <i>f</i>(<i>x</i>, <i>y</i>)</span> in space, and the line integral gives the (signed) <a href="/wiki/Cross-section_(geometry)" class="mw-redirect" title="Cross-section (geometry)">cross-sectional</a> area bounded by 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 {\mathcal {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7b3edab7022ca9e2976651bc59c489513ee9019" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.239ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}}"></span> and the graph of <span class="texhtml mvar" style="font-style:italic;">f</span>. See the animation to the right. </p> <div class="mw-heading mw-heading4"><h4 id="Derivation">Derivation</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Line_integral&action=edit&section=4" title="Edit section: Derivation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For a line integral over a scalar field, the integral can be constructed from a <a href="/wiki/Riemann_sum" title="Riemann sum">Riemann sum</a> using the above definitions of <span class="texhtml mvar" style="font-style:italic;">f</span>, <span class="texhtml mvar" style="font-style:italic;">C</span> and a parametrization <span class="texhtml"><b>r</b></span> of <span class="texhtml mvar" style="font-style:italic;">C</span>. This can be done by partitioning the <a href="/wiki/Interval_(mathematics)" title="Interval (mathematics)">interval</a> <span class="texhtml">[<i>a</i>, <i>b</i>]</span> into <span class="texhtml mvar" style="font-style:italic;">n</span> sub-intervals <span class="texhtml">[<i>t</i><sub><i>i</i>−1</sub>, <i>t</i><sub><i>i</i></sub>]</span> of length <span class="texhtml">Δ<i>t</i> = (<i>b</i> − <i>a</i>)/<i>n</i></span>, then <span class="texhtml"><b>r</b>(<i>t</i><sub><i>i</i></sub>)</span> denotes some point, call it a sample point, on the curve <span class="texhtml mvar" style="font-style:italic;">C</span>. We can use the <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> of sample points <span class="texhtml">{<b>r</b>(<i>t</i><sub><i>i</i></sub>): 1 ≤ <i>i</i> ≤ <i>n</i>}</span> to approximate the curve <span class="texhtml mvar" style="font-style:italic;">C</span> as a <a href="/wiki/Polygonal_path" class="mw-redirect" title="Polygonal path">polygonal path</a> by introducing the straight line piece between each of the sample points <span class="texhtml"><b>r</b>(<i>t</i><sub><i>i</i>−1</sub>)</span> and <span class="texhtml"><b>r</b>(<i>t</i><sub><i>i</i></sub>)</span>. (The approximation of a curve to a polygonal path is called <i>rectification of a curve,</i> see <a href="/wiki/Arc_length" title="Arc length">here</a> for more details.) We then label the distance of the line segment between adjacent sample points on the curve as <span class="texhtml">Δ<i>s</i><sub><i>i</i></sub></span>. The product of <span class="texhtml"><i>f</i>(<b>r</b>(<i>t</i><sub><i>i</i></sub>))</span> and <span class="texhtml">Δ<i>s</i><sub><i>i</i></sub></span> can be associated with the signed area of a rectangle with a height and width of <span class="texhtml"><i>f</i>(<b>r</b>(<i>t</i><sub><i>i</i></sub>))</span> and <span class="texhtml">Δ<i>s</i><sub><i>i</i></sub></span>, respectively. Taking the <a href="/wiki/Limit_of_a_sequence" title="Limit of a sequence">limit</a> of the <a href="/wiki/Summation" title="Summation">sum</a> of the terms as the length of the partitions approaches zero gives us <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 I=\lim _{\Delta s_{i}\to 0}\sum _{i=1}^{n}f(\mathbf {r} (t_{i}))\,\Delta s_{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Δ</mi> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <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> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi mathvariant="normal">Δ</mi> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I=\lim _{\Delta s_{i}\to 0}\sum _{i=1}^{n}f(\mathbf {r} (t_{i}))\,\Delta s_{i}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85a192f293130034b5a3d2a1b43f36da032832e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.128ex; height:6.843ex;" alt="{\displaystyle I=\lim _{\Delta s_{i}\to 0}\sum _{i=1}^{n}f(\mathbf {r} (t_{i}))\,\Delta s_{i}.}"></span> </p><p>By the <a href="/wiki/Mean_value_theorem" title="Mean value theorem">mean value theorem</a>, the distance between subsequent points on the curve, is <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 \Delta s_{i}=\left|\mathbf {r} (t_{i}+\Delta t)-\mathbf {r} (t_{i})\right|\approx \left|\mathbf {r} '(t_{i})\Delta t\right|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mo>≈</mo> <mrow> <mo>|</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mrow> <mo>|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta s_{i}=\left|\mathbf {r} (t_{i}+\Delta t)-\mathbf {r} (t_{i})\right|\approx \left|\mathbf {r} '(t_{i})\Delta t\right|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edeec3b7689fc395f0d18df564ccd7340ddfec11" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.179ex; height:3.009ex;" alt="{\displaystyle \Delta s_{i}=\left|\mathbf {r} (t_{i}+\Delta t)-\mathbf {r} (t_{i})\right|\approx \left|\mathbf {r} '(t_{i})\Delta t\right|}"></span> </p><p>Substituting this in the above Riemann sum yields <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 I=\lim _{\Delta t\to 0}\sum _{i=1}^{n}f(\mathbf {r} (t_{i}))\left|\mathbf {r} '(t_{i})\right|\Delta t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <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> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mrow> <mo>|</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I=\lim _{\Delta t\to 0}\sum _{i=1}^{n}f(\mathbf {r} (t_{i}))\left|\mathbf {r} '(t_{i})\right|\Delta t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/680a75e25b4f4b3a762860fdfc4212ff99081523" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:30.544ex; height:6.843ex;" alt="{\displaystyle I=\lim _{\Delta t\to 0}\sum _{i=1}^{n}f(\mathbf {r} (t_{i}))\left|\mathbf {r} '(t_{i})\right|\Delta t}"></span> which is the Riemann sum for the integral <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 I=\int _{a}^{b}f(\mathbf {r} (t))\left|\mathbf {r} '(t)\right|dt.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mrow> <mo>|</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mi>d</mi> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I=\int _{a}^{b}f(\mathbf {r} (t))\left|\mathbf {r} '(t)\right|dt.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79d896c9f15bed2e25669e6f075582af73b836d0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.104ex; height:6.343ex;" alt="{\displaystyle I=\int _{a}^{b}f(\mathbf {r} (t))\left|\mathbf {r} '(t)\right|dt.}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Line_integral_of_a_vector_field">Line integral of a vector field</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Line_integral&action=edit&section=5" title="Edit section: Line integral of a vector field"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Definition_2">Definition</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Line_integral&action=edit&section=6" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For a <a href="/wiki/Vector_field" title="Vector field">vector field</a> <span class="texhtml"><b>F</b>: <i>U</i> ⊆ <b>R</b><sup><i>n</i></sup> → <b>R</b><sup><i>n</i></sup></span>, the line integral along a <a href="/wiki/Piecewise_smooth" class="mw-redirect" title="Piecewise smooth">piecewise smooth</a> <a href="/wiki/Curve" title="Curve">curve</a> <span class="texhtml"><i>C</i> ⊂ <i>U</i></span>, in the direction of <b>r</b>, 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 _{C}\mathbf {F} (\mathbf {r} )\cdot d\mathbf {r} =\int _{a}^{b}\mathbf {F} (\mathbf {r} (t))\cdot \mathbf {r} '(t)\,dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</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> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>⋅</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{C}\mathbf {F} (\mathbf {r} )\cdot d\mathbf {r} =\int _{a}^{b}\mathbf {F} (\mathbf {r} (t))\cdot \mathbf {r} '(t)\,dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d317ca76dbfae168bf1eb9cea2f843a0549a4789" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:34.44ex; height:6.343ex;" alt="{\displaystyle \int _{C}\mathbf {F} (\mathbf {r} )\cdot d\mathbf {r} =\int _{a}^{b}\mathbf {F} (\mathbf {r} (t))\cdot \mathbf {r} '(t)\,dt}"></span> where <span class="texhtml">·</span> is the <a href="/wiki/Dot_product" title="Dot product">dot product</a>, and <span class="texhtml"><b>r</b>: [<i>a</i>, <i>b</i>] → <i>C</i></span> is a regular <a href="/wiki/Parametric_equation" title="Parametric equation">parametrization</a> (i.e: <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} '(t)||\neq 0\;\;\forall t\in [a,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≠</mo> <mn>0</mn> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi mathvariant="normal">∀</mi> <mi>t</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ||\mathbf {r} '(t)||\neq 0\;\;\forall t\in [a,b]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a845d5294bb81bb2d50c44b4747fd6d6ea9689d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.102ex; height:3.009ex;" alt="{\displaystyle ||\mathbf {r} '(t)||\neq 0\;\;\forall t\in [a,b]}"></span>) of the curve <i>C</i> such that <span class="texhtml"><b>r</b>(<i>a</i>)</span> and <span class="texhtml"><b>r</b>(<i>b</i>)</span> give the endpoints of <i>C</i>. </p><p>A line integral of a scalar field is thus a line integral of a vector field, where the vectors are always <a href="/wiki/Tangential" class="mw-redirect" title="Tangential">tangential</a> to the line of the integration. </p><p>Line integrals of vector fields are independent of the parametrization <b>r</b> in <a href="/wiki/Absolute_value" title="Absolute value">absolute value</a>, but they do depend on its <a href="/wiki/Curve_orientation" title="Curve orientation">orientation</a>. Specifically, a reversal in the orientation of the parametrization changes the sign of the line integral.<sup id="cite_ref-:1_2-1" class="reference"><a href="#cite_note-:1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>From the viewpoint of <a href="/wiki/Differential_geometry" title="Differential geometry">differential geometry</a>, the line integral of a vector field along a curve is the integral of the corresponding 1-form under the <a href="/wiki/Musical_isomorphism" title="Musical isomorphism">musical isomorphism</a> (which takes the vector field to the corresponding <a href="/wiki/Covector" class="mw-redirect" title="Covector">covector</a> field), over the curve considered as an <a href="/wiki/Immersion_(mathematics)" title="Immersion (mathematics)">immersed</a> 1-manifold. </p> <div class="mw-heading mw-heading4"><h4 id="Derivation_2">Derivation</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Line_integral&action=edit&section=7" title="Edit section: Derivation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Line_integral_of_vector_field.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/b/b0/Line_integral_of_vector_field.gif" decoding="async" width="450" height="410" class="mw-file-element" data-file-width="450" data-file-height="410" /></a><figcaption>The trajectory of a particle (in red) along a curve inside a vector field. Starting from <i>a</i>, the particle traces the path <i>C</i> along the vector field <i>F</i>. The dot product (green line) of its tangent vector (red arrow) and the field vector (blue arrow) defines an area under a curve, which is equivalent to the path's line integral. (Click on image for a detailed description.)</figcaption></figure> <p>The line integral of a vector field can be derived in a manner very similar to the case of a scalar field, but this time with the inclusion of a dot product. Again using the above definitions of <span class="texhtml"><b>F</b></span>, <span class="texhtml mvar" style="font-style:italic;">C</span> and its parametrization <span class="texhtml"><b>r</b>(<i>t</i>)</span>, we construct the integral from a <a href="/wiki/Riemann_sum" title="Riemann sum">Riemann sum</a>. We partition the <a href="/wiki/Interval_(mathematics)" title="Interval (mathematics)">interval</a> <span class="texhtml">[<i>a</i>, <i>b</i>]</span> (which is the range of the values of the <a href="/wiki/Parameter" title="Parameter">parameter</a> <span class="texhtml mvar" style="font-style:italic;">t</span>) into <span class="texhtml mvar" style="font-style:italic;">n</span> intervals of length <span class="texhtml">Δ<i>t</i> = (<i>b</i> − <i>a</i>)/<i>n</i></span>. Letting <span class="texhtml"><i>t<sub>i</sub></i></span> be the <span class="texhtml mvar" style="font-style:italic;">i</span>th point on <span class="texhtml">[<i>a</i>, <i>b</i>]</span>, then <span class="texhtml"><b>r</b>(<i>t</i><sub><i>i</i></sub>)</span> gives us the position of the <span class="texhtml mvar" style="font-style:italic;">i</span>th point on the curve. However, instead of calculating up the distances between subsequent points, we need to calculate their <a href="/wiki/Displacement_(vector)" class="mw-redirect" title="Displacement (vector)">displacement</a> vectors, <span class="texhtml">Δ<b>r</b><sub><i>i</i></sub></span>. As before, evaluating <span class="texhtml"><b>F</b></span> at all the points on the curve and taking the dot product with each displacement vector gives us the <a href="/wiki/Infinitesimal" title="Infinitesimal">infinitesimal</a> contribution of each partition of <span class="texhtml"><b>F</b></span> on <span class="texhtml mvar" style="font-style:italic;">C</span>. Letting the size of the partitions go to zero gives us a sum <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 I=\lim _{\Delta t\to 0}\sum _{i=1}^{n}\mathbf {F} (\mathbf {r} (t_{i}))\cdot \Delta \mathbf {r} _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi mathvariant="normal">Δ</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I=\lim _{\Delta t\to 0}\sum _{i=1}^{n}\mathbf {F} (\mathbf {r} (t_{i}))\cdot \Delta \mathbf {r} _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/376597a2ceb7c98255401bbf1c19dbff6a7d13b9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.386ex; height:6.843ex;" alt="{\displaystyle I=\lim _{\Delta t\to 0}\sum _{i=1}^{n}\mathbf {F} (\mathbf {r} (t_{i}))\cdot \Delta \mathbf {r} _{i}}"></span> </p><p>By the <a href="/wiki/Mean_value_theorem" title="Mean value theorem">mean value theorem</a>, we see that the displacement vector between adjacent points on the curve is <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 \Delta \mathbf {r} _{i}=\mathbf {r} (t_{i}+\Delta t)-\mathbf {r} (t_{i})\approx \mathbf {r} '(t_{i})\,\Delta t.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>≈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta \mathbf {r} _{i}=\mathbf {r} (t_{i}+\Delta t)-\mathbf {r} (t_{i})\approx \mathbf {r} '(t_{i})\,\Delta t.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdfa95407800597e5b86905d7caf335bf89f6aa0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.637ex; height:3.009ex;" alt="{\displaystyle \Delta \mathbf {r} _{i}=\mathbf {r} (t_{i}+\Delta t)-\mathbf {r} (t_{i})\approx \mathbf {r} '(t_{i})\,\Delta t.}"></span> </p><p>Substituting this in the above Riemann sum yields <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 I=\lim _{\Delta t\to 0}\sum _{i=1}^{n}\mathbf {F} (\mathbf {r} (t_{i}))\cdot \mathbf {r} '(t_{i})\,\Delta t,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>⋅</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I=\lim _{\Delta t\to 0}\sum _{i=1}^{n}\mathbf {F} (\mathbf {r} (t_{i}))\cdot \mathbf {r} '(t_{i})\,\Delta t,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d78d5e663fea70ca88b45f0293e3741a3fc7acef" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:31.594ex; height:6.843ex;" alt="{\displaystyle I=\lim _{\Delta t\to 0}\sum _{i=1}^{n}\mathbf {F} (\mathbf {r} (t_{i}))\cdot \mathbf {r} '(t_{i})\,\Delta t,}"></span> </p><p>which is the Riemann sum for the integral defined above. </p> <div class="mw-heading mw-heading3"><h3 id="Path_independence">Path independence</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Line_integral&action=edit&section=8" title="Edit section: Path independence"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Gradient_theorem" title="Gradient theorem">Gradient theorem</a></div> <p>If a vector field <span class="texhtml"><b>F</b></span> is the <a href="/wiki/Gradient" title="Gradient">gradient</a> of a <a href="/wiki/Scalar_field" title="Scalar field">scalar field</a> <span class="texhtml"><i>G</i></span> (i.e. if <span class="texhtml"><b>F</b></span> is <a href="/wiki/Conservative_vector_field" title="Conservative vector field">conservative</a>), that is, <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 \mathbf {F} =\nabla G,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mi>G</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} =\nabla G,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/199b3b0d079b447f7bdbb88274e52a208bf9c9a5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.191ex; height:2.509ex;" alt="{\displaystyle \mathbf {F} =\nabla G,}"></span> then by the <a href="/wiki/Chain_rule#Multivariable_case" title="Chain rule">multivariable chain rule</a> the <a href="/wiki/Derivative" title="Derivative">derivative</a> of the <a href="/wiki/Function_composition" title="Function composition">composition</a> of <span class="texhtml"><i>G</i></span> and <span class="texhtml"><b>r</b>(<i>t</i>)</span> is <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 {\frac {dG(\mathbf {r} (t))}{dt}}=\nabla G(\mathbf {r} (t))\cdot \mathbf {r} '(t)=\mathbf {F} (\mathbf {r} (t))\cdot \mathbf {r} '(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>G</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mi>G</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>⋅</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>⋅</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dG(\mathbf {r} (t))}{dt}}=\nabla G(\mathbf {r} (t))\cdot \mathbf {r} '(t)=\mathbf {F} (\mathbf {r} (t))\cdot \mathbf {r} '(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a594ef5792d8090938048752b9458568c7797471" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:44.431ex; height:5.843ex;" alt="{\displaystyle {\frac {dG(\mathbf {r} (t))}{dt}}=\nabla G(\mathbf {r} (t))\cdot \mathbf {r} '(t)=\mathbf {F} (\mathbf {r} (t))\cdot \mathbf {r} '(t)}"></span> which happens to be the integrand for the line integral of <span class="texhtml"><b>F</b></span> on <span class="texhtml"><b>r</b>(<i>t</i>)</span>. It follows, given a path <i>C</i>, 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 \int _{C}\mathbf {F} (\mathbf {r} )\cdot d\mathbf {r} =\int _{a}^{b}\mathbf {F} (\mathbf {r} (t))\cdot \mathbf {r} '(t)\,dt=\int _{a}^{b}{\frac {dG(\mathbf {r} (t))}{dt}}\,dt=G(\mathbf {r} (b))-G(\mathbf {r} (a)).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</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> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>⋅</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>G</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mi>G</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>−</mo> <mi>G</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{C}\mathbf {F} (\mathbf {r} )\cdot d\mathbf {r} =\int _{a}^{b}\mathbf {F} (\mathbf {r} (t))\cdot \mathbf {r} '(t)\,dt=\int _{a}^{b}{\frac {dG(\mathbf {r} (t))}{dt}}\,dt=G(\mathbf {r} (b))-G(\mathbf {r} (a)).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53fa5d1a2d56abfccec4cbb31587b5cab8f7759a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:75.116ex; height:6.343ex;" alt="{\displaystyle \int _{C}\mathbf {F} (\mathbf {r} )\cdot d\mathbf {r} =\int _{a}^{b}\mathbf {F} (\mathbf {r} (t))\cdot \mathbf {r} '(t)\,dt=\int _{a}^{b}{\frac {dG(\mathbf {r} (t))}{dt}}\,dt=G(\mathbf {r} (b))-G(\mathbf {r} (a)).}"></span> </p><p>In other words, the integral of <span class="texhtml"><b>F</b></span> over <i>C</i> depends solely on the values of <span class="texhtml"><i>G</i></span> at the points <span class="texhtml"><b>r</b>(<i>b</i>)</span> and <span class="texhtml"><b>r</b>(<i>a</i>)</span>, and is thus independent of the path between them. For this reason, a line integral of a conservative vector field is called <i>path independent</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Applications">Applications</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Line_integral&action=edit&section=9" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The line integral has many uses in physics. For example, the <a href="/wiki/Work_(physics)" title="Work (physics)">work</a> done on a particle traveling on a curve <i>C</i> inside a force field represented as a vector field <span class="texhtml"><b>F</b></span> is the line integral of <span class="texhtml"><b>F</b></span> on <i>C</i>.<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> <div class="mw-heading mw-heading2"><h2 id="Flow_across_a_curve">Flow across a curve</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Line_integral&action=edit&section=10" title="Edit section: Flow across a curve"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For a <a href="/wiki/Vector_field" title="Vector field">vector field</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 \mathbf {F} \colon U\subseteq \mathbb {R} ^{2}\to \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>:</mo> <mi>U</mi> <mo>⊆</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">→</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} \colon U\subseteq \mathbb {R} ^{2}\to \mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70798a4bbae70cabba481760ac949f6a38608413" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.676ex; height:2.843ex;" alt="{\displaystyle \mathbf {F} \colon U\subseteq \mathbb {R} ^{2}\to \mathbb {R} ^{2}}"></span>, <span class="texhtml"><b>F</b>(<i>x</i>, <i>y</i>) = (<i>P</i>(<i>x</i>, <i>y</i>), <i>Q</i>(<i>x</i>, <i>y</i>))</span>, the <b>line integral across a curve</b> <i>C</i> ⊂ <i>U</i>, also called the <a href="/wiki/Flux" title="Flux"><i>flux integral</i></a>, is defined in terms of a <a href="/wiki/Piecewise_smooth" class="mw-redirect" title="Piecewise smooth">piecewise smooth</a> <a href="/wiki/Parametric_equation" title="Parametric equation">parametrization</a> <span class="texhtml"><b>r</b>: [<i>a</i>,<i>b</i>] → <i>C</i></span>, <span class="texhtml"><b>r</b>(<i>t</i>) = (<i>x</i>(<i>t</i>), <i>y</i>(<i>t</i>))</span>, 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 _{C}\mathbf {F} (\mathbf {r} )\cdot d\mathbf {r} ^{\perp }=\int _{a}^{b}{\begin{bmatrix}P{\big (}x(t),y(t){\big )}\\Q{\big (}x(t),y(t){\big )}\end{bmatrix}}\cdot {\begin{bmatrix}y'(t)\\-x'(t)\end{bmatrix}}~dt=\int _{a}^{b}\left(-Q~dx+P~dy\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>d</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>⊥</mo> </mrow> </msup> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>y</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>y</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mtext> </mtext> <mi>d</mi> <mi>t</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mi>Q</mi> <mtext> </mtext> <mi>d</mi> <mi>x</mi> <mo>+</mo> <mi>P</mi> <mtext> </mtext> <mi>d</mi> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{C}\mathbf {F} (\mathbf {r} )\cdot d\mathbf {r} ^{\perp }=\int _{a}^{b}{\begin{bmatrix}P{\big (}x(t),y(t){\big )}\\Q{\big (}x(t),y(t){\big )}\end{bmatrix}}\cdot {\begin{bmatrix}y'(t)\\-x'(t)\end{bmatrix}}~dt=\int _{a}^{b}\left(-Q~dx+P~dy\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/648ad32b3545893968b09e1fa0768cf2098cb92e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:74.431ex; height:7.509ex;" alt="{\displaystyle \int _{C}\mathbf {F} (\mathbf {r} )\cdot d\mathbf {r} ^{\perp }=\int _{a}^{b}{\begin{bmatrix}P{\big (}x(t),y(t){\big )}\\Q{\big (}x(t),y(t){\big )}\end{bmatrix}}\cdot {\begin{bmatrix}y'(t)\\-x'(t)\end{bmatrix}}~dt=\int _{a}^{b}\left(-Q~dx+P~dy\right).}"></span> </p><p>Here <span class="texhtml">⋅</span> is the dot product, 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 \mathbf {r} '(t)^{\perp }=(y'(t),-x'(t))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>⊥</mo> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mo>−</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} '(t)^{\perp }=(y'(t),-x'(t))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b0661407cd0a57ff967d3b9bad4db5fd59c2f37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.854ex; height:3.176ex;" alt="{\displaystyle \mathbf {r} '(t)^{\perp }=(y'(t),-x'(t))}"></span> is the clockwise perpendicular of the velocity vector <span class="nowrap"><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} '(t)=(x'(t),y'(t))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} '(t)=(x'(t),y'(t))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1761cded64bc1422c874a509a3704ae97152cc6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.535ex; height:3.009ex;" alt="{\displaystyle \mathbf {r} '(t)=(x'(t),y'(t))}"></span>.</span> </p><p>The flow is computed in an oriented sense: the curve <span class="texhtml mvar" style="font-style:italic;">C</span> has a specified forward direction from <span class="texhtml"><b>r</b>(<i>a</i>)</span> to <span class="texhtml"><b>r</b>(<i>b</i>)</span>, and the flow is counted as positive when <span class="texhtml"><b>F</b>(<b>r</b>(<i>t</i>))</span> is on the clockwise side of the forward velocity vector <span class="texhtml"><b>r'</b>(<i>t</i>)</span>. </p> <div class="mw-heading mw-heading2"><h2 id="Complex_line_integral">Complex line integral</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Line_integral&action=edit&section=11" title="Edit section: Complex line integral"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Complex_analysis" title="Complex analysis">complex analysis</a>, the line integral is defined in terms of <a href="/wiki/Complex_number#Multiplication" title="Complex number">multiplication</a> and <a href="/wiki/Complex_number#Addition_and_subtraction" title="Complex number">addition</a> of complex numbers. Suppose <i>U</i> is an <a href="/wiki/Open_subset" class="mw-redirect" title="Open subset">open subset</a> of the <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a> <b>C</b>, <span class="texhtml"><i>f</i> : <i>U</i> → <b>C</b></span> is a function, 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 L\subset U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>⊂</mo> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L\subset U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38ccc5e372e3f9b86bbfb6a0bac51f2f7b274ea6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.464ex; height:2.176ex;" alt="{\displaystyle L\subset U}"></span> is a curve of finite length, parametrized by <span class="texhtml"><i>γ</i>: [<i>a</i>,<i>b</i>] → <i>L</i></span>, where <span class="texhtml"><i>γ</i>(<i>t</i>) = <i>x</i>(<i>t</i>) + <i>iy</i>(<i>t</i>)</span>. The line integral <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 _{L}f(z)\,dz}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{L}f(z)\,dz}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6687361b95b6d95787ecd4f1ce97d8d34c8a6fe" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:9.898ex; height:5.676ex;" alt="{\displaystyle \int _{L}f(z)\,dz}"></span> may be defined by subdividing the <a href="/wiki/Interval_(mathematics)" title="Interval (mathematics)">interval</a> [<i>a</i>, <i>b</i>] into <span class="nowrap"><i>a</i> = <i>t</i><sub>0</sub> < <i>t</i><sub>1</sub> < ... < <i>t</i><sub><i>n</i></sub> = <i>b</i></span> and considering the expression <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 \sum _{k=1}^{n}f(\gamma (t_{k}))\,[\gamma (t_{k})-\gamma (t_{k-1})]=\sum _{k=1}^{n}f(\gamma _{k})\,\Delta \gamma _{k}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>f</mi> <mo stretchy="false">(</mo> <mi>γ</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo stretchy="false">[</mo> <mi>γ</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>γ</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi mathvariant="normal">Δ</mi> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{n}f(\gamma (t_{k}))\,[\gamma (t_{k})-\gamma (t_{k-1})]=\sum _{k=1}^{n}f(\gamma _{k})\,\Delta \gamma _{k}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2bb22683cc64b9d75f8c5c6c3a2d60bde08a515" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:45.936ex; height:6.843ex;" alt="{\displaystyle \sum _{k=1}^{n}f(\gamma (t_{k}))\,[\gamma (t_{k})-\gamma (t_{k-1})]=\sum _{k=1}^{n}f(\gamma _{k})\,\Delta \gamma _{k}.}"></span> </p><p>The integral is then the limit of this <a href="/wiki/Riemann_sum" title="Riemann sum">Riemann sum</a> as the lengths of the subdivision intervals approach zero. </p><p>If the parametrization <span class="texhtml mvar" style="font-style:italic;">γ</span> is <a href="/wiki/Differentiable_function#Differentiability_classes" title="Differentiable function">continuously differentiable</a>, the line integral can be evaluated as an integral of a function of a real variable: <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 _{L}f(z)\,dz=\int _{a}^{b}f(\gamma (t))\gamma '(t)\,dt.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>z</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <msup> <mi>γ</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{L}f(z)\,dz=\int _{a}^{b}f(\gamma (t))\gamma '(t)\,dt.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3d3465068f61ba875b97a58b0a3c21f85705f1d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:31.488ex; height:6.343ex;" alt="{\displaystyle \int _{L}f(z)\,dz=\int _{a}^{b}f(\gamma (t))\gamma '(t)\,dt.}"></span> </p><p>When <span class="texhtml mvar" style="font-style:italic;">L</span> is a closed curve (initial and final points coincide), the line integral is often denoted <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 \oint _{L}f(z)\,dz,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>z</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \oint _{L}f(z)\,dz,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dedd2e21de79ca198fc043777090b3079a47228" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.35ex; height:3.176ex;" alt="{\textstyle \oint _{L}f(z)\,dz,}"></span> sometimes referred to in engineering as a <i>cyclic integral</i>. </p><p>To establish a complete analogy with the line integral of a vector field, one must go back to the definition of differentiability in multivariable calculus. The gradient is defined from <a href="/wiki/Riesz_representation_theorem" title="Riesz representation theorem">Riesz representation theorem</a>, and inner products in complex analysis involve conjugacy (the gradient of a 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 \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span> at 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 z\in \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z\in \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/169fae60c23a2027ece2aa7fd4b5047492887e91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.607ex; height:2.176ex;" alt="{\displaystyle z\in \mathbb {C} }"></span> would be <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 {\overline {\gamma '(z)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msup> <mi>γ</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\gamma '(z)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ef194c6d8469036a6d89cea595dbb4c41f12047" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.977ex; height:3.676ex;" alt="{\displaystyle {\overline {\gamma '(z)}}}"></span>, and the complex inner product would attribute twice a conjugate 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 \gamma '}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>γ</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma '}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0e2b9e9e12e56bd62af445be6803ec4843e919a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.965ex; height:3.009ex;" alt="{\displaystyle \gamma '}"></span> in the vector field definition of a line integral). </p><p>The line integral with respect to the conjugate complex differential <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 {\overline {dz}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>d</mi> <mi>z</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {dz}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f89d5d601573e933f7f08769e6a8e7ca78692436" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.424ex; height:3.009ex;" alt="{\displaystyle {\overline {dz}}}"></span> is defined<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> to be <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 _{L}f(z){\overline {dz}}:={\overline {\int _{L}{\overline {f(z)}}\,dz}}=\int _{a}^{b}f(\gamma (t)){\overline {\gamma '(t)}}\,dt.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>d</mi> <mi>z</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>z</mi> </mrow> <mo accent="false">¯</mo> </mover> </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>f</mi> <mo stretchy="false">(</mo> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msup> <mi>γ</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{L}f(z){\overline {dz}}:={\overline {\int _{L}{\overline {f(z)}}\,dz}}=\int _{a}^{b}f(\gamma (t)){\overline {\gamma '(t)}}\,dt.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13d1aaa179397451052608d1e9b7ac88d99131ee" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:45.213ex; height:6.509ex;" alt="{\displaystyle \int _{L}f(z){\overline {dz}}:={\overline {\int _{L}{\overline {f(z)}}\,dz}}=\int _{a}^{b}f(\gamma (t)){\overline {\gamma '(t)}}\,dt.}"></span> </p><p>The line integrals of complex functions can be evaluated using a number of techniques. The most direct is to split into real and imaginary parts, reducing the problem to evaluating two real-valued line integrals. The <a href="/wiki/Cauchy_integral_theorem" class="mw-redirect" title="Cauchy integral theorem">Cauchy integral theorem</a> may be used to equate the line integral of an <a href="/wiki/Analytic_function" title="Analytic function">analytic function</a> to the same integral over a more convenient curve. It also implies that over a closed curve enclosing a region where <span class="texhtml"><i>f</i>(<i>z</i>)</span> is analytic without <a href="/wiki/Mathematical_singularity" class="mw-redirect" title="Mathematical singularity">singularities</a>, the value of the integral is simply zero, or in case the region includes singularities, the <a href="/wiki/Residue_theorem" title="Residue theorem">residue theorem</a> computes the integral in terms of the singularities. This also implies the path independence of complex line integral for analytic functions. </p> <div class="mw-heading mw-heading3"><h3 id="Example">Example</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Line_integral&action=edit&section=12" title="Edit section: Example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Consider the function <span class="texhtml"><i>f</i>(<i>z</i>) = 1/<i>z</i></span>, and let the contour <i>L</i> be the counterclockwise <a href="/wiki/Unit_circle" title="Unit circle">unit circle</a> about 0, parametrized by <span class="texhtml"><i>z</i>(<i>t</i>) = <i>e</i><sup><i>it</i></sup></span> with <span class="texhtml mvar" style="font-style:italic;">t</span> in <span class="texhtml">[0, 2<i>π</i>]</span> using the <a href="/wiki/Euler%27s_formula" title="Euler's formula">complex exponential</a>. Substituting, we find: <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}\oint _{L}{\frac {1}{z}}\,dz&=\int _{0}^{2\pi }{\frac {1}{e^{it}}}ie^{it}\,dt=i\int _{0}^{2\pi }e^{-it}e^{it}\,dt\\&=i\int _{0}^{2\pi }dt=i(2\pi -0)=2\pi i.\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> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>z</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>z</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>t</mi> </mrow> </msup> </mfrac> </mrow> <mi>i</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>t</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mi>i</mi> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>t</mi> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>t</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>i</mi> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> </mrow> </msubsup> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mi>i</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mo>−</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mi>i</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\oint _{L}{\frac {1}{z}}\,dz&=\int _{0}^{2\pi }{\frac {1}{e^{it}}}ie^{it}\,dt=i\int _{0}^{2\pi }e^{-it}e^{it}\,dt\\&=i\int _{0}^{2\pi }dt=i(2\pi -0)=2\pi i.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5366d542b5b6582deeb49aed6e19002471687356" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:43.263ex; height:12.843ex;" alt="{\displaystyle {\begin{aligned}\oint _{L}{\frac {1}{z}}\,dz&=\int _{0}^{2\pi }{\frac {1}{e^{it}}}ie^{it}\,dt=i\int _{0}^{2\pi }e^{-it}e^{it}\,dt\\&=i\int _{0}^{2\pi }dt=i(2\pi -0)=2\pi i.\end{aligned}}}"></span> </p><p>This is a typical result of <a href="/wiki/Cauchy%27s_integral_formula" title="Cauchy's integral formula">Cauchy's integral formula</a> and the <a href="/wiki/Residue_theorem" title="Residue theorem">residue theorem</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Relation_of_complex_line_integral_and_line_integral_of_vector_field">Relation of complex line integral and line integral of vector field</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Line_integral&action=edit&section=13" title="Edit section: Relation of complex line integral and line integral of vector field"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Viewing complex numbers as 2-dimensional <a href="/wiki/Euclidean_vector" title="Euclidean vector">vectors</a>, the line integral of a complex-valued 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 f(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8dd568d570b390c337c0a911f0a1c5c214e8240" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.176ex; height:2.843ex;" alt="{\displaystyle f(z)}"></span> has real and complex parts equal to the line integral and the flux integral of the vector field corresponding to the <a href="/wiki/Complex_conjugate" title="Complex conjugate">conjugate</a> 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 {\overline {f(z)}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {f(z)}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cf07f37ed4db05d851bca17e5ad0e58030437bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.938ex; height:3.676ex;" alt="{\displaystyle {\overline {f(z)}}.}"></span> Specifically, 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 \mathbf {r} (t)=(x(t),y(t))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>y</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} (t)=(x(t),y(t))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00cd8d50d7668a850c7fa784958009e08ab87eb3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.476ex; height:2.843ex;" alt="{\displaystyle \mathbf {r} (t)=(x(t),y(t))}"></span> parametrizes <i>L</i>, 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 f(z)=u(z)+iv(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>u</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>i</mi> <mi>v</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(z)=u(z)+iv(z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aadcde493bbff896c524f30d2b09a14cb5f0523f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.169ex; height:2.843ex;" alt="{\displaystyle f(z)=u(z)+iv(z)}"></span> corresponds to 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 \mathbf {F} (x,y)={\overline {f(x+iy)}}=(u(x+iy),-v(x+iy)),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mo>−</mo> <mi>v</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} (x,y)={\overline {f(x+iy)}}=(u(x+iy),-v(x+iy)),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a93167e5519e698ef0fb83dff48c917b89f501ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.169ex; height:3.676ex;" alt="{\displaystyle \mathbf {F} (x,y)={\overline {f(x+iy)}}=(u(x+iy),-v(x+iy)),}"></span> then: <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}\int _{L}f(z)\,dz&=\int _{L}(u+iv)(dx+i\,dy)\\&=\int _{L}(u,-v)\cdot (dx,dy)+i\int _{L}(u,-v)\cdot (dy,-dx)\\&=\int _{L}\mathbf {F} (\mathbf {r} )\cdot d\mathbf {r} +i\int _{L}\mathbf {F} (\mathbf {r} )\cdot d\mathbf {r} ^{\perp }.\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> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>z</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>u</mi> <mo>+</mo> <mi>i</mi> <mi>v</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>d</mi> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mo>−</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>d</mi> <mi>x</mi> <mo>,</mo> <mi>d</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>i</mi> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mo>−</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>d</mi> <mi>y</mi> <mo>,</mo> <mo>−</mo> <mi>d</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>+</mo> <mi>i</mi> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>d</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>⊥</mo> </mrow> </msup> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\int _{L}f(z)\,dz&=\int _{L}(u+iv)(dx+i\,dy)\\&=\int _{L}(u,-v)\cdot (dx,dy)+i\int _{L}(u,-v)\cdot (dy,-dx)\\&=\int _{L}\mathbf {F} (\mathbf {r} )\cdot d\mathbf {r} +i\int _{L}\mathbf {F} (\mathbf {r} )\cdot d\mathbf {r} ^{\perp }.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55be47e9c9ebc76a500b08037b0ace94214752f8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.171ex; width:57.97ex; height:17.509ex;" alt="{\displaystyle {\begin{aligned}\int _{L}f(z)\,dz&=\int _{L}(u+iv)(dx+i\,dy)\\&=\int _{L}(u,-v)\cdot (dx,dy)+i\int _{L}(u,-v)\cdot (dy,-dx)\\&=\int _{L}\mathbf {F} (\mathbf {r} )\cdot d\mathbf {r} +i\int _{L}\mathbf {F} (\mathbf {r} )\cdot d\mathbf {r} ^{\perp }.\end{aligned}}}"></span> </p><p>By <a href="/wiki/Cauchy%27s_integral_theorem" title="Cauchy's integral theorem">Cauchy's theorem</a>, the left-hand integral is zero when <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(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8dd568d570b390c337c0a911f0a1c5c214e8240" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.176ex; height:2.843ex;" alt="{\displaystyle f(z)}"></span> is analytic (satisfying the <a href="/wiki/Cauchy%E2%80%93Riemann_equations" title="Cauchy–Riemann equations">Cauchy–Riemann equations</a>) for any smooth closed curve L. Correspondingly, by <a href="/wiki/Green%27s_theorem" title="Green's theorem">Green's theorem</a>, the right-hand integrals are zero when <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 {F} ={\overline {f(z)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} ={\overline {f(z)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29eec9c6295c30e3c282ae12505df8774f3dcbc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.072ex; height:3.676ex;" alt="{\displaystyle \mathbf {F} ={\overline {f(z)}}}"></span> is <a href="/wiki/Conservative_vector_field" title="Conservative vector field">irrotational</a> (<a href="/wiki/Curl_(mathematics)" title="Curl (mathematics)">curl</a>-free) and <a href="/wiki/Incompressible_flow" title="Incompressible flow">incompressible</a> (<a href="/wiki/Divergence" title="Divergence">divergence</a>-free). In fact, the Cauchy-Riemann equations 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 f(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8dd568d570b390c337c0a911f0a1c5c214e8240" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.176ex; height:2.843ex;" alt="{\displaystyle f(z)}"></span> are identical to the vanishing of curl and divergence for <span class="texhtml"><b>F</b></span>. </p><p>By <a href="/wiki/Green%27s_theorem" title="Green's theorem">Green's theorem</a>, the area of a region enclosed by a smooth, closed, positively oriented 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 L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span> is given by the integral <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 {\frac {1}{2i}}\int _{L}{\overline {z}}\,dz.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>i</mi> </mrow> </mfrac> </mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>z</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {1}{2i}}\int _{L}{\overline {z}}\,dz.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/daf66ee4a41dfa444e30e277654b7f30501bd0c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:9.994ex; height:3.509ex;" alt="{\textstyle {\frac {1}{2i}}\int _{L}{\overline {z}}\,dz.}"></span> This fact is used, for example, in the proof of the <a href="/wiki/Area_theorem_(conformal_mapping)" title="Area theorem (conformal mapping)">area theorem</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Quantum_mechanics">Quantum mechanics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Line_integral&action=edit&section=14" title="Edit section: Quantum mechanics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Path_integral_formulation" title="Path integral formulation">path integral formulation</a> of <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> actually refers not to path integrals in this sense but to <a href="/wiki/Functional_integration" title="Functional integration">functional integrals</a>, that is, integrals over a space of paths, of a function <i>of</i> a possible path. However, path integrals in the sense of this article are important in quantum mechanics; for example, complex contour integration is often used in evaluating <a href="/wiki/Probability_amplitude" title="Probability amplitude">probability amplitudes</a> in quantum <a href="/wiki/Scattering" title="Scattering">scattering</a> theory. </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=Line_integral&action=edit&section=15" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 25em;"> <ul><li><a href="/wiki/Divergence_theorem" title="Divergence theorem">Divergence theorem</a></li> <li><a href="/wiki/Gradient_theorem" title="Gradient theorem">Gradient theorem</a></li> <li><a href="/wiki/Methods_of_contour_integration" class="mw-redirect" title="Methods of contour integration">Methods of contour integration</a></li> <li><a href="/wiki/Nachbin%27s_theorem" title="Nachbin's theorem">Nachbin's theorem</a></li> <li><a href="/wiki/Line_element" title="Line element">Line element</a></li> <li><a href="/wiki/Surface_integral" title="Surface integral">Surface integral</a></li> <li><a href="/wiki/Volume_element" title="Volume element">Volume element</a></li> <li><a href="/wiki/Volume_integral" title="Volume integral">Volume integral</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=Line_integral&action=edit&section=16" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-Tang2006-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-Tang2006_1-0">^</a></b></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="CITEREFKwong-Tin_Tang2006" class="citation book cs1">Kwong-Tin Tang (30 November 2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=kR8TqQRxbeoC&q=%22line+integral%22"><i>Mathematical Methods for Engineers and Scientists 2: Vector Analysis, Ordinary Differential Equations and Laplace Transforms</i></a>. Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-30268-1" title="Special:BookSources/978-3-540-30268-1"><bdi>978-3-540-30268-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Methods+for+Engineers+and+Scientists+2%3A+Vector+Analysis%2C+Ordinary+Differential+Equations+and+Laplace+Transforms&rft.pub=Springer+Science+%26+Business+Media&rft.date=2006-11-30&rft.isbn=978-3-540-30268-1&rft.au=Kwong-Tin+Tang&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DkR8TqQRxbeoC%26q%3D%2522line%2Bintegral%2522&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALine+integral" class="Z3988"></span></span> </li> <li id="cite_note-:1-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-:1_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:1_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="CITEREFNykamp" class="citation web cs1">Nykamp, Duane. <a rel="nofollow" class="external text" href="https://mathinsight.org/line_integral_independent_parametrization">"Line integrals are independent of parametrization"</a>. <i>Math Insight</i><span class="reference-accessdate">. Retrieved <span class="nowrap">September 18,</span> 2020</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Math+Insight&rft.atitle=Line+integrals+are+independent+of+parametrization&rft.aulast=Nykamp&rft.aufirst=Duane&rft_id=https%3A%2F%2Fmathinsight.org%2Fline_integral_independent_parametrization&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALine+integral" 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 class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.whitman.edu/mathematics/calculus_online/section16.02.html">"16.2 Line Integrals"</a>. <i>www.whitman.edu</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-09-18</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.whitman.edu&rft.atitle=16.2+Line+Integrals&rft_id=https%3A%2F%2Fwww.whitman.edu%2Fmathematics%2Fcalculus_online%2Fsection16.02.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALine+integral" 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="CITEREFAhlfors1966" class="citation book cs1">Ahlfors, Lars (1966). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=RfYK28TcZEwC&pg=PA103"><i>Complex Analysis</i></a> (2nd ed.). New York: McGraw-Hill. p. 103.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Complex+Analysis&rft.place=New+York&rft.pages=103&rft.edition=2nd&rft.pub=McGraw-Hill&rft.date=1966&rft.aulast=Ahlfors&rft.aufirst=Lars&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DRfYK28TcZEwC%26pg%3DPA103&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALine+integral" class="Z3988"></span></span> </li> </ol></div></div> <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=Line_integral&action=edit&section=17" title="Edit section: External links"><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 class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Integral_over_trajectories">"Integral over trajectories"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Integral+over+trajectories&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DIntegral_over_trajectories&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALine+integral" class="Z3988"></span></li> <li><a href="/wiki/Khan_Academy" title="Khan Academy">Khan Academy</a> modules: <ul><li><a rel="nofollow" class="external text" href="https://www.khanacademy.org/video/introduction-to-the-line-integral?playlist=Calculus">"Introduction to the Line Integral"</a></li> <li><a rel="nofollow" class="external text" href="https://www.khanacademy.org/video/line-integral-example-1?playlist=Calculus">"Line Integral Example 1"</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20111219141618/http://www.khanacademy.org/video/line-integral-example-2--part-1?playlist=Calculus">"Line Integral Example 2 (part 1)"</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20111219142026/http://www.khanacademy.org/video/line-integral-example-2--part-2?playlist=Calculus">"Line Integral Example 2 (part 2)"</a></li></ul></li> <li><a rel="nofollow" class="external text" href="https://planetmath.org/LineIntegral">Path integral</a> at <a href="/wiki/PlanetMath" title="PlanetMath">PlanetMath</a>.</li> <li><a rel="nofollow" class="external text" href="http://mathinsight.org/line_integral_vector_field_introduction">Line integral of a vector field – Interactive</a></li></ul> <div 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<li><a href="/wiki/Finite_difference" title="Finite difference">Finite difference</a></li> <li><a href="/wiki/Free_variables_and_bound_variables" title="Free variables and bound variables">Free variables and bound variables</a></li> <li><a href="/wiki/Graph_of_a_function" title="Graph of a function">Graph of a function</a></li> <li><a href="/wiki/Linear_function" title="Linear function">Linear function</a></li> <li><a href="/wiki/Radian" title="Radian">Radian</a></li> <li><a href="/wiki/Rolle%27s_theorem" title="Rolle's theorem">Rolle's theorem</a></li> <li><a href="/wiki/Secant_line" title="Secant line">Secant</a></li> <li><a href="/wiki/Slope" title="Slope">Slope</a></li> <li><a href="/wiki/Tangent" title="Tangent">Tangent</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Limit_(mathematics)" title="Limit (mathematics)">Limits</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Indeterminate_form" title="Indeterminate form">Indeterminate form</a></li> <li><a href="/wiki/Limit_of_a_function" title="Limit of a function">Limit of a function</a> <ul><li><a href="/wiki/One-sided_limit" title="One-sided limit">One-sided limit</a></li></ul></li> <li><a href="/wiki/Limit_of_a_sequence" title="Limit of a sequence">Limit of a sequence</a></li> <li><a href="/wiki/Order_of_approximation" title="Order of approximation">Order of approximation</a></li> <li><a href="/wiki/(%CE%B5,_%CE%B4)-definition_of_limit" class="mw-redirect" title="(ε, δ)-definition of limit">(ε, δ)-definition of limit</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Differential_calculus" title="Differential calculus">Differential calculus</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Derivative" title="Derivative">Derivative</a></li> <li><a href="/wiki/Second_derivative" title="Second derivative">Second derivative</a></li> <li><a href="/wiki/Partial_derivative" title="Partial derivative">Partial derivative</a></li> <li><a href="/wiki/Differential_(mathematics)" title="Differential (mathematics)">Differential</a></li> <li><a href="/wiki/Differential_operator" title="Differential operator">Differential operator</a></li> <li><a href="/wiki/Mean_value_theorem" title="Mean value theorem">Mean value theorem</a></li> <li><a href="/wiki/Notation_for_differentiation" title="Notation for differentiation">Notation</a> <ul><li><a href="/wiki/Leibniz%27s_notation" title="Leibniz's notation">Leibniz's notation</a></li> <li><a href="/wiki/Newton%27s_notation_for_differentiation" class="mw-redirect" title="Newton's notation for differentiation">Newton's notation</a></li></ul></li> <li><a href="/wiki/Differentiation_rules" title="Differentiation rules">Rules of differentiation</a> <ul><li><a href="/wiki/Linearity_of_differentiation" title="Linearity of differentiation">linearity</a></li> <li><a href="/wiki/Power_rule" title="Power rule">Power</a></li> <li><a href="/wiki/Sum_rule_in_differentiation" class="mw-redirect" title="Sum rule in differentiation">Sum</a></li> <li><a href="/wiki/Chain_rule" title="Chain rule">Chain</a></li> <li><a href="/wiki/L%27H%C3%B4pital%27s_rule" title="L'Hôpital's rule">L'Hôpital's</a></li> <li><a href="/wiki/Product_rule" title="Product rule">Product</a> <ul><li><a href="/wiki/General_Leibniz_rule" title="General Leibniz rule">General Leibniz's rule</a></li></ul></li> <li><a href="/wiki/Quotient_rule" title="Quotient rule">Quotient</a></li></ul></li> <li>Other techniques <ul><li><a href="/wiki/Implicit_differentiation" class="mw-redirect" title="Implicit differentiation">Implicit differentiation</a></li> <li><a href="/wiki/Inverse_functions_and_differentiation" class="mw-redirect" title="Inverse functions and differentiation">Inverse functions and differentiation</a></li> <li><a href="/wiki/Logarithmic_derivative" title="Logarithmic derivative">Logarithmic derivative</a></li> <li><a href="/wiki/Related_rates" title="Related rates">Related rates</a></li></ul></li> <li><a href="/wiki/Stationary_point" title="Stationary point">Stationary points</a> <ul><li><a href="/wiki/First_derivative_test" class="mw-redirect" title="First derivative test">First derivative test</a></li> <li><a href="/wiki/Second_derivative_test" class="mw-redirect" title="Second derivative test">Second derivative test</a></li> <li><a href="/wiki/Extreme_value_theorem" title="Extreme value theorem">Extreme value theorem</a></li> <li><a href="/wiki/Maximum_and_minimum" title="Maximum and minimum">Maximum and minimum</a></li></ul></li> <li>Further applications <ul><li><a href="/wiki/Newton%27s_method" title="Newton's method">Newton's method</a></li> <li><a href="/wiki/Taylor%27s_theorem" title="Taylor's theorem">Taylor's theorem</a></li></ul></li> <li><a href="/wiki/Differential_equation" title="Differential equation">Differential equation</a> <ul><li><a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">Ordinary differential equation</a></li> <li><a href="/wiki/Partial_differential_equation" title="Partial differential equation">Partial differential equation</a></li> <li><a href="/wiki/Stochastic_differential_equation" title="Stochastic differential equation">Stochastic differential equation</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Integral_calculus" class="mw-redirect" title="Integral calculus">Integral calculus</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Antiderivative" title="Antiderivative">Antiderivative</a></li> <li><a href="/wiki/Arc_length" title="Arc length">Arc length</a></li> <li><a href="/wiki/Riemann_integral" title="Riemann integral">Riemann integral</a></li> <li><a href="/wiki/Integral#Properties" title="Integral">Basic properties</a></li> <li><a href="/wiki/Constant_of_integration" title="Constant of integration">Constant of integration</a></li> <li><a href="/wiki/Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus">Fundamental theorem of calculus</a> <ul><li><a href="/wiki/Leibniz_integral_rule" title="Leibniz integral rule">Differentiating under the integral sign</a></li></ul></li> <li><a href="/wiki/Integration_by_parts" title="Integration by parts">Integration by parts</a></li> <li><a href="/wiki/Integration_by_substitution" title="Integration by substitution">Integration by substitution</a> <ul><li><a href="/wiki/Trigonometric_substitution" title="Trigonometric substitution">trigonometric</a></li> <li><a href="/wiki/Euler_substitution" title="Euler substitution">Euler</a></li> <li><a href="/wiki/Tangent_half-angle_substitution" title="Tangent half-angle substitution">Tangent half-angle substitution</a></li></ul></li> <li><a href="/wiki/Partial_fractions_in_integration" class="mw-redirect" title="Partial fractions in integration">Partial fractions in integration</a> <ul><li><a href="/wiki/Quadratic_integral" title="Quadratic integral">Quadratic integral</a></li></ul></li> <li><a href="/wiki/Trapezoidal_rule" title="Trapezoidal rule">Trapezoidal rule</a></li> <li>Volumes <ul><li><a href="/wiki/Disc_integration" title="Disc integration">Washer method</a></li> <li><a href="/wiki/Shell_integration" title="Shell integration">Shell method</a></li></ul></li> <li><a href="/wiki/Integral_equation" title="Integral equation">Integral equation</a></li> <li><a href="/wiki/Integro-differential_equation" title="Integro-differential equation">Integro-differential equation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Vector_calculus" title="Vector calculus">Vector calculus</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>Derivatives <ul><li><a href="/wiki/Curl_(mathematics)" title="Curl (mathematics)">Curl</a></li> <li><a href="/wiki/Directional_derivative" title="Directional derivative">Directional derivative</a></li> <li><a href="/wiki/Divergence" title="Divergence">Divergence</a></li> <li><a href="/wiki/Gradient" title="Gradient">Gradient</a></li> <li><a href="/wiki/Laplace_operator" title="Laplace operator">Laplacian</a></li></ul></li> <li>Basic theorems <ul><li><a href="/wiki/Fundamental_Theorem_of_Line_Integrals" class="mw-redirect" title="Fundamental Theorem of Line Integrals">Line integrals</a></li> <li><a href="/wiki/Green%27s_theorem" title="Green's theorem">Green's</a></li> <li><a href="/wiki/Stokes%27_theorem" title="Stokes' theorem">Stokes'</a></li> <li><a href="/wiki/Divergence_theorem" title="Divergence theorem">Gauss'</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Multivariable_calculus" title="Multivariable calculus">Multivariable calculus</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Divergence_theorem" title="Divergence theorem">Divergence theorem</a></li> <li><a href="/wiki/Geometric_calculus" title="Geometric calculus">Geometric</a></li> <li><a href="/wiki/Hessian_matrix" title="Hessian matrix">Hessian matrix</a></li> <li><a href="/wiki/Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian matrix and determinant</a></li> <li><a href="/wiki/Lagrange_multiplier" title="Lagrange multiplier">Lagrange multiplier</a></li> <li><a class="mw-selflink selflink">Line integral</a></li> <li><a href="/wiki/Matrix_calculus" title="Matrix calculus">Matrix</a></li> <li><a href="/wiki/Multiple_integral" title="Multiple integral">Multiple integral</a></li> <li><a href="/wiki/Partial_derivative" title="Partial derivative">Partial derivative</a></li> <li><a href="/wiki/Surface_integral" title="Surface integral">Surface integral</a></li> <li><a href="/wiki/Volume_integral" title="Volume integral">Volume integral</a></li> <li>Advanced topics <ul><li><a href="/wiki/Differential_form" title="Differential form">Differential forms</a></li> <li><a href="/wiki/Exterior_derivative" title="Exterior derivative">Exterior derivative</a></li> <li><a href="/wiki/Generalized_Stokes%27_theorem" class="mw-redirect" title="Generalized Stokes' theorem">Generalized Stokes' theorem</a></li> <li><a href="/wiki/Tensor_calculus" class="mw-redirect" title="Tensor calculus">Tensor calculus</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Sequences and series</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Arithmetico-geometric_sequence" title="Arithmetico-geometric sequence">Arithmetico-geometric sequence</a></li> <li>Types of series <ul><li><a href="/wiki/Alternating_series" title="Alternating series">Alternating</a></li> <li><a href="/wiki/Binomial_series" title="Binomial series">Binomial</a></li> <li><a href="/wiki/Fourier_series" title="Fourier series">Fourier</a></li> <li><a href="/wiki/Geometric_series" title="Geometric series">Geometric</a></li> <li><a href="/wiki/Harmonic_series_(mathematics)" title="Harmonic series (mathematics)">Harmonic</a></li> <li><a href="/wiki/Infinite_series" class="mw-redirect" title="Infinite series">Infinite</a></li> <li><a href="/wiki/Power_series" title="Power series">Power</a> <ul><li><a href="/wiki/Maclaurin_series" class="mw-redirect" title="Maclaurin series">Maclaurin</a></li> <li><a href="/wiki/Taylor_series" title="Taylor series">Taylor</a></li></ul></li> <li><a href="/wiki/Telescoping_series" title="Telescoping series">Telescoping</a></li></ul></li> <li>Tests of convergence <ul><li><a href="/wiki/Abel%27s_test" title="Abel's test">Abel's</a></li> <li><a href="/wiki/Alternating_series_test" title="Alternating series test">Alternating series</a></li> <li><a href="/wiki/Cauchy_condensation_test" title="Cauchy condensation test">Cauchy condensation</a></li> <li><a href="/wiki/Direct_comparison_test" title="Direct comparison test">Direct comparison</a></li> <li><a href="/wiki/Dirichlet%27s_test" title="Dirichlet's test">Dirichlet's</a></li> <li><a href="/wiki/Integral_test_for_convergence" title="Integral test for convergence">Integral</a></li> <li><a href="/wiki/Limit_comparison_test" title="Limit comparison test">Limit comparison</a></li> <li><a href="/wiki/Ratio_test" title="Ratio test">Ratio</a></li> <li><a href="/wiki/Root_test" title="Root test">Root</a></li> <li><a href="/wiki/Term_test" class="mw-redirect" title="Term test">Term</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Special functions<br />and numbers</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bernoulli_number" title="Bernoulli number">Bernoulli numbers</a></li> <li><a href="/wiki/E_(mathematical_constant)" title="E (mathematical constant)">e (mathematical constant)</a></li> <li><a href="/wiki/Exponential_function" title="Exponential function">Exponential function</a></li> <li><a href="/wiki/Natural_logarithm" title="Natural logarithm">Natural logarithm</a></li> <li><a href="/wiki/Stirling%27s_approximation" title="Stirling's approximation">Stirling's approximation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/History_of_calculus" title="History of calculus">History of calculus</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Adequality" title="Adequality">Adequality</a></li> <li><a href="/wiki/Brook_Taylor" title="Brook Taylor">Brook Taylor</a></li> <li><a href="/wiki/Colin_Maclaurin" title="Colin Maclaurin">Colin Maclaurin</a></li> <li><a href="/wiki/Generality_of_algebra" title="Generality of algebra">Generality of algebra</a></li> <li><a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a></li> <li><a href="/wiki/Infinitesimal" title="Infinitesimal">Infinitesimal</a></li> <li><a href="/wiki/Infinitesimal_calculus" class="mw-redirect" title="Infinitesimal calculus">Infinitesimal calculus</a></li> <li><a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a></li> <li><a href="/wiki/Fluxion" title="Fluxion">Fluxion</a></li> <li><a href="/wiki/Law_of_Continuity" class="mw-redirect" title="Law of Continuity">Law of Continuity</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a></li> <li><i><a href="/wiki/Method_of_Fluxions" title="Method of Fluxions">Method of Fluxions</a></i></li> <li><i><a href="/wiki/The_Method_of_Mechanical_Theorems" title="The Method of Mechanical Theorems">The Method of Mechanical Theorems</a></i></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Lists</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Integrals" scope="row" class="navbox-group" style="width:1%;text-align:left"><a href="/wiki/Lists_of_integrals" title="Lists of integrals">Integrals</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_integrals_of_rational_functions" title="List of integrals of rational functions">rational functions</a></li> <li><a href="/wiki/List_of_integrals_of_irrational_functions" title="List of integrals of irrational functions">irrational functions</a></li> <li><a href="/wiki/List_of_integrals_of_exponential_functions" title="List of integrals of exponential functions">exponential functions</a></li> <li><a href="/wiki/List_of_integrals_of_logarithmic_functions" title="List of integrals of logarithmic functions">logarithmic functions</a></li> <li><a href="/wiki/List_of_integrals_of_hyperbolic_functions" title="List of integrals of hyperbolic functions">hyperbolic functions</a> <ul><li><a href="/wiki/List_of_integrals_of_inverse_hyperbolic_functions" title="List of integrals of inverse hyperbolic functions">inverse</a></li></ul></li> <li><a href="/wiki/List_of_integrals_of_trigonometric_functions" title="List of integrals of trigonometric functions">trigonometric functions</a> <ul><li><a href="/wiki/List_of_integrals_of_inverse_trigonometric_functions" title="List of integrals of inverse trigonometric functions">inverse</a></li> <li><a href="/wiki/Integral_of_the_secant_function" title="Integral of the secant function">Secant</a></li> <li><a href="/wiki/Integral_of_secant_cubed" title="Integral of secant cubed">Secant cubed</a></li></ul></li></ul> </div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_limits" title="List of limits">List of limits</a></li> <li><a href="/wiki/Differentiation_rules" title="Differentiation rules">List of derivatives</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Miscellaneous topics</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>Complex calculus <ul><li><a href="/wiki/Contour_integral" class="mw-redirect" title="Contour integral">Contour integral</a></li></ul></li> <li>Differential geometry <ul><li><a href="/wiki/Manifold" title="Manifold">Manifold</a></li> <li><a href="/wiki/Curvature" title="Curvature">Curvature</a></li> <li><a href="/wiki/Differential_geometry_of_curves" class="mw-redirect" title="Differential geometry of curves">of curves</a></li> <li><a href="/wiki/Differential_geometry_of_surfaces" title="Differential geometry of surfaces">of surfaces</a></li> <li><a href="/wiki/Tensor" title="Tensor">Tensor</a></li></ul></li> <li><a href="/wiki/Euler%E2%80%93Maclaurin_formula" title="Euler–Maclaurin formula">Euler–Maclaurin formula</a></li> <li><a href="/wiki/Gabriel%27s_horn" title="Gabriel's horn">Gabriel's horn</a></li> <li><a href="/wiki/Integration_Bee" title="Integration Bee">Integration Bee</a></li> <li><a href="/wiki/Proof_that_22/7_exceeds_%CF%80" title="Proof that 22/7 exceeds π">Proof that 22/7 exceeds π</a></li> <li><a href="/wiki/Regiomontanus%27_angle_maximization_problem" title="Regiomontanus' angle maximization problem">Regiomontanus' angle maximization problem</a></li> <li><a href="/wiki/Steinmetz_solid" title="Steinmetz solid">Steinmetz solid</a></li></ul> </div></td></tr></tbody></table></div> <div 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 navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Help:Authority_control" title="Help:Authority control">Authority control databases</a>: 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Line integral - Wikipedia