{"id":112,"date":"2021-07-30T17:14:56","date_gmt":"2021-07-30T17:14:56","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=112"},"modified":"2022-11-01T05:34:59","modified_gmt":"2022-11-01T05:34:59","slug":"summary-of-stokes-theorem","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/summary-of-stokes-theorem\/","title":{"raw":"Summary of Stokes' Theorem","rendered":"Summary of Stokes&#8217; Theorem"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul>\r\n \t<li>Stokes\u2019 theorem relates a flux integral over a surface to a line integral around the boundary of the surface. Stokes\u2019 theorem is a higher dimensional version of Green\u2019s theorem, and therefore is another version of the Fundamental Theorem of Calculus in higher dimensions.<\/li>\r\n \t<li>Stokes\u2019 theorem can be used to transform a difficult surface integral into an easier line integral, or a difficult line integral into an easier surface integral.<\/li>\r\n \t<li>Through Stokes\u2019 theorem, line integrals can be evaluated using the simplest surface with boundary\u00a0[latex]C[\/latex].<\/li>\r\n \t<li>Faraday\u2019s law relates the curl of an electric field to the rate of change of the corresponding magnetic field. Stokes\u2019 theorem can be used to derive Faraday\u2019s law.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Key Equations<\/h2>\r\n<ul>\r\n \t<li><strong>Stokes' theorem\r\n<\/strong>[latex]\\displaystyle\\int_{C} {\\bf{F}}\\cdot{d{\\bf{r}}}=\\displaystyle\\iint_{S}\\text{curl } {\\bf{F}}\\cdot{d{\\bf{S}}}[\/latex]<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl class=\"definition\">\r\n \t<dt>Stokes' theorem<\/dt>\r\n \t<dd><span style=\"font-size: 1em;\">relates the flux integral over a surface <\/span>[latex]S[\/latex]<span style=\"font-size: 1em;\"> to a line integral around the boundary <\/span>[latex]C[\/latex]<span style=\"font-size: 1em;\"> of the surface <\/span>[latex]S[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>surface independent<\/dt>\r\n \t<dd><span style=\"font-size: 1em;\">flux integrals of curl vector fields are surface independent if their evaluation does not depend on the surface but only on the boundary of the surface<\/span><\/dd>\r\n<\/dl>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul>\n<li>Stokes\u2019 theorem relates a flux integral over a surface to a line integral around the boundary of the surface. Stokes\u2019 theorem is a higher dimensional version of Green\u2019s theorem, and therefore is another version of the Fundamental Theorem of Calculus in higher dimensions.<\/li>\n<li>Stokes\u2019 theorem can be used to transform a difficult surface integral into an easier line integral, or a difficult line integral into an easier surface integral.<\/li>\n<li>Through Stokes\u2019 theorem, line integrals can be evaluated using the simplest surface with boundary\u00a0[latex]C[\/latex].<\/li>\n<li>Faraday\u2019s law relates the curl of an electric field to the rate of change of the corresponding magnetic field. Stokes\u2019 theorem can be used to derive Faraday\u2019s law.<\/li>\n<\/ul>\n<\/div>\n<h2>Key Equations<\/h2>\n<ul>\n<li><strong>Stokes&#8217; theorem<br \/>\n<\/strong>[latex]\\displaystyle\\int_{C} {\\bf{F}}\\cdot{d{\\bf{r}}}=\\displaystyle\\iint_{S}\\text{curl } {\\bf{F}}\\cdot{d{\\bf{S}}}[\/latex]<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl class=\"definition\">\n<dt>Stokes&#8217; theorem<\/dt>\n<dd><span style=\"font-size: 1em;\">relates the flux integral over a surface <\/span>[latex]S[\/latex]<span style=\"font-size: 1em;\"> to a line integral around the boundary <\/span>[latex]C[\/latex]<span style=\"font-size: 1em;\"> of the surface <\/span>[latex]S[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>surface independent<\/dt>\n<dd><span style=\"font-size: 1em;\">flux integrals of curl vector fields are surface independent if their evaluation does not depend on the surface but only on the boundary of the surface<\/span><\/dd>\n<\/dl>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-112\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Calculus Volume 3. <strong>Authored by<\/strong>: Gilbert Strang, Edwin (Jed) Herman. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\">https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":33,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 3\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-112","chapter","type-chapter","status-publish","hentry"],"part":24,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/112","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":11,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/112\/revisions"}],"predecessor-version":[{"id":3794,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/112\/revisions\/3794"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/24"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/112\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=112"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=112"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=112"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=112"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}