{"id":106,"date":"2021-07-30T17:14:18","date_gmt":"2021-07-30T17:14:18","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=106"},"modified":"2022-11-01T05:21:04","modified_gmt":"2022-11-01T05:21:04","slug":"summary-of-greens-theorem","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/summary-of-greens-theorem\/","title":{"raw":"Summary of Green's Theorem","rendered":"Summary of Green&#8217;s Theorem"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul>\r\n \t<li>Green\u2019s theorem relates the integral over a connected region to an integral over the boundary of the region. Green\u2019s theorem is a version of the Fundamental Theorem of Calculus in one higher dimension.<\/li>\r\n \t<li>Green\u2019s Theorem comes in two forms: a circulation form and a flux form. In the circulation form, the integrand is [latex]{\\bf{F}}\\cdot{\\bf{T}}[\/latex]. In the flux form, the integrand is\u00a0[latex]{\\bf{F}}\\cdot{\\bf{N}}[\/latex].<\/li>\r\n \t<li>Green\u2019s theorem can be used to transform a difficult line integral into an easier double integral, or to transform a difficult double integral into an easier line integral.<\/li>\r\n \t<li>A vector field is source free if it has a stream function. The flux of a source-free vector field across a closed curve is zero, just as the circulation of a conservative vector field across a closed curve is zero.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1169736654976\">\r\n \t<li><strong data-effect=\"bold\">Green\u2019s theorem, circulation form<\/strong>\r\n[latex]\\displaystyle\\oint_{C} Pdx+Qdy=\\displaystyle\\iint_{D} Q_{x}-P_{y}dA[\/latex], where [latex]C[\/latex] is the boundary of\u00a0[latex]D[\/latex]<\/li>\r\n \t<li><strong data-effect=\"bold\">Green\u2019s theorem, flux form<\/strong>\r\n[latex]\\displaystyle\\oint_{C} {\\bf{F}}\\cdot {d{\\bf{r}}} = \\displaystyle\\iint_{D} Q_{x}-P_{y}dA[\/latex], where [latex]C[\/latex] is the boundary of\u00a0[latex]D[\/latex]<\/li>\r\n \t<li><strong data-effect=\"bold\">Green\u2019s theorem, extended version<\/strong>\r\n[latex]\\displaystyle\\oint_{\\partial{D}} {\\bf{F}}\\cdot {d{\\bf{r}}} = \\displaystyle\\iint_{D} Q_{x}-P_{y}dA[\/latex]<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl class=\"definition\">\r\n \t<dt>Green's theorem<\/dt>\r\n \t<dd><span style=\"font-size: 1em;\">relates the integral over a connected region to an integral over the boundary of the region<\/span><\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>stream function<\/dt>\r\n \t<dd>if [latex]{\\bf{F}} = {\\langle}P, Q{\\rangle}[\/latex] is a source-free vector field, then stream function <em data-effect=\"italics\">g<\/em> is a function such that [latex]P = g_{y}[\/latex], and\u00a0[latex]Q = -{g_{x}}[\/latex]<\/dd>\r\n<\/dl>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul>\n<li>Green\u2019s theorem relates the integral over a connected region to an integral over the boundary of the region. Green\u2019s theorem is a version of the Fundamental Theorem of Calculus in one higher dimension.<\/li>\n<li>Green\u2019s Theorem comes in two forms: a circulation form and a flux form. In the circulation form, the integrand is [latex]{\\bf{F}}\\cdot{\\bf{T}}[\/latex]. In the flux form, the integrand is\u00a0[latex]{\\bf{F}}\\cdot{\\bf{N}}[\/latex].<\/li>\n<li>Green\u2019s theorem can be used to transform a difficult line integral into an easier double integral, or to transform a difficult double integral into an easier line integral.<\/li>\n<li>A vector field is source free if it has a stream function. The flux of a source-free vector field across a closed curve is zero, just as the circulation of a conservative vector field across a closed curve is zero.<\/li>\n<\/ul>\n<\/div>\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1169736654976\">\n<li><strong data-effect=\"bold\">Green\u2019s theorem, circulation form<\/strong><br \/>\n[latex]\\displaystyle\\oint_{C} Pdx+Qdy=\\displaystyle\\iint_{D} Q_{x}-P_{y}dA[\/latex], where [latex]C[\/latex] is the boundary of\u00a0[latex]D[\/latex]<\/li>\n<li><strong data-effect=\"bold\">Green\u2019s theorem, flux form<\/strong><br \/>\n[latex]\\displaystyle\\oint_{C} {\\bf{F}}\\cdot {d{\\bf{r}}} = \\displaystyle\\iint_{D} Q_{x}-P_{y}dA[\/latex], where [latex]C[\/latex] is the boundary of\u00a0[latex]D[\/latex]<\/li>\n<li><strong data-effect=\"bold\">Green\u2019s theorem, extended version<\/strong><br \/>\n[latex]\\displaystyle\\oint_{\\partial{D}} {\\bf{F}}\\cdot {d{\\bf{r}}} = \\displaystyle\\iint_{D} Q_{x}-P_{y}dA[\/latex]<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl class=\"definition\">\n<dt>Green&#8217;s theorem<\/dt>\n<dd><span style=\"font-size: 1em;\">relates the integral over a connected region to an integral over the boundary of the region<\/span><\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>stream function<\/dt>\n<dd>if [latex]{\\bf{F}} = {\\langle}P, Q{\\rangle}[\/latex] is a source-free vector field, then stream function <em data-effect=\"italics\">g<\/em> is a function such that [latex]P = g_{y}[\/latex], and\u00a0[latex]Q = -{g_{x}}[\/latex]<\/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-106\">\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":19,"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-106","chapter","type-chapter","status-publish","hentry"],"part":24,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/106","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":13,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/106\/revisions"}],"predecessor-version":[{"id":3790,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/106\/revisions\/3790"}],"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\/106\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=106"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=106"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=106"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=106"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}