{"id":794,"date":"2025-10-09T15:39:39","date_gmt":"2025-10-09T15:39:39","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=794"},"modified":"2025-11-10T17:00:09","modified_gmt":"2025-11-10T17:00:09","slug":"the-divergence-theorem-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/the-divergence-theorem-learn-it-1\/","title":{"raw":"The Divergence Theorem: Learn It 1","rendered":"The Divergence Theorem: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Understand the divergence theorem<\/li>\r\n \t<li>Calculate flux using the divergence theorem<\/li>\r\n \t<li>Apply the divergence theorem to physical problems<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 data-type=\"title\">Overview of Theorems<\/h2>\r\n<div data-type=\"note\"><section id=\"fs-id1167793470788\" data-depth=\"1\">\r\n<p id=\"fs-id1167793965594\">Before examining the divergence theorem, it is helpful to begin with an overview of the versions of the Fundamental Theorem of Calculus we have discussed. Each theorem relates an integral of a \"derivative\" over a domain to an integral over the boundary of that domain.<\/p>\r\n\r\n<\/section><\/div>\r\n<div data-type=\"note\">\r\n<ol>\r\n \t<li><strong data-effect=\"bold\">The Fundamental Theorem of Calculus<\/strong>:<span data-type=\"newline\">\r\n<\/span>\r\n<div id=\"fs-id1167794118999\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\r\n<p style=\"text-align: center;\">[latex]\\large{\\displaystyle\\int_a^bf^\\prime(x)dx=f(b)-f(a)}[\/latex].<\/p>\r\n\r\n<\/div>\r\n<span data-type=\"newline\">\r\n<\/span>This theorem relates the integral of derivative [latex]f'[\/latex] over line segment [latex][a,b][\/latex] along the [latex]x[\/latex]-axis to a difference of [latex]f[\/latex] evaluated on the boundary.<\/li>\r\n \t<li><strong data-effect=\"bold\">The Fundamental Theorem for Line Integrals<\/strong>:<span data-type=\"newline\">\r\n<\/span>\r\n<div id=\"fs-id1167793604124\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\r\n<p style=\"text-align: center;\">[latex]\\large{\\displaystyle\\int_C\\nabla f\\cdot d{\\bf{r}}=f(P_1)-f(P_0)}[\/latex].<\/p>\r\n\r\n<\/div>\r\n<span data-type=\"newline\">\r\n<\/span>where [latex]P_0[\/latex] is the initial point of [latex]C[\/latex] and [latex]P_1[\/latex] is the terminal point of [latex]C[\/latex]. The\u00a0<span id=\"d77de5bc-07e0-40d3-a789-c2f01549dd9b_term287\" class=\"no-emphasis\" data-type=\"term\">Fundamental Theorem for Line Integrals<\/span>\u00a0allows path [latex]C[\/latex] to be a path in a plane or in space, not just a line segment on the [latex]x[\/latex]-axis. If we think of the gradient as a derivative, then this theorem relates an integral of derivative [latex]\\nabla f[\/latex] over path [latex]C[\/latex] to a difference of [latex]f[\/latex] evaluated on the boundary of [latex]C[\/latex].<\/li>\r\n \t<li><strong data-effect=\"bold\">Green\u2019s theorem, circulation form<\/strong>:<span data-type=\"newline\">\r\n<\/span>\r\n<div id=\"fs-id1167793878210\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\r\n<p style=\"text-align: center;\">[latex]\\large{\\displaystyle\\iint_D(Q_x-P_y)dA=\\displaystyle\\int_C{\\bf{F}}\\cdot d{\\bf{r}}}[\/latex].<\/p>\r\n\r\n<\/div>\r\n<span data-type=\"newline\">\r\n<\/span>Since [latex]Q_x-P_y=\\text{curl }{\\bf{F}}\\cdot{\\bf{k}}[\/latex] and curl is a derivative of sorts,\u00a0<span id=\"d77de5bc-07e0-40d3-a789-c2f01549dd9b_term288\" class=\"no-emphasis\" data-type=\"term\">Green\u2019s theorem<\/span>\u00a0relates the integral of derivative curl[latex]{\\bf{F}}[\/latex]\u00a0over planar region [latex]D[\/latex] to an integral of\u00a0[latex]{\\bf{F}}[\/latex]\u00a0over the boundary of [latex]D[\/latex].<\/li>\r\n \t<li><strong data-effect=\"bold\">Green\u2019s theorem, flux form<\/strong>:<span data-type=\"newline\">\r\n<\/span>\r\n<div id=\"fs-id1167793401271\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\r\n<p style=\"text-align: center;\">[latex]\\large{\\displaystyle\\iint_D(P_x+Q_y)dA=\\displaystyle\\int_C{\\bf{F}}\\cdot{\\bf{N}}ds}[\/latex].<\/p>\r\n\r\n<\/div>\r\n<span data-type=\"newline\">\r\n<\/span>Since [latex]P_x+Q_y=\\text{div }{\\bf{F}}[\/latex] and divergence is a derivative of sorts, the flux form of Green\u2019s theorem relates the integral of derivative div\u00a0[latex]{\\bf{F}}[\/latex]\u00a0over planar region [latex]D[\/latex] to an integral of\u00a0[latex]{\\bf{F}}[\/latex]\u00a0over the boundary of [latex]D[\/latex].<\/li>\r\n \t<li><strong data-effect=\"bold\">Stokes\u2019 theorem<\/strong>:<span data-type=\"newline\">\r\n<\/span>\r\n<div id=\"fs-id1167793959488\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\large{\\displaystyle\\iint_S\\text{curl }{\\bf{F}}\\cdot ds=\\displaystyle\\int_C{\\bf{F}}\\cdot d{\\bf{r}}}[\/latex]<\/div>\r\n<span data-type=\"newline\">\r\n<\/span>If we think of the curl as a derivative of sorts, then\u00a0<span id=\"d77de5bc-07e0-40d3-a789-c2f01549dd9b_term289\" class=\"no-emphasis\" data-type=\"term\">Stokes\u2019 theorem<\/span>\u00a0relates the integral of derivative curl[latex]{\\bf{F}}[\/latex]\u00a0over surface [latex]S[\/latex] (not necessarily planar) to an integral of\u00a0[latex]{\\bf{F}}[\/latex]\u00a0over the boundary of [latex]S[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Understand the divergence theorem<\/li>\n<li>Calculate flux using the divergence theorem<\/li>\n<li>Apply the divergence theorem to physical problems<\/li>\n<\/ul>\n<\/section>\n<h2 data-type=\"title\">Overview of Theorems<\/h2>\n<div data-type=\"note\">\n<section id=\"fs-id1167793470788\" data-depth=\"1\">\n<p id=\"fs-id1167793965594\">Before examining the divergence theorem, it is helpful to begin with an overview of the versions of the Fundamental Theorem of Calculus we have discussed. Each theorem relates an integral of a &#8220;derivative&#8221; over a domain to an integral over the boundary of that domain.<\/p>\n<\/section>\n<\/div>\n<div data-type=\"note\">\n<ol>\n<li><strong data-effect=\"bold\">The Fundamental Theorem of Calculus<\/strong>:<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<div id=\"fs-id1167794118999\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\n<p style=\"text-align: center;\">[latex]\\large{\\displaystyle\\int_a^bf^\\prime(x)dx=f(b)-f(a)}[\/latex].<\/p>\n<\/div>\n<p><span data-type=\"newline\"><br \/>\n<\/span>This theorem relates the integral of derivative [latex]f'[\/latex] over line segment [latex][a,b][\/latex] along the [latex]x[\/latex]-axis to a difference of [latex]f[\/latex] evaluated on the boundary.<\/li>\n<li><strong data-effect=\"bold\">The Fundamental Theorem for Line Integrals<\/strong>:<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<div id=\"fs-id1167793604124\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\n<p style=\"text-align: center;\">[latex]\\large{\\displaystyle\\int_C\\nabla f\\cdot d{\\bf{r}}=f(P_1)-f(P_0)}[\/latex].<\/p>\n<\/div>\n<p><span data-type=\"newline\"><br \/>\n<\/span>where [latex]P_0[\/latex] is the initial point of [latex]C[\/latex] and [latex]P_1[\/latex] is the terminal point of [latex]C[\/latex]. The\u00a0<span id=\"d77de5bc-07e0-40d3-a789-c2f01549dd9b_term287\" class=\"no-emphasis\" data-type=\"term\">Fundamental Theorem for Line Integrals<\/span>\u00a0allows path [latex]C[\/latex] to be a path in a plane or in space, not just a line segment on the [latex]x[\/latex]-axis. If we think of the gradient as a derivative, then this theorem relates an integral of derivative [latex]\\nabla f[\/latex] over path [latex]C[\/latex] to a difference of [latex]f[\/latex] evaluated on the boundary of [latex]C[\/latex].<\/li>\n<li><strong data-effect=\"bold\">Green\u2019s theorem, circulation form<\/strong>:<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<div id=\"fs-id1167793878210\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\n<p style=\"text-align: center;\">[latex]\\large{\\displaystyle\\iint_D(Q_x-P_y)dA=\\displaystyle\\int_C{\\bf{F}}\\cdot d{\\bf{r}}}[\/latex].<\/p>\n<\/div>\n<p><span data-type=\"newline\"><br \/>\n<\/span>Since [latex]Q_x-P_y=\\text{curl }{\\bf{F}}\\cdot{\\bf{k}}[\/latex] and curl is a derivative of sorts,\u00a0<span id=\"d77de5bc-07e0-40d3-a789-c2f01549dd9b_term288\" class=\"no-emphasis\" data-type=\"term\">Green\u2019s theorem<\/span>\u00a0relates the integral of derivative curl[latex]{\\bf{F}}[\/latex]\u00a0over planar region [latex]D[\/latex] to an integral of\u00a0[latex]{\\bf{F}}[\/latex]\u00a0over the boundary of [latex]D[\/latex].<\/li>\n<li><strong data-effect=\"bold\">Green\u2019s theorem, flux form<\/strong>:<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<div id=\"fs-id1167793401271\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\n<p style=\"text-align: center;\">[latex]\\large{\\displaystyle\\iint_D(P_x+Q_y)dA=\\displaystyle\\int_C{\\bf{F}}\\cdot{\\bf{N}}ds}[\/latex].<\/p>\n<\/div>\n<p><span data-type=\"newline\"><br \/>\n<\/span>Since [latex]P_x+Q_y=\\text{div }{\\bf{F}}[\/latex] and divergence is a derivative of sorts, the flux form of Green\u2019s theorem relates the integral of derivative div\u00a0[latex]{\\bf{F}}[\/latex]\u00a0over planar region [latex]D[\/latex] to an integral of\u00a0[latex]{\\bf{F}}[\/latex]\u00a0over the boundary of [latex]D[\/latex].<\/li>\n<li><strong data-effect=\"bold\">Stokes\u2019 theorem<\/strong>:<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<div id=\"fs-id1167793959488\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\large{\\displaystyle\\iint_S\\text{curl }{\\bf{F}}\\cdot ds=\\displaystyle\\int_C{\\bf{F}}\\cdot d{\\bf{r}}}[\/latex]<\/div>\n<p><span data-type=\"newline\"><br \/>\n<\/span>If we think of the curl as a derivative of sorts, then\u00a0<span id=\"d77de5bc-07e0-40d3-a789-c2f01549dd9b_term289\" class=\"no-emphasis\" data-type=\"term\">Stokes\u2019 theorem<\/span>\u00a0relates the integral of derivative curl[latex]{\\bf{F}}[\/latex]\u00a0over surface [latex]S[\/latex] (not necessarily planar) to an integral of\u00a0[latex]{\\bf{F}}[\/latex]\u00a0over the boundary of [latex]S[\/latex].<\/li>\n<\/ol>\n<\/div>\n","protected":false},"author":15,"menu_order":22,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-794","chapter","type-chapter","status-publish","hentry"],"part":438,"module-header":"learn_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/794","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":4,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/794\/revisions"}],"predecessor-version":[{"id":2111,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/794\/revisions\/2111"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/438"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/794\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=794"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=794"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=794"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=794"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}