{"id":114,"date":"2021-07-30T17:15:11","date_gmt":"2021-07-30T17:15:11","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=114"},"modified":"2022-11-01T05:37:57","modified_gmt":"2022-11-01T05:37:57","slug":"summary-of-the-divergence-theorem","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/summary-of-the-divergence-theorem\/","title":{"raw":"Summary of the Divergence Theorem","rendered":"Summary of the Divergence Theorem"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul>\r\n \t<li>The divergence theorem relates a surface integral across closed surface\u00a0[latex]S[\/latex]\u00a0to a triple integral over the solid enclosed by [latex]S[\/latex]. The divergence theorem is a higher dimensional version of the flux form of Green\u2019s theorem, and is therefore a higher dimensional version of the Fundamental Theorem of Calculus.<\/li>\r\n \t<li>The divergence theorem can be used to transform a difficult flux integral into an easier triple integral and vice versa.<\/li>\r\n \t<li>The divergence theorem can be used to derive Gauss\u2019 law, a fundamental law in electrostatics.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Key Equations<\/h2>\r\n<ul>\r\n \t<li><strong>Divergence theorem\r\n<\/strong>[latex]\\displaystyle\\iiint_{E} \\text{div }{\\bf{F}}dV=\\underset{S}{\\displaystyle\\iint}{\\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>divergence theorem<\/dt>\r\n \t<dd><span style=\"font-size: 1em;\">a theorem used to transform a difficult flux integral into an easier triple integral and vice versa<\/span><\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>Gauss' law<\/dt>\r\n \t<dd>if [latex]S[\/latex] is a piecewise, smooth closed surface in a vacuum and [latex]Q[\/latex] is the total stationary charge inside of [latex]S[\/latex], then the flux of electrostatic field [latex]\\bf{E}[\/latex] across [latex]S[\/latex]\u00a0is [latex]Q|{\\varepsilon}_{0}[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>inverse-square law<\/dt>\r\n \t<dd><span style=\"font-size: 1em;\">the electrostatic force at a given point is inversely proportional to the square of the distance from the source of the charge<\/span><\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>The Fundamental Theorem for Line Integrals<\/dt>\r\n \t<dd>the value of the line integral [latex]\\displaystyle\\int_{C} {\\nabla}{f}\\cdot{d{\\bf{r}}}[\/latex]\u00a0depends only on the value of [latex]f[\/latex]\u00a0at the endpoints of [latex]C[\/latex]:\u00a0[latex]\\displaystyle\\int_{C} {\\nabla}{f}\\cdot{d{\\bf{r}}}=f({\\bf{r}}(b)))-f({\\bf{r}}(a))[\/latex]<\/dd>\r\n<\/dl>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul>\n<li>The divergence theorem relates a surface integral across closed surface\u00a0[latex]S[\/latex]\u00a0to a triple integral over the solid enclosed by [latex]S[\/latex]. The divergence theorem is a higher dimensional version of the flux form of Green\u2019s theorem, and is therefore a higher dimensional version of the Fundamental Theorem of Calculus.<\/li>\n<li>The divergence theorem can be used to transform a difficult flux integral into an easier triple integral and vice versa.<\/li>\n<li>The divergence theorem can be used to derive Gauss\u2019 law, a fundamental law in electrostatics.<\/li>\n<\/ul>\n<\/div>\n<h2>Key Equations<\/h2>\n<ul>\n<li><strong>Divergence theorem<br \/>\n<\/strong>[latex]\\displaystyle\\iiint_{E} \\text{div }{\\bf{F}}dV=\\underset{S}{\\displaystyle\\iint}{\\bf{F}}\\cdot{d{\\bf{S}}}[\/latex]<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl class=\"definition\">\n<dt>divergence theorem<\/dt>\n<dd><span style=\"font-size: 1em;\">a theorem used to transform a difficult flux integral into an easier triple integral and vice versa<\/span><\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>Gauss&#8217; law<\/dt>\n<dd>if [latex]S[\/latex] is a piecewise, smooth closed surface in a vacuum and [latex]Q[\/latex] is the total stationary charge inside of [latex]S[\/latex], then the flux of electrostatic field [latex]\\bf{E}[\/latex] across [latex]S[\/latex]\u00a0is [latex]Q|{\\varepsilon}_{0}[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>inverse-square law<\/dt>\n<dd><span style=\"font-size: 1em;\">the electrostatic force at a given point is inversely proportional to the square of the distance from the source of the charge<\/span><\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>The Fundamental Theorem for Line Integrals<\/dt>\n<dd>the value of the line integral [latex]\\displaystyle\\int_{C} {\\nabla}{f}\\cdot{d{\\bf{r}}}[\/latex]\u00a0depends only on the value of [latex]f[\/latex]\u00a0at the endpoints of [latex]C[\/latex]:\u00a0[latex]\\displaystyle\\int_{C} {\\nabla}{f}\\cdot{d{\\bf{r}}}=f({\\bf{r}}(b)))-f({\\bf{r}}(a))[\/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-114\">\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 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