{"id":110,"date":"2021-07-30T17:14:41","date_gmt":"2021-07-30T17:14:41","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=110"},"modified":"2022-11-01T05:32:09","modified_gmt":"2022-11-01T05:32:09","slug":"summary-of-surface-integrals","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/summary-of-surface-integrals\/","title":{"raw":"Summary of Surface Integrals","rendered":"Summary of Surface Integrals"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul>\r\n \t<li>Surfaces can be parameterized, just as curves can be parameterized. In general, surfaces must be parameterized with two parameters.<\/li>\r\n \t<li>Surfaces can sometimes be oriented, just as curves can be oriented. Some surfaces, such as a M\u00f6bius strip, cannot be oriented.<\/li>\r\n \t<li>A surface integral is like a line integral in one higher dimension. The domain of integration of a surface integral is a surface in a plane or space, rather than a curve in a plane or space.<\/li>\r\n \t<li>The integrand of a surface integral can be a scalar function or a vector field. To calculate a surface integral with an integrand that is a function, use [latex]\\displaystyle\\iint_{S} f(x,y,z)dS=\\displaystyle\\iint_{D}f ({\\bf{r}}(u,v)) \\Arrowvert{{\\bf{t}}_{u}\\times{\\bf{t}}_{v}} \\Arrowvert{dA}[\/latex]. To calculate a surface integral with an integrand that is a vector field, use\u00a0[latex]\\displaystyle\\iint_{S} {\\bf{F}}\\cdot {d{\\bf{S}}}=\\displaystyle\\iint_{S} {\\bf{F}}\\cdot {\\bf{N}}dS[\/latex]<\/li>\r\n \t<li>If\u00a0[latex]S[\/latex]\u00a0is a surface, then the area of [latex]S[\/latex]\u00a0is [latex]\\displaystyle\\iint_{S}dS[\/latex]<\/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\">Scalar surface integral<\/strong>\r\n[latex]\\displaystyle\\iint_{S} f(x,y,z)dS=\\displaystyle\\iint_{D}f ({\\bf{r}}(u,v)) \\Arrowvert{{\\bf{t}}_{u}\\times{\\bf{t}}_{v}} \\Arrowvert{dA}[\/latex]<\/li>\r\n \t<li><strong>Flux integral <\/strong>\r\n[latex]\\displaystyle\\iint_{S} {\\bf{F}}\\cdot {\\bf{N}}dS=\\displaystyle\\iint_{S} {\\bf{F}}\\cdot {d{\\bf{S}}}=\\displaystyle\\iint_{S} {\\bf{F}}({\\bf{r}}(u,v))\\cdot\\left({\\bf{t}}_{u}\\times{\\bf{t}}_{v}\\right)dA[\/latex]<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl class=\"definition\">\r\n \t<dt>flux integral<\/dt>\r\n \t<dd><span style=\"font-size: 1em;\">another name for a surface integral of a vector field; the preferred term in physics and engineering<\/span><\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>grid curves<\/dt>\r\n \t<dd><span style=\"font-size: 1em;\">curves on a surface that are parallel to grid lines in a coordinate plane<\/span><\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>heat flow<\/dt>\r\n \t<dd><span style=\"font-size: 1em;\">a vector field proportional to the negative temperature gradient in an object<\/span><\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>mass flux<\/dt>\r\n \t<dd><span style=\"font-size: 1em;\">the rate of mass flow of a fluid per unit area, measured in mass per unit time per unit area<\/span><\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>orientation of a surface<\/dt>\r\n \t<dd><span style=\"font-size: 1em;\">if a surface has an \u201cinner\u201d side and an \u201couter\u201d side, then an orientation is a choice of the inner or the outer side; the surface could also have \u201cupward\u201d and \u201cdownward\u201d orientations<\/span><\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>parameter domain (parameter space)<\/dt>\r\n \t<dd>the region of the <em data-effect=\"italics\">uv<\/em> plane over which the parameters <em data-effect=\"italics\">u<\/em> and <em data-effect=\"italics\">v<\/em> vary for parameterization [latex]{\\bf{r}}(u, v) = {\\langle} x (u, v), y (u, v), z (u, v) {\\rangle}[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>parameterized surface<\/dt>\r\n \t<dd>a surface given by a description of the form [latex]{\\bf{r}}(u, v) = {\\langle} x (u, v), y (u, v), z (u, v) {\\rangle}[\/latex] ,\u00a0where the parameters <em data-effect=\"italics\">u<\/em> and <em data-effect=\"italics\">v<\/em> vary over a parameter domain in the <em data-effect=\"italics\">uv<\/em>-plane<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>regular parameterization<\/dt>\r\n \t<dd>parameterization [latex]{\\bf{r}}(u, v) = {\\langle} x (u, v), y (u, v), z (u, v) {\\rangle}[\/latex]\u00a0such that [latex]{\\bf{r}}_{u}{\\times}{\\bf{r}}_{v}[\/latex] is not zero for point [latex](u, v)[\/latex] in the parameter domain<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>surface area<\/dt>\r\n \t<dd>the area of surface <em data-effect=\"italics\">S<\/em> given by the surface integral [latex]\\displaystyle{\\int_{} {\\int_{S} d{\\bf{S}}}} [\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>surface integral of a scalar-valued function<\/dt>\r\n \t<dd><span style=\"font-size: 1em;\">a surface integral in which the integrand is a scalar function<\/span><\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>surface integral of a vector field<\/dt>\r\n \t<dd><span style=\"font-size: 1em;\">a surface integral in which the integrand is a vector field<\/span><\/dd>\r\n<\/dl>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul>\n<li>Surfaces can be parameterized, just as curves can be parameterized. In general, surfaces must be parameterized with two parameters.<\/li>\n<li>Surfaces can sometimes be oriented, just as curves can be oriented. Some surfaces, such as a M\u00f6bius strip, cannot be oriented.<\/li>\n<li>A surface integral is like a line integral in one higher dimension. The domain of integration of a surface integral is a surface in a plane or space, rather than a curve in a plane or space.<\/li>\n<li>The integrand of a surface integral can be a scalar function or a vector field. To calculate a surface integral with an integrand that is a function, use [latex]\\displaystyle\\iint_{S} f(x,y,z)dS=\\displaystyle\\iint_{D}f ({\\bf{r}}(u,v)) \\Arrowvert{{\\bf{t}}_{u}\\times{\\bf{t}}_{v}} \\Arrowvert{dA}[\/latex]. To calculate a surface integral with an integrand that is a vector field, use\u00a0[latex]\\displaystyle\\iint_{S} {\\bf{F}}\\cdot {d{\\bf{S}}}=\\displaystyle\\iint_{S} {\\bf{F}}\\cdot {\\bf{N}}dS[\/latex]<\/li>\n<li>If\u00a0[latex]S[\/latex]\u00a0is a surface, then the area of [latex]S[\/latex]\u00a0is [latex]\\displaystyle\\iint_{S}dS[\/latex]<\/li>\n<\/ul>\n<\/div>\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1169736654976\">\n<li><strong data-effect=\"bold\">Scalar surface integral<\/strong><br \/>\n[latex]\\displaystyle\\iint_{S} f(x,y,z)dS=\\displaystyle\\iint_{D}f ({\\bf{r}}(u,v)) \\Arrowvert{{\\bf{t}}_{u}\\times{\\bf{t}}_{v}} \\Arrowvert{dA}[\/latex]<\/li>\n<li><strong>Flux integral <\/strong><br \/>\n[latex]\\displaystyle\\iint_{S} {\\bf{F}}\\cdot {\\bf{N}}dS=\\displaystyle\\iint_{S} {\\bf{F}}\\cdot {d{\\bf{S}}}=\\displaystyle\\iint_{S} {\\bf{F}}({\\bf{r}}(u,v))\\cdot\\left({\\bf{t}}_{u}\\times{\\bf{t}}_{v}\\right)dA[\/latex]<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl class=\"definition\">\n<dt>flux integral<\/dt>\n<dd><span style=\"font-size: 1em;\">another name for a surface integral of a vector field; the preferred term in physics and engineering<\/span><\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>grid curves<\/dt>\n<dd><span style=\"font-size: 1em;\">curves on a surface that are parallel to grid lines in a coordinate plane<\/span><\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>heat flow<\/dt>\n<dd><span style=\"font-size: 1em;\">a vector field proportional to the negative temperature gradient in an object<\/span><\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>mass flux<\/dt>\n<dd><span style=\"font-size: 1em;\">the rate of mass flow of a fluid per unit area, measured in mass per unit time per unit area<\/span><\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>orientation of a surface<\/dt>\n<dd><span style=\"font-size: 1em;\">if a surface has an \u201cinner\u201d side and an \u201couter\u201d side, then an orientation is a choice of the inner or the outer side; the surface could also have \u201cupward\u201d and \u201cdownward\u201d orientations<\/span><\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>parameter domain (parameter space)<\/dt>\n<dd>the region of the <em data-effect=\"italics\">uv<\/em> plane over which the parameters <em data-effect=\"italics\">u<\/em> and <em data-effect=\"italics\">v<\/em> vary for parameterization [latex]{\\bf{r}}(u, v) = {\\langle} x (u, v), y (u, v), z (u, v) {\\rangle}[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>parameterized surface<\/dt>\n<dd>a surface given by a description of the form [latex]{\\bf{r}}(u, v) = {\\langle} x (u, v), y (u, v), z (u, v) {\\rangle}[\/latex] ,\u00a0where the parameters <em data-effect=\"italics\">u<\/em> and <em data-effect=\"italics\">v<\/em> vary over a parameter domain in the <em data-effect=\"italics\">uv<\/em>-plane<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>regular parameterization<\/dt>\n<dd>parameterization [latex]{\\bf{r}}(u, v) = {\\langle} x (u, v), y (u, v), z (u, v) {\\rangle}[\/latex]\u00a0such that [latex]{\\bf{r}}_{u}{\\times}{\\bf{r}}_{v}[\/latex] is not zero for point [latex](u, v)[\/latex] in the parameter domain<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>surface area<\/dt>\n<dd>the area of surface <em data-effect=\"italics\">S<\/em> given by the surface integral [latex]\\displaystyle{\\int_{} {\\int_{S} d{\\bf{S}}}}[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>surface integral of a scalar-valued function<\/dt>\n<dd><span style=\"font-size: 1em;\">a surface integral in which the integrand is a scalar function<\/span><\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>surface integral of a vector field<\/dt>\n<dd><span style=\"font-size: 1em;\">a surface integral in which the integrand is a vector field<\/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-110\">\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":29,"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 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