{"id":774,"date":"2025-10-09T15:38:00","date_gmt":"2025-10-09T15:38:00","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=774"},"modified":"2025-10-17T17:00:03","modified_gmt":"2025-10-17T17:00:03","slug":"advanced-vector-calculus-cheat-sheet","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/advanced-vector-calculus-cheat-sheet\/","title":{"raw":"Advanced Vector Calculus: Cheat Sheet","rendered":"Advanced Vector Calculus: Cheat Sheet"},"content":{"raw":"<h2>Essential Concepts<\/h2>\r\n<strong>Divergence and Curl<\/strong>\r\n<ul>\r\n \t<li>The divergence of a vector field is a scalar function. Divergence measures the \u201coutflowing-ness\u201d of a vector field. If\u00a0[latex]{\\bf{v}}[\/latex]\u00a0is the velocity field of a fluid, then the divergence of\u00a0[latex]{\\bf{v}}[\/latex]\u00a0at a point is the outflow of the fluid less the inflow at the point.<\/li>\r\n \t<li>The curl of a vector field is a vector field. The curl of a vector field at point\u00a0[latex]P[\/latex]\u00a0measures the tendency of particles at\u00a0[latex]P[\/latex]\u00a0to rotate about the axis that points in the direction of the curl at\u00a0[latex]P[\/latex].<\/li>\r\n \t<li>A vector field with a simply connected domain is conservative if and only if its curl is zero.<\/li>\r\n<\/ul>\r\n<strong>Surface Integrals<\/strong>\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<strong>Stokes\u2019 Theorem<\/strong>\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<strong>The Divergence Theorem<\/strong>\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<h2>Key Equations<\/h2>\r\n<ul>\r\n \t<li><strong>Curl\r\n<\/strong>[latex]\\nabla\\times{\\bf{F}}=(R_{y}-Q_{z}){\\bf{i}}+(P_{z}-R_{x}){\\bf{j}}+(Q_{x}+P_{y}){\\bf{k}}[\/latex]<\/li>\r\n \t<li><strong>Divergence\r\n<\/strong>[latex]\\nabla\\cdot{\\bf{F}}=P_{x}+Q_{y}+R_{z}[\/latex]<\/li>\r\n \t<li><strong>Divergence fo curl is zero\r\n<\/strong>[latex]\\nabla\\cdot(\\nabla\\times{\\bf{F}})=0[\/latex]<\/li>\r\n \t<li><strong>Curl of a gradient is the zero vector\r\n<\/strong>[latex]\\nabla\\times(\\nabla{f})=0[\/latex]<\/li>\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 \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 \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>curl<\/dt>\r\n \t<dd>the curl of vector field [latex]{\\bf{F}}=\\langle{P,Q,R}\\rangle[\/latex], denoted [latex]\\nabla\\times{\\bf{F}}[\/latex]\u00a0is the \u201cdeterminant\u201d of the matrix [latex]\\begin{vmatrix}{\\bf{i}} &amp; {\\bf{j}} &amp; {\\bf{k}}\\\\ \\frac{d}{dx} &amp; \\frac{d}{dy} &amp; \\frac{d}{dz}\\\\P &amp; Q &amp; R\\end{vmatrix}[\/latex] and is given by the expression\u00a0[latex](R_{y}-Q_{z}){\\bf{i}}+(P_{z}-R_{x}){\\bf{j}}+(Q_{x}+P_{y}){\\bf{k}}[\/latex];\u00a0it measures the tendency of particles at a point to rotate about the axis that points in the direction of the curl at the point<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>divergence<\/dt>\r\n \t<dd>the divergence of a vector field [latex]{\\bf{F}}=\\langle{P,Q,R}\\rangle[\/latex], denoted [latex]\\nabla\\times{\\bf{F}}[\/latex] is [latex]P_{x}+Q_{y}+R_{z}[\/latex]; it measures the \u201coutflowing-ness\u201d of a vector field<\/dd>\r\n<\/dl>\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>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>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>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>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>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>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 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 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>\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>\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":"<h2>Essential Concepts<\/h2>\n<p><strong>Divergence and Curl<\/strong><\/p>\n<ul>\n<li>The divergence of a vector field is a scalar function. Divergence measures the \u201coutflowing-ness\u201d of a vector field. If\u00a0[latex]{\\bf{v}}[\/latex]\u00a0is the velocity field of a fluid, then the divergence of\u00a0[latex]{\\bf{v}}[\/latex]\u00a0at a point is the outflow of the fluid less the inflow at the point.<\/li>\n<li>The curl of a vector field is a vector field. The curl of a vector field at point\u00a0[latex]P[\/latex]\u00a0measures the tendency of particles at\u00a0[latex]P[\/latex]\u00a0to rotate about the axis that points in the direction of the curl at\u00a0[latex]P[\/latex].<\/li>\n<li>A vector field with a simply connected domain is conservative if and only if its curl is zero.<\/li>\n<\/ul>\n<p><strong>Surface Integrals<\/strong><\/p>\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<p><strong>Stokes\u2019 Theorem<\/strong><\/p>\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<p><strong>The Divergence Theorem<\/strong><\/p>\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<h2>Key Equations<\/h2>\n<ul>\n<li><strong>Curl<br \/>\n<\/strong>[latex]\\nabla\\times{\\bf{F}}=(R_{y}-Q_{z}){\\bf{i}}+(P_{z}-R_{x}){\\bf{j}}+(Q_{x}+P_{y}){\\bf{k}}[\/latex]<\/li>\n<li><strong>Divergence<br \/>\n<\/strong>[latex]\\nabla\\cdot{\\bf{F}}=P_{x}+Q_{y}+R_{z}[\/latex]<\/li>\n<li><strong>Divergence fo curl is zero<br \/>\n<\/strong>[latex]\\nabla\\cdot(\\nabla\\times{\\bf{F}})=0[\/latex]<\/li>\n<li><strong>Curl of a gradient is the zero vector<br \/>\n<\/strong>[latex]\\nabla\\times(\\nabla{f})=0[\/latex]<\/li>\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<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<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>curl<\/dt>\n<dd>the curl of vector field [latex]{\\bf{F}}=\\langle{P,Q,R}\\rangle[\/latex], denoted [latex]\\nabla\\times{\\bf{F}}[\/latex]\u00a0is the \u201cdeterminant\u201d of the matrix [latex]\\begin{vmatrix}{\\bf{i}} & {\\bf{j}} & {\\bf{k}}\\\\ \\frac{d}{dx} & \\frac{d}{dy} & \\frac{d}{dz}\\\\P & Q & R\\end{vmatrix}[\/latex] and is given by the expression\u00a0[latex](R_{y}-Q_{z}){\\bf{i}}+(P_{z}-R_{x}){\\bf{j}}+(Q_{x}+P_{y}){\\bf{k}}[\/latex];\u00a0it measures the tendency of particles at a point to rotate about the axis that points in the direction of the curl at the point<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>divergence<\/dt>\n<dd>the divergence of a vector field [latex]{\\bf{F}}=\\langle{P,Q,R}\\rangle[\/latex], denoted [latex]\\nabla\\times{\\bf{F}}[\/latex] is [latex]P_{x}+Q_{y}+R_{z}[\/latex]; it measures the \u201coutflowing-ness\u201d of a vector field<\/dd>\n<\/dl>\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>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>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>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>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>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>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 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 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<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<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","protected":false},"author":15,"menu_order":1,"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-774","chapter","type-chapter","status-publish","hentry"],"part":438,"module-header":"cheat_sheet","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\/774","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\/774\/revisions"}],"predecessor-version":[{"id":1546,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/774\/revisions\/1546"}],"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\/774\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=774"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=774"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=774"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=774"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}