{"id":738,"date":"2025-10-09T15:35:41","date_gmt":"2025-10-09T15:35:41","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=738"},"modified":"2025-10-17T15:03:07","modified_gmt":"2025-10-17T15:03:07","slug":"vector-calculus-and-vector-fields-cheat-sheet","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/vector-calculus-and-vector-fields-cheat-sheet\/","title":{"raw":"Vector Calculus and Vector Fields: Cheat Sheet","rendered":"Vector Calculus and Vector Fields: Cheat Sheet"},"content":{"raw":"<h2>Essential Concepts<\/h2>\r\n<strong>Introduction to Vector Fields<\/strong>\r\n<ul>\r\n \t<li>A vector field assigns a vector [latex]{\\bf{F}}(x,y)[\/latex] to each point\u00a0[latex](x,y)[\/latex] in a subset [latex]D[\/latex] of\u00a0[latex]\\mathbb{R}^{2}[\/latex] or\u00a0[latex]\\mathbb{R}^{3}[\/latex].\u00a0[latex]{\\bf{F}}(x,y,z)[\/latex] to each point\u00a0[latex](x,y,z)[\/latex] in a subset [latex]D[\/latex] of\u00a0[latex]\\mathbb{R}^{3}[\/latex].<\/li>\r\n \t<li>Vector fields can describe the distribution of vector quantities such as forces or velocities over a region of the plane or of space. They are in common use in such areas as physics, engineering, meteorology, oceanography.<\/li>\r\n \t<li>We can sketch a vector field by examining its defining equation to determine relative magnitudes in various locations and then drawing enough vectors to determine a pattern.<\/li>\r\n \t<li>A vector field [latex]{\\bf{F}}[\/latex] is called conservative if there exists a scalar function [latex]f[\/latex] such that [latex]\\nabla{f}={\\bf{F}}[\/latex].<\/li>\r\n<\/ul>\r\n<strong>Line Integrals<\/strong>\r\n<ul>\r\n \t<li>Line integrals generalize the notion of a single-variable integral to higher dimensions. The domain of integration in a single-variable integral is a line segment along the\u00a0<em data-effect=\"italics\">x<\/em>-axis, but the domain of integration in a line integral is a curve in a plane or in space.<\/li>\r\n \t<li>If [latex]C[\/latex]\u00a0is a curve, then the length of\u00a0[latex]C[\/latex] is [latex]\\displaystyle\\int_{C} ds[\/latex].<\/li>\r\n \t<li>There are two kinds of line integral: scalar line integrals and vector line integrals. Scalar line integrals can be used to calculate the mass of a wire; vector line integrals can be used to calculate the work done on a particle traveling through a field.<\/li>\r\n \t<li>Scalar line integrals can be calculated using[latex]\\displaystyle\\int_{C} f(x,y,z)ds=\\displaystyle\\int_{a}^{b} f({\\bf{r}}(t))\\sqrt{(x^{\\prime}(t))^{2}+(y^{\\prime}(t))^{2}+(z^{\\prime}(t))^{2}}dt[\/latex]; vector line integrals can be calculated using\u00a0[latex]\\displaystyle\\int_{C} {\\bf{F}}\\cdot{ds}=\\displaystyle\\int_{C}{\\bf{F}}\\cdot{\\bf{T}}ds=\\displaystyle\\int_{a}^{b}{\\bf{F}}({\\bf{r}}(t))\\cdot{\\bf{r}}^{\\prime}(t)dt[\/latex].<\/li>\r\n \t<li>Two key concepts expressed in terms of line integrals are flux and circulation. Flux measures the rate that a field crosses a given line; circulation measures the tendency of a field to move in the same direction as a given closed curve.<\/li>\r\n<\/ul>\r\n<strong>Conservative Vector Fields<\/strong>\r\n<ul>\r\n \t<li>The theorems in this section require curves that are closed, simple, or both, and regions that are connected or simply connected.<\/li>\r\n \t<li>The line integral of a conservative vector field can be calculated using the Fundamental Theorem for Line Integrals. This theorem is a generalization of the Fundamental Theorem of Calculus in higher dimensions. Using this theorem usually makes the calculation of the line integral easier.<\/li>\r\n \t<li>Conservative fields are independent of path. The line integral of a conservative field depends only on the value of the potential function at the endpoints of the domain curve.<\/li>\r\n \t<li>Given vector field\u00a0[latex]{\\bf{F}}[\/latex], we can test whether\u00a0[latex]{\\bf{F}}[\/latex] is conservative by using the cross-partial property. If\u00a0[latex]{\\bf{F}}[\/latex]\u00a0has the cross-partial property and the domain is simply connected, then\u00a0[latex]{\\bf{F}}[\/latex]\u00a0is conservative (and thus has a potential function). If\u00a0[latex]{\\bf{F}}[\/latex]\u00a0is conservative, we can find a potential function by using the Problem-Solving Strategy.<\/li>\r\n \t<li>The circulation of a conservative vector field on a simply connected domain over a closed curve is zero<\/li>\r\n<\/ul>\r\n<strong>Green\u2019s Theorem<\/strong>\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<h2>Key Equations<\/h2>\r\n<ul>\r\n \t<li><strong>Vector field in [latex]\\mathbb{R}^{2}[\/latex]\r\n<\/strong>[latex]{\\bf{F}}(x,y)=\\langle{P(x,y),Q(x,y)}\\rangle[\/latex]\u00a0 or\u00a0\u00a0[latex]{\\bf{F}}(x,y)=P(x,y){\\bf{i}},Q(x,y){\\bf{j}}[\/latex]<\/li>\r\n \t<li><strong>Vector field in [latex]\\mathbb{R}^{3}[\/latex]\r\n<\/strong>[latex]{\\bf{F}}(x,y,z)=\\langle{P(x,y,z),Q(x,y,z),R(x,y,z)}\\rangle[\/latex]\u00a0 or\u00a0\u00a0[latex]{\\bf{F}}(x,y,z)=P(x,y,z){\\bf{i}},Q(x,y,z){\\bf{j}},R(x,y,z){\\bf{k}}[\/latex]<\/li>\r\n \t<li><strong data-effect=\"bold\">Calculating a scalar line integral<\/strong>\r\n[latex]\\displaystyle\\int_{C} f(x,y,z)ds=\\displaystyle\\int_{a}^{b} f({\\bf{r}}(t))\\sqrt{(x^{\\prime}(t))^{2}+(y^{\\prime}(t))^{2}+(z^{\\prime}(t))^{2}}dt[\/latex]<\/li>\r\n \t<li><strong>Calculating a vector line integral <\/strong>\r\n[latex]\\displaystyle\\int_{C} {\\bf{F}}\\cdot{ds}=\\displaystyle\\int_{C}{\\bf{F}}\\cdot{\\bf{T}}ds=\\displaystyle\\int_{a}^{b}{\\bf{F}}({\\bf{r}}(t))\\cdot{\\bf{r}}^{\\prime}(t)dt[\/latex]\r\nor\r\n[latex]{\\displaystyle\\int_{C}} Pdx+Qdy+Rdz= {\\displaystyle\\int_{a}^{b}}\\left(P({\\bf{r}}(t))\\frac{dx}{dt}+Q({\\bf{r}}(t))\\frac{dy}{dt}+R({\\bf{r}}(t))\\frac{dz}{dt}\\right)dt[\/latex]<\/li>\r\n \t<li><strong data-effect=\"bold\">Calculating flux<\/strong>\r\n[latex]\\displaystyle\\int_{C} {\\bf{F}}\\cdot{\\frac{{\\bf{n}}(t)}{\\Arrowvert{\\bf{n}}(t)\\Arrowvert}}ds=\\displaystyle\\int_{a}^{b}{\\bf{F}}({\\bf{r}}(t))\\cdot{\\bf{n}}(t)dt[\/latex]<\/li>\r\n \t<li><strong data-effect=\"bold\">Fundamental Theorem for Line Integrals<\/strong>\r\n[latex]\\displaystyle\\int_{C}\\nabla {f}\\cdot {d{\\bf{r}}}=f({\\bf{r}}(b))-f({\\bf{r}}(a))[\/latex]<\/li>\r\n \t<li><strong>Circulation of a conservative field over curve\u00a0[latex]C[\/latex]\u00a0that encloses a simply connected region\u00a0<\/strong>\r\n[latex]\\displaystyle\\oint_{C}\\nabla {f}\\cdot {d{\\bf{r}}}=0[\/latex]<\/li>\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>circulation<\/dt>\r\n \t<dd>the tendency of a fluid to move in the direction of curve [latex]C[\/latex]. If [latex]C[\/latex] is a closed curve, then the circulation of [latex]{\\bf{F}}[\/latex]\u00a0along [latex]C[\/latex] is line integral [latex]\\displaystyle\\int_{C} {\\bf{F}}\\cdot{\\bf{T}}ds[\/latex], which we also denote\u00a0[latex]\\displaystyle\\oint_{C} {\\bf{F}}\\cdot{\\bf{T}}ds[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>closed curve<\/dt>\r\n \t<dd><span style=\"font-size: 1em;\">a curve that begins and ends at the same point<\/span><\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>closed curve<\/dt>\r\n \t<dd>a curve for which there exists a parameterization [latex]{\\bf{r}}(t),a\\le{t}\\le{b}[\/latex], such that [latex]{\\bf{r}}(a)={\\bf{r}}(b)[\/latex],\u00a0and the curve is traversed exactly once<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>connected region<\/dt>\r\n \t<dd><span style=\"font-size: 1em;\">a region in which any two points can be connected by a path with a trace contained entirely inside the region<\/span><\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>conservative field<\/dt>\r\n \t<dd>a vector field for which there exists a scalar function [latex]f[\/latex]\u00a0such that [latex]\\nabla{f}={\\bf{F}}[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>flux<\/dt>\r\n \t<dd>the rate of a fluid flowing across a curve in a vector field; the flux of vector field [latex]{\\bf{F}}[\/latex]\u00a0across plane curve [latex]C[\/latex] is line integral [latex]\\displaystyle\\int_{C} {\\bf{F}}\\cdot{\\frac{{\\bf{n}}(t)}{\\Arrowvert{\\bf{n}}(t)\\Arrowvert}}ds[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>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>\r\n<dl class=\"definition\">\r\n \t<dt>gradient field<\/dt>\r\n \t<dd>a vector field [latex]{\\bf{F}}[\/latex]\u00a0for which there exists a scalar function [latex]f[\/latex]\u00a0such that [latex]\\nabla{f}={\\bf{F}}[\/latex]\u00a0in other words, a vector field that is the gradient of a function; such vector fields are also called <em data-effect=\"italics\">conservative<\/em><\/dd>\r\n<\/dl>\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>independent of path (path independent)<\/dt>\r\n \t<dd>a vector field [latex]{\\bf{F}}[\/latex] has path independence if [latex]\\displaystyle\\int_{C_{1}} {\\bf{F}}\\cdot{d{\\bf{r}}}=\\displaystyle\\int_{C_{2}} {\\bf{F}}\\cdot{d{\\bf{r}}}[\/latex]\u00a0for any curves [latex]C_{1}[\/latex]\u00a0and [latex]C_{2}[\/latex]\u00a0in the domain of [latex]{\\bf{F}}[\/latex] with the same initial points and terminal points<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>line integral<\/dt>\r\n \t<dd><span style=\"font-size: 1em;\">the integral of a function along a curve in a plane or in space<\/span><\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>orientation of a curve<\/dt>\r\n \t<dd><span style=\"font-size: 1em;\">the orientation of a curve [latex]C[\/latex]<\/span><span style=\"font-size: 1em;\">\u00a0is a specified direction of [latex]C[\/latex]<\/span><\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>piecewise smooth curve<\/dt>\r\n \t<dd><span style=\"font-size: 1em;\">an oriented curve that is not smooth, but can be written as the union of finitely many smooth curves<\/span><\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>potential function<\/dt>\r\n \t<dd>a scalar function [latex]f[\/latex]\u00a0such that [latex]\\nabla{f}={\\bf{F}}[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>radial field<\/dt>\r\n \t<dd><span style=\"font-size: 1em;\">a vector field in which all vectors either point directly toward or directly away from the origin; the magnitude of any vector depends only on its distance from the origin<\/span><\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>rotational field<\/dt>\r\n \t<dd>a vector field in which the vector at point\u00a0<span style=\"font-size: 1em;\">[latex](x,y)[\/latex]<\/span>\u00a0is tangent to a circle with radius\u00a0<span style=\"font-size: 1em;\">[latex]r=\\sqrt{x^{2}+y^{2}}[\/latex]<\/span>\u00a0in a rotational field, all vectors flow either clockwise or counterclockwise, and the magnitude of a vector depends only on its distance from the origin<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>scalar line integral<\/dt>\r\n \t<dd>the scalar line integral of a function\u00a0[latex]f[\/latex]\u00a0along a curve [latex]C[\/latex] with respect to arc length is the integral [latex]\\displaystyle\\int_C \\! f\\, \\mathrm{d}s[\/latex],\u00a0it is the integral of a scalar function [latex]f[\/latex] along a curve in a plane or in space; such an integral is defined in terms of a Riemann sum, as is a single-variable integral<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>simple curve<\/dt>\r\n \t<dd><span style=\"font-size: 1em;\">a curve that does not cross itself<\/span><\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>simply connected region<\/dt>\r\n \t<dd><span style=\"font-size: 1em;\">a region that is connected and has the property that any closed curve that lies entirely inside the region encompasses points that are entirely inside 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>\r\n<dl class=\"definition\">\r\n \t<dt>unit vector field<\/dt>\r\n \t<dd><span style=\"font-size: 1em;\">a vector field in which the magnitude of every vector is [latex]1[\/latex]<\/span><\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>vector field<\/dt>\r\n \t<dd>measured in [latex]\\mathbb{R}^{2}[\/latex],\u00a0an assignment of a vector <span style=\"font-size: 1em;\">[latex]{\\bf{F}}(x,y)[\/latex]<\/span>\u00a0to each point\u00a0<span style=\"font-size: 1em;\">[latex](x,y)[\/latex]<\/span>\u00a0of a subset\u00a0<span style=\"font-size: 1em;\">[latex]D[\/latex]<\/span>\u00a0of [latex]\\mathbb{R}^{2}[\/latex]; in [latex]\\mathbb{R}^{3}[\/latex],\u00a0an assignment of a vector <span style=\"font-size: 1em;\">[latex]{\\bf{F}}(x,y,z)[\/latex]<\/span> to each point\u00a0<span style=\"font-size: 1em;\">[latex](x,y,z)[\/latex]<\/span>\u00a0of a subset\u00a0<span style=\"font-size: 1em;\">[latex]D[\/latex]<\/span>\u00a0of [latex]\\mathbb{R}^{3}[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>vector line integral<\/dt>\r\n \t<dd>the vector line integral of vector field [latex]{\\bf{F}}[\/latex] along curve [latex]C[\/latex] is the integral of the dot product of [latex]{\\bf{F}}[\/latex]\u00a0with unit tangent vector [latex]{\\bf{T}}[\/latex]\u00a0of [latex]C[\/latex] with respect to arc length, [latex]\\displaystyle\\int_{C} {\\bf{F}}\\cdot{\\bf{T}}ds[\/latex];\u00a0such an integral is defined in terms of a Riemann sum, similar to a single-variable integral<\/dd>\r\n<\/dl>","rendered":"<h2>Essential Concepts<\/h2>\n<p><strong>Introduction to Vector Fields<\/strong><\/p>\n<ul>\n<li>A vector field assigns a vector [latex]{\\bf{F}}(x,y)[\/latex] to each point\u00a0[latex](x,y)[\/latex] in a subset [latex]D[\/latex] of\u00a0[latex]\\mathbb{R}^{2}[\/latex] or\u00a0[latex]\\mathbb{R}^{3}[\/latex].\u00a0[latex]{\\bf{F}}(x,y,z)[\/latex] to each point\u00a0[latex](x,y,z)[\/latex] in a subset [latex]D[\/latex] of\u00a0[latex]\\mathbb{R}^{3}[\/latex].<\/li>\n<li>Vector fields can describe the distribution of vector quantities such as forces or velocities over a region of the plane or of space. They are in common use in such areas as physics, engineering, meteorology, oceanography.<\/li>\n<li>We can sketch a vector field by examining its defining equation to determine relative magnitudes in various locations and then drawing enough vectors to determine a pattern.<\/li>\n<li>A vector field [latex]{\\bf{F}}[\/latex] is called conservative if there exists a scalar function [latex]f[\/latex] such that [latex]\\nabla{f}={\\bf{F}}[\/latex].<\/li>\n<\/ul>\n<p><strong>Line Integrals<\/strong><\/p>\n<ul>\n<li>Line integrals generalize the notion of a single-variable integral to higher dimensions. The domain of integration in a single-variable integral is a line segment along the\u00a0<em data-effect=\"italics\">x<\/em>-axis, but the domain of integration in a line integral is a curve in a plane or in space.<\/li>\n<li>If [latex]C[\/latex]\u00a0is a curve, then the length of\u00a0[latex]C[\/latex] is [latex]\\displaystyle\\int_{C} ds[\/latex].<\/li>\n<li>There are two kinds of line integral: scalar line integrals and vector line integrals. Scalar line integrals can be used to calculate the mass of a wire; vector line integrals can be used to calculate the work done on a particle traveling through a field.<\/li>\n<li>Scalar line integrals can be calculated using[latex]\\displaystyle\\int_{C} f(x,y,z)ds=\\displaystyle\\int_{a}^{b} f({\\bf{r}}(t))\\sqrt{(x^{\\prime}(t))^{2}+(y^{\\prime}(t))^{2}+(z^{\\prime}(t))^{2}}dt[\/latex]; vector line integrals can be calculated using\u00a0[latex]\\displaystyle\\int_{C} {\\bf{F}}\\cdot{ds}=\\displaystyle\\int_{C}{\\bf{F}}\\cdot{\\bf{T}}ds=\\displaystyle\\int_{a}^{b}{\\bf{F}}({\\bf{r}}(t))\\cdot{\\bf{r}}^{\\prime}(t)dt[\/latex].<\/li>\n<li>Two key concepts expressed in terms of line integrals are flux and circulation. Flux measures the rate that a field crosses a given line; circulation measures the tendency of a field to move in the same direction as a given closed curve.<\/li>\n<\/ul>\n<p><strong>Conservative Vector Fields<\/strong><\/p>\n<ul>\n<li>The theorems in this section require curves that are closed, simple, or both, and regions that are connected or simply connected.<\/li>\n<li>The line integral of a conservative vector field can be calculated using the Fundamental Theorem for Line Integrals. This theorem is a generalization of the Fundamental Theorem of Calculus in higher dimensions. Using this theorem usually makes the calculation of the line integral easier.<\/li>\n<li>Conservative fields are independent of path. The line integral of a conservative field depends only on the value of the potential function at the endpoints of the domain curve.<\/li>\n<li>Given vector field\u00a0[latex]{\\bf{F}}[\/latex], we can test whether\u00a0[latex]{\\bf{F}}[\/latex] is conservative by using the cross-partial property. If\u00a0[latex]{\\bf{F}}[\/latex]\u00a0has the cross-partial property and the domain is simply connected, then\u00a0[latex]{\\bf{F}}[\/latex]\u00a0is conservative (and thus has a potential function). If\u00a0[latex]{\\bf{F}}[\/latex]\u00a0is conservative, we can find a potential function by using the Problem-Solving Strategy.<\/li>\n<li>The circulation of a conservative vector field on a simply connected domain over a closed curve is zero<\/li>\n<\/ul>\n<p><strong>Green\u2019s Theorem<\/strong><\/p>\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<h2>Key Equations<\/h2>\n<ul>\n<li><strong>Vector field in [latex]\\mathbb{R}^{2}[\/latex]<br \/>\n<\/strong>[latex]{\\bf{F}}(x,y)=\\langle{P(x,y),Q(x,y)}\\rangle[\/latex]\u00a0 or\u00a0\u00a0[latex]{\\bf{F}}(x,y)=P(x,y){\\bf{i}},Q(x,y){\\bf{j}}[\/latex]<\/li>\n<li><strong>Vector field in [latex]\\mathbb{R}^{3}[\/latex]<br \/>\n<\/strong>[latex]{\\bf{F}}(x,y,z)=\\langle{P(x,y,z),Q(x,y,z),R(x,y,z)}\\rangle[\/latex]\u00a0 or\u00a0\u00a0[latex]{\\bf{F}}(x,y,z)=P(x,y,z){\\bf{i}},Q(x,y,z){\\bf{j}},R(x,y,z){\\bf{k}}[\/latex]<\/li>\n<li><strong data-effect=\"bold\">Calculating a scalar line integral<\/strong><br \/>\n[latex]\\displaystyle\\int_{C} f(x,y,z)ds=\\displaystyle\\int_{a}^{b} f({\\bf{r}}(t))\\sqrt{(x^{\\prime}(t))^{2}+(y^{\\prime}(t))^{2}+(z^{\\prime}(t))^{2}}dt[\/latex]<\/li>\n<li><strong>Calculating a vector line integral <\/strong><br \/>\n[latex]\\displaystyle\\int_{C} {\\bf{F}}\\cdot{ds}=\\displaystyle\\int_{C}{\\bf{F}}\\cdot{\\bf{T}}ds=\\displaystyle\\int_{a}^{b}{\\bf{F}}({\\bf{r}}(t))\\cdot{\\bf{r}}^{\\prime}(t)dt[\/latex]<br \/>\nor<br \/>\n[latex]{\\displaystyle\\int_{C}} Pdx+Qdy+Rdz= {\\displaystyle\\int_{a}^{b}}\\left(P({\\bf{r}}(t))\\frac{dx}{dt}+Q({\\bf{r}}(t))\\frac{dy}{dt}+R({\\bf{r}}(t))\\frac{dz}{dt}\\right)dt[\/latex]<\/li>\n<li><strong data-effect=\"bold\">Calculating flux<\/strong><br \/>\n[latex]\\displaystyle\\int_{C} {\\bf{F}}\\cdot{\\frac{{\\bf{n}}(t)}{\\Arrowvert{\\bf{n}}(t)\\Arrowvert}}ds=\\displaystyle\\int_{a}^{b}{\\bf{F}}({\\bf{r}}(t))\\cdot{\\bf{n}}(t)dt[\/latex]<\/li>\n<li><strong data-effect=\"bold\">Fundamental Theorem for Line Integrals<\/strong><br \/>\n[latex]\\displaystyle\\int_{C}\\nabla {f}\\cdot {d{\\bf{r}}}=f({\\bf{r}}(b))-f({\\bf{r}}(a))[\/latex]<\/li>\n<li><strong>Circulation of a conservative field over curve\u00a0[latex]C[\/latex]\u00a0that encloses a simply connected region\u00a0<\/strong><br \/>\n[latex]\\displaystyle\\oint_{C}\\nabla {f}\\cdot {d{\\bf{r}}}=0[\/latex]<\/li>\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>circulation<\/dt>\n<dd>the tendency of a fluid to move in the direction of curve [latex]C[\/latex]. If [latex]C[\/latex] is a closed curve, then the circulation of [latex]{\\bf{F}}[\/latex]\u00a0along [latex]C[\/latex] is line integral [latex]\\displaystyle\\int_{C} {\\bf{F}}\\cdot{\\bf{T}}ds[\/latex], which we also denote\u00a0[latex]\\displaystyle\\oint_{C} {\\bf{F}}\\cdot{\\bf{T}}ds[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>closed curve<\/dt>\n<dd><span style=\"font-size: 1em;\">a curve that begins and ends at the same point<\/span><\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>closed curve<\/dt>\n<dd>a curve for which there exists a parameterization [latex]{\\bf{r}}(t),a\\le{t}\\le{b}[\/latex], such that [latex]{\\bf{r}}(a)={\\bf{r}}(b)[\/latex],\u00a0and the curve is traversed exactly once<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>connected region<\/dt>\n<dd><span style=\"font-size: 1em;\">a region in which any two points can be connected by a path with a trace contained entirely inside the region<\/span><\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>conservative field<\/dt>\n<dd>a vector field for which there exists a scalar function [latex]f[\/latex]\u00a0such that [latex]\\nabla{f}={\\bf{F}}[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>flux<\/dt>\n<dd>the rate of a fluid flowing across a curve in a vector field; the flux of vector field [latex]{\\bf{F}}[\/latex]\u00a0across plane curve [latex]C[\/latex] is line integral [latex]\\displaystyle\\int_{C} {\\bf{F}}\\cdot{\\frac{{\\bf{n}}(t)}{\\Arrowvert{\\bf{n}}(t)\\Arrowvert}}ds[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>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<dl class=\"definition\">\n<dt>gradient field<\/dt>\n<dd>a vector field [latex]{\\bf{F}}[\/latex]\u00a0for which there exists a scalar function [latex]f[\/latex]\u00a0such that [latex]\\nabla{f}={\\bf{F}}[\/latex]\u00a0in other words, a vector field that is the gradient of a function; such vector fields are also called <em data-effect=\"italics\">conservative<\/em><\/dd>\n<\/dl>\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>independent of path (path independent)<\/dt>\n<dd>a vector field [latex]{\\bf{F}}[\/latex] has path independence if [latex]\\displaystyle\\int_{C_{1}} {\\bf{F}}\\cdot{d{\\bf{r}}}=\\displaystyle\\int_{C_{2}} {\\bf{F}}\\cdot{d{\\bf{r}}}[\/latex]\u00a0for any curves [latex]C_{1}[\/latex]\u00a0and [latex]C_{2}[\/latex]\u00a0in the domain of [latex]{\\bf{F}}[\/latex] with the same initial points and terminal points<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>line integral<\/dt>\n<dd><span style=\"font-size: 1em;\">the integral of a function along a curve in a plane or in space<\/span><\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>orientation of a curve<\/dt>\n<dd><span style=\"font-size: 1em;\">the orientation of a curve [latex]C[\/latex]<\/span><span style=\"font-size: 1em;\">\u00a0is a specified direction of [latex]C[\/latex]<\/span><\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>piecewise smooth curve<\/dt>\n<dd><span style=\"font-size: 1em;\">an oriented curve that is not smooth, but can be written as the union of finitely many smooth curves<\/span><\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>potential function<\/dt>\n<dd>a scalar function [latex]f[\/latex]\u00a0such that [latex]\\nabla{f}={\\bf{F}}[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>radial field<\/dt>\n<dd><span style=\"font-size: 1em;\">a vector field in which all vectors either point directly toward or directly away from the origin; the magnitude of any vector depends only on its distance from the origin<\/span><\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>rotational field<\/dt>\n<dd>a vector field in which the vector at point\u00a0<span style=\"font-size: 1em;\">[latex](x,y)[\/latex]<\/span>\u00a0is tangent to a circle with radius\u00a0<span style=\"font-size: 1em;\">[latex]r=\\sqrt{x^{2}+y^{2}}[\/latex]<\/span>\u00a0in a rotational field, all vectors flow either clockwise or counterclockwise, and the magnitude of a vector depends only on its distance from the origin<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>scalar line integral<\/dt>\n<dd>the scalar line integral of a function\u00a0[latex]f[\/latex]\u00a0along a curve [latex]C[\/latex] with respect to arc length is the integral [latex]\\displaystyle\\int_C \\! f\\, \\mathrm{d}s[\/latex],\u00a0it is the integral of a scalar function [latex]f[\/latex] along a curve in a plane or in space; such an integral is defined in terms of a Riemann sum, as is a single-variable integral<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>simple curve<\/dt>\n<dd><span style=\"font-size: 1em;\">a curve that does not cross itself<\/span><\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>simply connected region<\/dt>\n<dd><span style=\"font-size: 1em;\">a region that is connected and has the property that any closed curve that lies entirely inside the region encompasses points that are entirely inside 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<dl class=\"definition\">\n<dt>unit vector field<\/dt>\n<dd><span style=\"font-size: 1em;\">a vector field in which the magnitude of every vector is [latex]1[\/latex]<\/span><\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>vector field<\/dt>\n<dd>measured in [latex]\\mathbb{R}^{2}[\/latex],\u00a0an assignment of a vector <span style=\"font-size: 1em;\">[latex]{\\bf{F}}(x,y)[\/latex]<\/span>\u00a0to each point\u00a0<span style=\"font-size: 1em;\">[latex](x,y)[\/latex]<\/span>\u00a0of a subset\u00a0<span style=\"font-size: 1em;\">[latex]D[\/latex]<\/span>\u00a0of [latex]\\mathbb{R}^{2}[\/latex]; in [latex]\\mathbb{R}^{3}[\/latex],\u00a0an assignment of a vector <span style=\"font-size: 1em;\">[latex]{\\bf{F}}(x,y,z)[\/latex]<\/span> to each point\u00a0<span style=\"font-size: 1em;\">[latex](x,y,z)[\/latex]<\/span>\u00a0of a subset\u00a0<span style=\"font-size: 1em;\">[latex]D[\/latex]<\/span>\u00a0of [latex]\\mathbb{R}^{3}[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>vector line integral<\/dt>\n<dd>the vector line integral of vector field [latex]{\\bf{F}}[\/latex] along curve [latex]C[\/latex] is the integral of the dot product of [latex]{\\bf{F}}[\/latex]\u00a0with unit tangent vector [latex]{\\bf{T}}[\/latex]\u00a0of [latex]C[\/latex] with respect to arc length, [latex]\\displaystyle\\int_{C} {\\bf{F}}\\cdot{\\bf{T}}ds[\/latex];\u00a0such an integral is defined in terms of a Riemann sum, similar to a single-variable integral<\/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-738","chapter","type-chapter","status-publish","hentry"],"part":436,"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\/738","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\/738\/revisions"}],"predecessor-version":[{"id":1427,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/738\/revisions\/1427"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/436"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/738\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=738"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=738"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=738"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=738"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}