{"id":778,"date":"2025-10-09T15:38:10","date_gmt":"2025-10-09T15:38:10","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=778"},"modified":"2025-11-06T14:53:35","modified_gmt":"2025-11-06T14:53:35","slug":"divergence-and-curl-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/divergence-and-curl-learn-it-1\/","title":{"raw":"Divergence and Curl: Learn It 1","rendered":"Divergence and Curl: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Determine divergence from the formula for a given vector field<\/li>\r\n \t<li>Determine curl from the formula for a given vector field<\/li>\r\n \t<li>Use divergence and curl to identify conservative fields<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 id=\"fs-id1167793571517\" class=\" \">Divergence<\/h2>\r\n<p class=\"whitespace-normal break-words\">Divergence is an operation on a vector field that tells us how the field behaves toward or away from a point. Locally, the divergence of a vector field [latex]\\mathbf{F}[\/latex] in [latex]\\mathbb{R}^2[\/latex] or [latex]\\mathbb{R}^3[\/latex] at a particular point [latex]P[\/latex] is a measure of the \"outflowing-ness\" of the vector field at [latex]P[\/latex].<\/p>\r\n<p class=\"whitespace-normal break-words\">If [latex]\\mathbf{F}[\/latex] represents the velocity of a fluid, then the divergence of [latex]\\mathbf{F}[\/latex] at [latex]P[\/latex] measures the net rate of change with respect to time of the amount of fluid flowing away from [latex]P[\/latex] (the tendency of the fluid to flow \"out of\" [latex]P[\/latex]). If the amount of fluid flowing into [latex]P[\/latex] is the same as the amount flowing out, then the divergence at [latex]P[\/latex] is zero.<\/p>\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>divergence<\/h3>\r\n<p id=\"fs-id1167793931072\">If [latex]{\\bf{F}}\\langle{P},Q,R\\rangle[\/latex] is a vector field in [latex]\\mathbb{R}^3[\/latex] and [latex]P_x[\/latex], [latex]Q_y[\/latex], and [latex]R_z[\/latex] all exist, then the <strong><span id=\"6f668e5d-300d-4e6c-aec3-70b59f71fe69_term265\" data-type=\"term\">divergence<\/span><\/strong> of\u00a0[latex]{\\bf{F}}[\/latex]\u00a0is defined by:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\text{div }{\\bf{F}}=P_x+Q_y+R_z=\\dfrac{\\partial{P}}{\\partial{x}}+\\dfrac{\\partial{Q}}{\\partial{y}}+\\dfrac{\\partial{R}}{\\partial{z}}[\/latex].<\/p>\r\n\r\n<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\">\r\n<p id=\"fs-id1167793272083\">Note the divergence of a vector field is not a vector field, but a scalar function. In terms of the gradient operator [latex]\\nabla=\\left\\langle\\frac{\\partial}{\\partial{x}},\\frac{\\partial}{\\partial{y}},\\frac{\\partial}{\\partial{z}}\\right\\rangle[\/latex], divergence can be written symbolically as the dot product:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\text{div }{\\bf{F}}=\\nabla\\cdot{\\bf{F}}[\/latex].<\/p>\r\n\r\n<\/section>Note this is merely helpful notation, because the dot product of a vector of operators and a vector of functions is not meaningfully defined given our current definition of dot product. If [latex]{\\bf{F}}\\langle{P},Q\\rangle[\/latex] is a vector field in [latex]\\mathbb{R}^2[\/latex], and [latex]P_x[\/latex] and [latex]Q_y[\/latex] both exist, then the divergence of [latex]{\\bf{F}}[\/latex] is defined similarly as:\r\n<p style=\"text-align: center;\">[latex]\\text{div }{\\bf{F}}=P_x+Q_y=\\dfrac{\\partial{P}}{\\partial{x}}+\\dfrac{\\partial{Q}}{\\partial{y}}=\\nabla\\cdot{\\bf{F}}[\/latex].<\/p>\r\nTo illustrate this point, consider the two vector fields in Figure 1.\r\n\r\n[caption id=\"attachment_1436\" align=\"aligncenter\" width=\"794\"]<img class=\"size-full wp-image-1436\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17151438\/52108a91d94bdbb0791e0c04baae71a55a6f9b3d.webp\" alt=\"Two images of vector fields A and B in two dimensions. Vector field A has arrows pointing up and to the right. They do not change in size or direction. It has zero divergence. Vector field B has arrows surrounding the origin in a counterclockwise direction. The arrows are larger the closer they are to the origin. It also has zero divergence.\" width=\"794\" height=\"1103\" \/> Figure 1. (a) Vector field [latex]\\langle1,2\\rangle[\/latex] has zero divergence. (b) Vector field [latex]\\langle-y,x\\rangle[\/latex] also has zero divergence.[\/caption]At any particular point, the amount flowing in is the same as the amount flowing out, so at every point the \"outflowing-ness\" of the field is zero. Therefore, we expect the divergence of both fields to be zero, and this is indeed the case, as:\r\n<p style=\"text-align: center;\">[latex]\\text{div }(\\langle1,2\\rangle)=\\dfrac{\\partial}{\\partial{x}}(1)+\\dfrac{\\partial}{\\partial{y}}(2)=0\\text{ and div }(\\langle-y,x\\rangle)=\\dfrac{\\partial}{\\partial{x}}(-y)+\\dfrac{\\partial}{\\partial{y}}(x)=0[\/latex].<\/p>\r\nBy contrast, consider radial vector field [latex]{\\bf{R}}(x,y)=\\langle-x,-y\\rangle[\/latex] in Figure 2.\r\n\r\n[caption id=\"attachment_1437\" align=\"aligncenter\" width=\"717\"]<img class=\"size-full wp-image-1437\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17151504\/f8745ea1aed0b6e1cf0169f5f882642308e3c174.webp\" alt=\"A vector field in two dimensions with negative divergence. The arrows point in towards the origin in a radial pattern. The closer the arrows are to the origin, the larger they are.\" width=\"717\" height=\"497\" \/> Figure 2. This vector field has negative divergence.[\/caption]\r\n\r\nAt any given point, more fluid is flowing in than is flowing out, and therefore the \u201coutgoingness\u201d of the field is negative. We expect the divergence of this field to be negative, and this is indeed the case, as [latex]\\text{div }({\\bf{R}})=\\frac{\\partial}{\\partial{x}}(-x)+\\frac{\\partial}{\\partial{y}}(-y)=-2[\/latex].\r\n\r\nTo get a global sense of what divergence is telling us, suppose that a vector field in [latex]\\mathbb{R}^2[\/latex] represents the velocity of a fluid. Imagine taking an elastic circle (a circle with a shape that can be changed by the vector field) and dropping it into a fluid. If the circle maintains its exact area as it flows through the fluid, then the divergence is zero. This would occur for both vector fields in Figure 1.\r\n<p class=\"whitespace-normal break-words\">On the other hand, if the circle's shape is distorted so that its area shrinks or expands, then the divergence is not zero. Imagine dropping such an elastic circle into the radial vector field in Figure 2 so that the center of the circle lands at point [latex](3, 3)[\/latex]. The circle would flow toward the origin, and as it did so the front of the circle would travel more slowly than the back, causing the circle to \"scrunch\" and lose area. This is how you can see a negative divergence.<\/p>\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">If [latex]{\\bf{F}}(x,y,z)=e^x{\\bf{i}}+yz{\\bf{j}}-y^2{\\bf{k}}[\/latex], then find the divergence of\u00a0[latex]{\\bf{F}}[\/latex]\u00a0at [latex](0, 2, -1)[\/latex].[reveal-answer q=\"943504823\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"943504823\"]<\/section>\r\n<p id=\"fs-id1167793887560\">The divergence of\u00a0[latex]{\\bf{F}}[\/latex]\u00a0is<\/p>\r\n<p style=\"text-align: center;\">[latex]\\large{\\frac{\\partial}{\\partial{x}}\\left(e^x\\right)+\\frac{\\partial}{\\partial{y}}(yz)-\\frac{\\partial}{\\partial{z}}\\left(yz^2\\right)=e^x+z-2yz}[\/latex].<\/p>\r\n<p id=\"fs-id1167793642080\">Therefore, the divergence at [latex](0, 2, -1)[\/latex] is [latex]e^0-1+4=4[\/latex]. If\u00a0[latex]{\\bf{F}}[\/latex]\u00a0represents the velocity of a fluid, then more fluid is flowing out than flowing in at point [latex](0, 2, -1)[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\nOne application for divergence occurs in physics when working with magnetic fields. A magnetic field is a vector field that models the influence of electric currents and magnetic materials. Physicists use divergence in Gauss's law for magnetism, which states that if [latex]\\mathbf{B}[\/latex] is a magnetic field, then [latex]\\nabla\\cdot\\mathbf{B}=0[\/latex]; in other words, the divergence of a magnetic field is zero.\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">Is it possible for [latex]{\\bf{F}}(x,y)=\\langle{x}^2y,y-xy^2\\rangle[\/latex] to be a magnetic field?[reveal-answer q=\"234057932\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"234057932\"]\r\n<p id=\"fs-id1167794064443\">If\u00a0[latex]{\\bf{F}}[\/latex]\u00a0were magnetic, then its divergence would be zero. The divergence of\u00a0[latex]{\\bf{F}}[\/latex]\u00a0is<\/p>\r\n<p style=\"text-align: center;\">[latex]\\large{\\frac{\\partial}{\\partial{x}}\\left(x^2y\\right)+\\frac{\\partial}{\\partial{y}}\\left(y-xy^2\\right)=2xy+1-2xy=1}[\/latex].<\/p>\r\n<p id=\"fs-id1167794059031\">and therefore\u00a0[latex]{\\bf{F}}[\/latex]\u00a0cannot model a magnetic field (Figure 3).<\/p>\r\n\r\n[caption id=\"attachment_1440\" align=\"aligncenter\" width=\"717\"]<img class=\"size-full wp-image-1440\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17151704\/8012ae03cada4cfa516ccb68746e2005e677e07c.webp\" alt=\"A vector field in two dimensions with divergence equal to 1. The arrows are quite flat near the x axis and vertical near the y axis. They seem to asymptotically approach the axes in quadrants 2 and 4, pointing up and to the right in quadrant 2 and down and to the left in quadrant 4. In quadrant 1, they start by pointing up and to the right close to the y axis, but they soon shift to pointing down and to the right. In quadrant 3, they start by pointing down and to the left close to the y axis, bu they soon shift to pointing up and to the left. The closer the arrows are to the origin, the shorter they are.\" width=\"717\" height=\"572\" \/> Figure 3:\u00a0 The divergence of vector field [latex]F(x,y)=\u27e8x^2y,y\u2212xy^2\u27e9[\/latex]<br \/>is one, so it cannot model a magnetic field.[\/caption][\/hidden-answer]<\/section>\r\n<p class=\"whitespace-normal break-words\">Another application for divergence is detecting whether a field is source free. Recall that a source-free field is a vector field that has a stream function; equivalently, a source-free field is a field with a flux that is zero along any closed curve. The next two theorems say that, under certain conditions, source-free vector fields are precisely the vector fields with zero divergence.<\/p>\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>divergence of a source-free vector field<\/h3>\r\nIf [latex]{\\bf{F}}=\\langle{P},Q\\rangle[\/latex] is a source-free continuous vector field with differentiable component functions, then [latex]\\text{div }{\\bf{F}}=0[\/latex].\r\n\r\n<\/section><section class=\"textbox connectIt\">\r\n<p style=\"text-align: center;\"><strong>Proof<\/strong><\/p>\r\n\r\n\r\n<hr \/>\r\n<p id=\"fs-id1167793628157\">Since\u00a0[latex]{\\bf{F}}[\/latex]\u00a0is source free, there is a function [latex]g(x, y)[\/latex] with [latex]g_y=P[\/latex] and [latex]-g_x=Q[\/latex]. Therefore, [latex]{\\bf{F}}=\\langle{g}_y,-g_x\\rangle[\/latex] and [latex]\\text{div }{\\bf{F}}=g_{yx}-g_{xy}=0[\/latex] by Clairaut\u2019s theorem.<\/p>\r\n[latex]_\\blacksquare[\/latex]\r\n\r\n<\/section>\r\n<p class=\"whitespace-normal break-words\">The converse of this theorem is true on simply connected regions, but the proof is too technical to include here. Thus, we have the following theorem, which can test whether a vector field in [latex]\\mathbb{R}^2[\/latex] is source free.<\/p>\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>divergence test for source-free vector fields<\/h3>\r\nLet [latex]{\\bf{F}}=\\langle{P},Q\\rangle[\/latex] be a continuous vector field with differentiable component functions with a domain that is simply connected. Then, [latex]\\text{div }{\\bf{F}}=0[\/latex] if and only if\u00a0[latex]{\\bf{F}}[\/latex]\u00a0is source free.\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Is field [latex]{\\bf{F}}(x,t)=\\langle{x}^2y,5-xy^2\\rangle[\/latex] source free?[reveal-answer q=\"132854489\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"132854489\"]\r\n<p id=\"fs-id1167793895785\">Note the domain of [latex]{\\bf{F}}[\/latex] is [latex]\\mathbb{R}^2[\/latex], which is simply connected. Furthermore, [latex]{\\bf{F}}[\/latex] is continuous with differentiable component functions. Therefore, we can use the divergence test for source-free vector fields to analyze [latex]{\\bf{F}}[\/latex]. The divergence of [latex]{\\bf{F}}[\/latex] is<\/p>\r\n<p style=\"text-align: center;\">[latex]\\large{\\frac{\\partial}{\\partial{x}}\\left(x^2y\\right)+\\frac{\\partial}{\\partial{y}}\\left(5-xy^2\\right)=2xy-2xy=0}[\/latex].<\/p>\r\n<p id=\"fs-id1167793579555\">Therefore, [latex]{\\bf{F}}[\/latex] is source free by the divergence test for source-free vector fields.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n<h3 class=\"text-lg font-bold text-text-100 mt-1 -mb-1.5\">Divergence and Green's Theorem<\/h3>\r\n<p id=\"fs-id1167793950331\">Recall that the flux form of Green\u2019s theorem says that<\/p>\r\n<p style=\"text-align: center;\">[latex]\\large{\\displaystyle\\oint_C{\\bf{F}}\\cdot{\\bf{N}}ds=\\displaystyle\\iint_DP_x+Q_ydA}[\/latex],<\/p>\r\n<p id=\"fs-id1167793618969\">where [latex]C[\/latex] is a simple closed curve and [latex]D[\/latex] is the region enclosed by [latex]C[\/latex]. Since [latex]P_x+Q_y=\\text{div }{\\bf{F}}[\/latex], Green\u2019s theorem is sometimes written as<\/p>\r\n<p style=\"text-align: center;\">[latex]\\large{\\displaystyle\\oint_C{\\bf{F}}\\cdot{\\bf{N}}ds=\\displaystyle\\iint_D\\text{div }{\\bf{F}}dA}[\/latex].<\/p>\r\n<p id=\"fs-id1167794168799\">Therefore, Green\u2019s theorem can be written in terms of divergence.<\/p>\r\nIf we think of divergence as a derivative of sorts, then Green\u2019s theorem says the \u201cderivative\u201d of\u00a0[latex]{\\bf{F}}[\/latex]\u00a0on a region can be translated into a line integral of\u00a0[latex]{\\bf{F}}[\/latex]\u00a0along the boundary of the region. This is analogous to the Fundamental Theorem of Calculus, in which the derivative of a function [latex]f[\/latex] on a line segment [latex][a,b][\/latex] can be translated into a statement about [latex]f[\/latex] on the boundary of [latex][a,b][\/latex]. Using divergence, we can see that Green\u2019s theorem is a higher-dimensional analog of the Fundamental Theorem of Calculus.\r\n<p id=\"fs-id1167793604548\">We can use all of what we have learned in the application of divergence.<\/p>\r\nLet\u00a0[latex]{\\bf{v}}[\/latex] be a vector field modeling the velocity of a fluid. Since the divergence of[latex]{\\bf{v}}[\/latex] at point [latex]P[\/latex] measures the \u201coutflowing-ness\u201d of the fluid at [latex]P[\/latex], [latex]\\text{div }{\\bf{v}}(P)&gt;0[\/latex] implies that more fluid is flowing out of [latex]P[\/latex] than flowing in. Similarly, [latex]\\text{div }{\\bf{v}}(P)&lt;0[\/latex] implies the more fluid is flowing in to [latex]P[\/latex] than is flowing out, and [latex]\\text{div }{\\bf{v}}(P)=0[\/latex] implies the same amount of fluid is flowing in as flowing out.\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">Suppose [latex]{\\bf{v}}(x,y)=\\langle-xy,y\\rangle[\/latex], [latex]y&gt;0[\/latex] models the flow of a fluid. Is more fluid flowing into point [latex](1, 4)[\/latex] than flowing out?[reveal-answer q=\"238742098\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"238742098\"]\r\n<p id=\"fs-id1167793912351\">To determine whether more fluid is flowing into [latex](1, 4)[\/latex] than is flowing out, we calculate the divergence of\u00a0[latex]{\\bf{v}}[\/latex]\u00a0at [latex](1, 4)[\/latex]:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\large{\\text{div }({\\bf{v}})=\\frac{\\partial}{\\partial{x}}(-xy)+\\frac{\\partial}{\\partial{y}}(y)=-y+1}[\/latex].<\/p>\r\n<p id=\"fs-id1167793424488\">To find the divergence at [latex](1, 4)[\/latex], substitute the point into the divergence: [latex]-4+1=-3[\/latex]. Since the divergence of\u00a0[latex]{\\bf{v}}[\/latex]\u00a0at [latex](1, 4)[\/latex] is negative, more fluid is flowing in than flowing out (Figure 4).<\/p>\r\n\r\n[caption id=\"attachment_1441\" align=\"aligncenter\" width=\"867\"]<img class=\"size-full wp-image-1441\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17151855\/c873a5e24f1beb8a62b6a931632bf9f238dc2475.webp\" alt=\"A vector field in two dimensions with negative divergence at (1,4). The arrows are very flat but become more vertical closer to the y axis. Above the x axis, the arrows point up and towards the y axis on either side of it. Below the x axis, the arrows point down and away from the y axis on either side of it.\" width=\"867\" height=\"497\" \/> Figure 4: Vector field [latex]v(x,y)=\u27e8\u2212xy,y\u27e9[\/latex]<br \/>has negative divergence at [latex](1,4)[\/latex].[\/caption][\/hidden-answer]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Determine divergence from the formula for a given vector field<\/li>\n<li>Determine curl from the formula for a given vector field<\/li>\n<li>Use divergence and curl to identify conservative fields<\/li>\n<\/ul>\n<\/section>\n<h2 id=\"fs-id1167793571517\" class=\"\">Divergence<\/h2>\n<p class=\"whitespace-normal break-words\">Divergence is an operation on a vector field that tells us how the field behaves toward or away from a point. Locally, the divergence of a vector field [latex]\\mathbf{F}[\/latex] in [latex]\\mathbb{R}^2[\/latex] or [latex]\\mathbb{R}^3[\/latex] at a particular point [latex]P[\/latex] is a measure of the &#8220;outflowing-ness&#8221; of the vector field at [latex]P[\/latex].<\/p>\n<p class=\"whitespace-normal break-words\">If [latex]\\mathbf{F}[\/latex] represents the velocity of a fluid, then the divergence of [latex]\\mathbf{F}[\/latex] at [latex]P[\/latex] measures the net rate of change with respect to time of the amount of fluid flowing away from [latex]P[\/latex] (the tendency of the fluid to flow &#8220;out of&#8221; [latex]P[\/latex]). If the amount of fluid flowing into [latex]P[\/latex] is the same as the amount flowing out, then the divergence at [latex]P[\/latex] is zero.<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>divergence<\/h3>\n<p id=\"fs-id1167793931072\">If [latex]{\\bf{F}}\\langle{P},Q,R\\rangle[\/latex] is a vector field in [latex]\\mathbb{R}^3[\/latex] and [latex]P_x[\/latex], [latex]Q_y[\/latex], and [latex]R_z[\/latex] all exist, then the <strong><span id=\"6f668e5d-300d-4e6c-aec3-70b59f71fe69_term265\" data-type=\"term\">divergence<\/span><\/strong> of\u00a0[latex]{\\bf{F}}[\/latex]\u00a0is defined by:<\/p>\n<p style=\"text-align: center;\">[latex]\\text{div }{\\bf{F}}=P_x+Q_y+R_z=\\dfrac{\\partial{P}}{\\partial{x}}+\\dfrac{\\partial{Q}}{\\partial{y}}+\\dfrac{\\partial{R}}{\\partial{z}}[\/latex].<\/p>\n<\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">\n<p id=\"fs-id1167793272083\">Note the divergence of a vector field is not a vector field, but a scalar function. In terms of the gradient operator [latex]\\nabla=\\left\\langle\\frac{\\partial}{\\partial{x}},\\frac{\\partial}{\\partial{y}},\\frac{\\partial}{\\partial{z}}\\right\\rangle[\/latex], divergence can be written symbolically as the dot product:<\/p>\n<p style=\"text-align: center;\">[latex]\\text{div }{\\bf{F}}=\\nabla\\cdot{\\bf{F}}[\/latex].<\/p>\n<\/section>\n<p>Note this is merely helpful notation, because the dot product of a vector of operators and a vector of functions is not meaningfully defined given our current definition of dot product. If [latex]{\\bf{F}}\\langle{P},Q\\rangle[\/latex] is a vector field in [latex]\\mathbb{R}^2[\/latex], and [latex]P_x[\/latex] and [latex]Q_y[\/latex] both exist, then the divergence of [latex]{\\bf{F}}[\/latex] is defined similarly as:<\/p>\n<p style=\"text-align: center;\">[latex]\\text{div }{\\bf{F}}=P_x+Q_y=\\dfrac{\\partial{P}}{\\partial{x}}+\\dfrac{\\partial{Q}}{\\partial{y}}=\\nabla\\cdot{\\bf{F}}[\/latex].<\/p>\n<p>To illustrate this point, consider the two vector fields in Figure 1.<\/p>\n<figure id=\"attachment_1436\" aria-describedby=\"caption-attachment-1436\" style=\"width: 794px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-1436\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17151438\/52108a91d94bdbb0791e0c04baae71a55a6f9b3d.webp\" alt=\"Two images of vector fields A and B in two dimensions. Vector field A has arrows pointing up and to the right. They do not change in size or direction. It has zero divergence. Vector field B has arrows surrounding the origin in a counterclockwise direction. The arrows are larger the closer they are to the origin. It also has zero divergence.\" width=\"794\" height=\"1103\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17151438\/52108a91d94bdbb0791e0c04baae71a55a6f9b3d.webp 794w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17151438\/52108a91d94bdbb0791e0c04baae71a55a6f9b3d-216x300.webp 216w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17151438\/52108a91d94bdbb0791e0c04baae71a55a6f9b3d-737x1024.webp 737w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17151438\/52108a91d94bdbb0791e0c04baae71a55a6f9b3d-768x1067.webp 768w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17151438\/52108a91d94bdbb0791e0c04baae71a55a6f9b3d-65x90.webp 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17151438\/52108a91d94bdbb0791e0c04baae71a55a6f9b3d-225x313.webp 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17151438\/52108a91d94bdbb0791e0c04baae71a55a6f9b3d-350x486.webp 350w\" sizes=\"auto, (max-width: 794px) 100vw, 794px\" \/><figcaption id=\"caption-attachment-1436\" class=\"wp-caption-text\">Figure 1. (a) Vector field [latex]\\langle1,2\\rangle[\/latex] has zero divergence. (b) Vector field [latex]\\langle-y,x\\rangle[\/latex] also has zero divergence.<\/figcaption><\/figure>\n<p>At any particular point, the amount flowing in is the same as the amount flowing out, so at every point the &#8220;outflowing-ness&#8221; of the field is zero. Therefore, we expect the divergence of both fields to be zero, and this is indeed the case, as:<\/p>\n<p style=\"text-align: center;\">[latex]\\text{div }(\\langle1,2\\rangle)=\\dfrac{\\partial}{\\partial{x}}(1)+\\dfrac{\\partial}{\\partial{y}}(2)=0\\text{ and div }(\\langle-y,x\\rangle)=\\dfrac{\\partial}{\\partial{x}}(-y)+\\dfrac{\\partial}{\\partial{y}}(x)=0[\/latex].<\/p>\n<p>By contrast, consider radial vector field [latex]{\\bf{R}}(x,y)=\\langle-x,-y\\rangle[\/latex] in Figure 2.<\/p>\n<figure id=\"attachment_1437\" aria-describedby=\"caption-attachment-1437\" style=\"width: 717px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-1437\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17151504\/f8745ea1aed0b6e1cf0169f5f882642308e3c174.webp\" alt=\"A vector field in two dimensions with negative divergence. The arrows point in towards the origin in a radial pattern. The closer the arrows are to the origin, the larger they are.\" width=\"717\" height=\"497\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17151504\/f8745ea1aed0b6e1cf0169f5f882642308e3c174.webp 717w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17151504\/f8745ea1aed0b6e1cf0169f5f882642308e3c174-300x208.webp 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17151504\/f8745ea1aed0b6e1cf0169f5f882642308e3c174-65x45.webp 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17151504\/f8745ea1aed0b6e1cf0169f5f882642308e3c174-225x156.webp 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17151504\/f8745ea1aed0b6e1cf0169f5f882642308e3c174-350x243.webp 350w\" sizes=\"auto, (max-width: 717px) 100vw, 717px\" \/><figcaption id=\"caption-attachment-1437\" class=\"wp-caption-text\">Figure 2. This vector field has negative divergence.<\/figcaption><\/figure>\n<p>At any given point, more fluid is flowing in than is flowing out, and therefore the \u201coutgoingness\u201d of the field is negative. We expect the divergence of this field to be negative, and this is indeed the case, as [latex]\\text{div }({\\bf{R}})=\\frac{\\partial}{\\partial{x}}(-x)+\\frac{\\partial}{\\partial{y}}(-y)=-2[\/latex].<\/p>\n<p>To get a global sense of what divergence is telling us, suppose that a vector field in [latex]\\mathbb{R}^2[\/latex] represents the velocity of a fluid. Imagine taking an elastic circle (a circle with a shape that can be changed by the vector field) and dropping it into a fluid. If the circle maintains its exact area as it flows through the fluid, then the divergence is zero. This would occur for both vector fields in Figure 1.<\/p>\n<p class=\"whitespace-normal break-words\">On the other hand, if the circle&#8217;s shape is distorted so that its area shrinks or expands, then the divergence is not zero. Imagine dropping such an elastic circle into the radial vector field in Figure 2 so that the center of the circle lands at point [latex](3, 3)[\/latex]. The circle would flow toward the origin, and as it did so the front of the circle would travel more slowly than the back, causing the circle to &#8220;scrunch&#8221; and lose area. This is how you can see a negative divergence.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">If [latex]{\\bf{F}}(x,y,z)=e^x{\\bf{i}}+yz{\\bf{j}}-y^2{\\bf{k}}[\/latex], then find the divergence of\u00a0[latex]{\\bf{F}}[\/latex]\u00a0at [latex](0, 2, -1)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q943504823\">Show Solution<\/button><\/p>\n<div id=\"q943504823\" class=\"hidden-answer\" style=\"display: none\"><\/section>\n<p id=\"fs-id1167793887560\">The divergence of\u00a0[latex]{\\bf{F}}[\/latex]\u00a0is<\/p>\n<p style=\"text-align: center;\">[latex]\\large{\\frac{\\partial}{\\partial{x}}\\left(e^x\\right)+\\frac{\\partial}{\\partial{y}}(yz)-\\frac{\\partial}{\\partial{z}}\\left(yz^2\\right)=e^x+z-2yz}[\/latex].<\/p>\n<p id=\"fs-id1167793642080\">Therefore, the divergence at [latex](0, 2, -1)[\/latex] is [latex]e^0-1+4=4[\/latex]. If\u00a0[latex]{\\bf{F}}[\/latex]\u00a0represents the velocity of a fluid, then more fluid is flowing out than flowing in at point [latex](0, 2, -1)[\/latex].<\/p>\n<\/div>\n<\/div>\n<p>One application for divergence occurs in physics when working with magnetic fields. A magnetic field is a vector field that models the influence of electric currents and magnetic materials. Physicists use divergence in Gauss&#8217;s law for magnetism, which states that if [latex]\\mathbf{B}[\/latex] is a magnetic field, then [latex]\\nabla\\cdot\\mathbf{B}=0[\/latex]; in other words, the divergence of a magnetic field is zero.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">Is it possible for [latex]{\\bf{F}}(x,y)=\\langle{x}^2y,y-xy^2\\rangle[\/latex] to be a magnetic field?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q234057932\">Show Solution<\/button><\/p>\n<div id=\"q234057932\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167794064443\">If\u00a0[latex]{\\bf{F}}[\/latex]\u00a0were magnetic, then its divergence would be zero. The divergence of\u00a0[latex]{\\bf{F}}[\/latex]\u00a0is<\/p>\n<p style=\"text-align: center;\">[latex]\\large{\\frac{\\partial}{\\partial{x}}\\left(x^2y\\right)+\\frac{\\partial}{\\partial{y}}\\left(y-xy^2\\right)=2xy+1-2xy=1}[\/latex].<\/p>\n<p id=\"fs-id1167794059031\">and therefore\u00a0[latex]{\\bf{F}}[\/latex]\u00a0cannot model a magnetic field (Figure 3).<\/p>\n<figure id=\"attachment_1440\" aria-describedby=\"caption-attachment-1440\" style=\"width: 717px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-1440\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17151704\/8012ae03cada4cfa516ccb68746e2005e677e07c.webp\" alt=\"A vector field in two dimensions with divergence equal to 1. The arrows are quite flat near the x axis and vertical near the y axis. They seem to asymptotically approach the axes in quadrants 2 and 4, pointing up and to the right in quadrant 2 and down and to the left in quadrant 4. In quadrant 1, they start by pointing up and to the right close to the y axis, but they soon shift to pointing down and to the right. In quadrant 3, they start by pointing down and to the left close to the y axis, bu they soon shift to pointing up and to the left. The closer the arrows are to the origin, the shorter they are.\" width=\"717\" height=\"572\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17151704\/8012ae03cada4cfa516ccb68746e2005e677e07c.webp 717w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17151704\/8012ae03cada4cfa516ccb68746e2005e677e07c-300x239.webp 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17151704\/8012ae03cada4cfa516ccb68746e2005e677e07c-65x52.webp 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17151704\/8012ae03cada4cfa516ccb68746e2005e677e07c-225x179.webp 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17151704\/8012ae03cada4cfa516ccb68746e2005e677e07c-350x279.webp 350w\" sizes=\"auto, (max-width: 717px) 100vw, 717px\" \/><figcaption id=\"caption-attachment-1440\" class=\"wp-caption-text\">Figure 3:\u00a0 The divergence of vector field [latex]F(x,y)=\u27e8x^2y,y\u2212xy^2\u27e9[\/latex]<br \/>is one, so it cannot model a magnetic field.<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<p class=\"whitespace-normal break-words\">Another application for divergence is detecting whether a field is source free. Recall that a source-free field is a vector field that has a stream function; equivalently, a source-free field is a field with a flux that is zero along any closed curve. The next two theorems say that, under certain conditions, source-free vector fields are precisely the vector fields with zero divergence.<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>divergence of a source-free vector field<\/h3>\n<p>If [latex]{\\bf{F}}=\\langle{P},Q\\rangle[\/latex] is a source-free continuous vector field with differentiable component functions, then [latex]\\text{div }{\\bf{F}}=0[\/latex].<\/p>\n<\/section>\n<section class=\"textbox connectIt\">\n<p style=\"text-align: center;\"><strong>Proof<\/strong><\/p>\n<hr \/>\n<p id=\"fs-id1167793628157\">Since\u00a0[latex]{\\bf{F}}[\/latex]\u00a0is source free, there is a function [latex]g(x, y)[\/latex] with [latex]g_y=P[\/latex] and [latex]-g_x=Q[\/latex]. Therefore, [latex]{\\bf{F}}=\\langle{g}_y,-g_x\\rangle[\/latex] and [latex]\\text{div }{\\bf{F}}=g_{yx}-g_{xy}=0[\/latex] by Clairaut\u2019s theorem.<\/p>\n<p>[latex]_\\blacksquare[\/latex]<\/p>\n<\/section>\n<p class=\"whitespace-normal break-words\">The converse of this theorem is true on simply connected regions, but the proof is too technical to include here. Thus, we have the following theorem, which can test whether a vector field in [latex]\\mathbb{R}^2[\/latex] is source free.<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>divergence test for source-free vector fields<\/h3>\n<p>Let [latex]{\\bf{F}}=\\langle{P},Q\\rangle[\/latex] be a continuous vector field with differentiable component functions with a domain that is simply connected. Then, [latex]\\text{div }{\\bf{F}}=0[\/latex] if and only if\u00a0[latex]{\\bf{F}}[\/latex]\u00a0is source free.<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Is field [latex]{\\bf{F}}(x,t)=\\langle{x}^2y,5-xy^2\\rangle[\/latex] source free?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q132854489\">Show Solution<\/button><\/p>\n<div id=\"q132854489\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793895785\">Note the domain of [latex]{\\bf{F}}[\/latex] is [latex]\\mathbb{R}^2[\/latex], which is simply connected. Furthermore, [latex]{\\bf{F}}[\/latex] is continuous with differentiable component functions. Therefore, we can use the divergence test for source-free vector fields to analyze [latex]{\\bf{F}}[\/latex]. The divergence of [latex]{\\bf{F}}[\/latex] is<\/p>\n<p style=\"text-align: center;\">[latex]\\large{\\frac{\\partial}{\\partial{x}}\\left(x^2y\\right)+\\frac{\\partial}{\\partial{y}}\\left(5-xy^2\\right)=2xy-2xy=0}[\/latex].<\/p>\n<p id=\"fs-id1167793579555\">Therefore, [latex]{\\bf{F}}[\/latex] is source free by the divergence test for source-free vector fields.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h3 class=\"text-lg font-bold text-text-100 mt-1 -mb-1.5\">Divergence and Green&#8217;s Theorem<\/h3>\n<p id=\"fs-id1167793950331\">Recall that the flux form of Green\u2019s theorem says that<\/p>\n<p style=\"text-align: center;\">[latex]\\large{\\displaystyle\\oint_C{\\bf{F}}\\cdot{\\bf{N}}ds=\\displaystyle\\iint_DP_x+Q_ydA}[\/latex],<\/p>\n<p id=\"fs-id1167793618969\">where [latex]C[\/latex] is a simple closed curve and [latex]D[\/latex] is the region enclosed by [latex]C[\/latex]. Since [latex]P_x+Q_y=\\text{div }{\\bf{F}}[\/latex], Green\u2019s theorem is sometimes written as<\/p>\n<p style=\"text-align: center;\">[latex]\\large{\\displaystyle\\oint_C{\\bf{F}}\\cdot{\\bf{N}}ds=\\displaystyle\\iint_D\\text{div }{\\bf{F}}dA}[\/latex].<\/p>\n<p id=\"fs-id1167794168799\">Therefore, Green\u2019s theorem can be written in terms of divergence.<\/p>\n<p>If we think of divergence as a derivative of sorts, then Green\u2019s theorem says the \u201cderivative\u201d of\u00a0[latex]{\\bf{F}}[\/latex]\u00a0on a region can be translated into a line integral of\u00a0[latex]{\\bf{F}}[\/latex]\u00a0along the boundary of the region. This is analogous to the Fundamental Theorem of Calculus, in which the derivative of a function [latex]f[\/latex] on a line segment [latex][a,b][\/latex] can be translated into a statement about [latex]f[\/latex] on the boundary of [latex][a,b][\/latex]. Using divergence, we can see that Green\u2019s theorem is a higher-dimensional analog of the Fundamental Theorem of Calculus.<\/p>\n<p id=\"fs-id1167793604548\">We can use all of what we have learned in the application of divergence.<\/p>\n<p>Let\u00a0[latex]{\\bf{v}}[\/latex] be a vector field modeling the velocity of a fluid. Since the divergence of[latex]{\\bf{v}}[\/latex] at point [latex]P[\/latex] measures the \u201coutflowing-ness\u201d of the fluid at [latex]P[\/latex], [latex]\\text{div }{\\bf{v}}(P)>0[\/latex] implies that more fluid is flowing out of [latex]P[\/latex] than flowing in. Similarly, [latex]\\text{div }{\\bf{v}}(P)<0[\/latex] implies the more fluid is flowing in to [latex]P[\/latex] than is flowing out, and [latex]\\text{div }{\\bf{v}}(P)=0[\/latex] implies the same amount of fluid is flowing in as flowing out.\n\n\n\n<section class=\"textbox example\" aria-label=\"Example\">Suppose [latex]{\\bf{v}}(x,y)=\\langle-xy,y\\rangle[\/latex], [latex]y>0[\/latex] models the flow of a fluid. Is more fluid flowing into point [latex](1, 4)[\/latex] than flowing out?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q238742098\">Show Solution<\/button><\/p>\n<div id=\"q238742098\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793912351\">To determine whether more fluid is flowing into [latex](1, 4)[\/latex] than is flowing out, we calculate the divergence of\u00a0[latex]{\\bf{v}}[\/latex]\u00a0at [latex](1, 4)[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]\\large{\\text{div }({\\bf{v}})=\\frac{\\partial}{\\partial{x}}(-xy)+\\frac{\\partial}{\\partial{y}}(y)=-y+1}[\/latex].<\/p>\n<p id=\"fs-id1167793424488\">To find the divergence at [latex](1, 4)[\/latex], substitute the point into the divergence: [latex]-4+1=-3[\/latex]. Since the divergence of\u00a0[latex]{\\bf{v}}[\/latex]\u00a0at [latex](1, 4)[\/latex] is negative, more fluid is flowing in than flowing out (Figure 4).<\/p>\n<figure id=\"attachment_1441\" aria-describedby=\"caption-attachment-1441\" style=\"width: 867px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-1441\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17151855\/c873a5e24f1beb8a62b6a931632bf9f238dc2475.webp\" alt=\"A vector field in two dimensions with negative divergence at (1,4). The arrows are very flat but become more vertical closer to the y axis. Above the x axis, the arrows point up and towards the y axis on either side of it. Below the x axis, the arrows point down and away from the y axis on either side of it.\" width=\"867\" height=\"497\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17151855\/c873a5e24f1beb8a62b6a931632bf9f238dc2475.webp 867w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17151855\/c873a5e24f1beb8a62b6a931632bf9f238dc2475-300x172.webp 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17151855\/c873a5e24f1beb8a62b6a931632bf9f238dc2475-768x440.webp 768w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17151855\/c873a5e24f1beb8a62b6a931632bf9f238dc2475-65x37.webp 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17151855\/c873a5e24f1beb8a62b6a931632bf9f238dc2475-225x129.webp 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/17151855\/c873a5e24f1beb8a62b6a931632bf9f238dc2475-350x201.webp 350w\" sizes=\"auto, (max-width: 867px) 100vw, 867px\" \/><figcaption id=\"caption-attachment-1441\" class=\"wp-caption-text\">Figure 4: Vector field [latex]v(x,y)=\u27e8\u2212xy,y\u27e9[\/latex]<br \/>has negative divergence at [latex](1,4)[\/latex].<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":15,"menu_order":5,"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-778","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\/778","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":10,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/778\/revisions"}],"predecessor-version":[{"id":2067,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/778\/revisions\/2067"}],"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\/778\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=778"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=778"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=778"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=778"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}