{"id":796,"date":"2025-10-09T15:39:44","date_gmt":"2025-10-09T15:39:44","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=796"},"modified":"2025-11-13T20:47:13","modified_gmt":"2025-11-13T20:47:13","slug":"the-divergence-theorem-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/the-divergence-theorem-fresh-take\/","title":{"raw":"The Divergence Theorem: Fresh Take","rendered":"The Divergence Theorem: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Understand the divergence theorem<\/li>\r\n \t<li>Calculate flux using the divergence theorem<\/li>\r\n \t<li>Apply the divergence theorem to physical problems<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 data-type=\"title\">Overview of Theorems<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">All the major theorems we've studied\u2014from the Fundamental Theorem of Calculus through Stokes' Theorem\u2014share a common structure: they relate an integral of a \"derivative\" over a region to an integral over the boundary of that region. Understanding this pattern reveals the deep unity underlying these seemingly different results.<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>The Common Pattern<\/strong><\/p>\r\n<p class=\"font-claude-response-body whitespace-pre-wrap break-words\">Each theorem follows the template: [latex]\\text{(Integral of derivative over region)} = \\text{(Integral over boundary)}[\/latex]<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>The Hierarchy<\/strong><\/p>\r\n\r\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>Fundamental Theorem of Calculus<\/strong>: [latex]\\int_a^b f'(x)\\,dx = f(b) - f(a)[\/latex]\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Derivative: [latex]f'[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Region: line segment [latex][a,b][\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Boundary: endpoints [latex]a[\/latex] and [latex]b[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Fundamental Theorem for Line Integrals<\/strong>: [latex]\\int_C \\nabla f \\cdot d\\mathbf{r} = f(P_1) - f(P_0)[\/latex]\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Derivative: gradient [latex]\\nabla f[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Region: curve [latex]C[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Boundary: endpoints [latex]P_0[\/latex] and [latex]P_1[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Green's Theorem (Circulation Form)<\/strong>: [latex]\\iint_D (Q_x - P_y)\\,dA = \\int_C \\mathbf{F} \\cdot d\\mathbf{r}[\/latex]\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Derivative: curl ([latex]Q_x - P_y = \\text{curl } \\mathbf{F} \\cdot \\mathbf{k}[\/latex])<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Region: planar region [latex]D[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Boundary: closed curve [latex]C[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Green's Theorem (Flux Form)<\/strong>: [latex]\\iint_D (P_x + Q_y)\\,dA = \\int_C \\mathbf{F} \\cdot \\mathbf{N}\\,ds[\/latex]\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Derivative: divergence ([latex]P_x + Q_y = \\text{div } \\mathbf{F}[\/latex])<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Region: planar region [latex]D[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Boundary: closed curve [latex]C[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Stokes' Theorem<\/strong>: [latex]\\iint_S \\text{curl } \\mathbf{F} \\cdot d\\mathbf{S} = \\int_C \\mathbf{F} \\cdot d\\mathbf{r}[\/latex]\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Derivative: curl<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Region: surface [latex]S[\/latex] (in space, not necessarily planar)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Boundary: closed curve [latex]C[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>The Big Picture<\/strong><\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">Each theorem generalizes the previous one to higher dimensions, extending the Fundamental Theorem of Calculus from one dimension to curves, surfaces, and eventually three-dimensional regions. The \"derivative\" changes (from [latex]f'[\/latex] to gradient to curl to divergence), but the fundamental relationship between interior and boundary remains constant.<\/p>\r\n\r\n<\/div>\r\n<h2 data-type=\"title\">Stating the Divergence Theorem<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">The Divergence Theorem completes the pattern of fundamental theorems by relating a triple integral of divergence over a solid region to a flux integral over its boundary surface. It's the three-dimensional analog of Green's Theorem (flux form), extending the relationship between divergence and flux to solid regions.<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>The Core Formula<\/strong><\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">For a closed surface [latex]S[\/latex] (oriented outward) enclosing solid region [latex]E[\/latex] and vector field [latex]\\mathbf{F}[\/latex]:<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\" style=\"text-align: center;\">[latex]\\iiint_E \\text{div } \\mathbf{F}\\,dV = \\iint_S \\mathbf{F} \\cdot d\\mathbf{S}[\/latex]<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">The left side integrates divergence throughout the interior volume. The right side measures flux across the boundary surface. This remarkable connection means you can calculate either integral by computing the other.<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>Understanding the Proof<\/strong><\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">Imagine dividing solid [latex]E[\/latex] into tiny boxes. For each small box with volume [latex]\\Delta V[\/latex], the flux out of the box is approximately [latex]\\text{div } \\mathbf{F} \\cdot \\Delta V[\/latex]. When you add up fluxes over all boxes, the fluxes across shared interior faces cancel (one box's outward flux equals the adjacent box's inward flux). Only fluxes across faces on the boundary of [latex]E[\/latex] survive. As box size shrinks to zero, this sum approaches both the triple integral of divergence and the surface integral of flux.<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>Physical Interpretation<\/strong><\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">The Divergence Theorem confirms that divergence measures \"outflow per unit volume.\" For a tiny ball [latex]B_r[\/latex] of radius [latex]r[\/latex] centered at point [latex]P[\/latex]:<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\" style=\"text-align: center;\">[latex]\\text{div } \\mathbf{F}(P) = \\lim_{r \\to 0} \\frac{1}{V(B_r)} \\iint_{S_r} \\mathbf{F} \\cdot d\\mathbf{S}[\/latex]<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">Divergence at [latex]P[\/latex] equals the net outward flux per unit volume in the limit as the region shrinks to [latex]P[\/latex].<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>Strategic Use<\/strong><\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">Computing flux across a complicated closed surface? Use the Divergence Theorem to convert to a (potentially simpler) triple integral of divergence over the interior.<\/p>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Verify the divergence theorem for vector field [latex]{\\bf{F}}(x,y,z)=\\langle x+y+z,y,2x-y\\rangle[\/latex] and surface [latex]S[\/latex] given by the cylinder [latex]x^{2}+y^{2}=1[\/latex], [latex]0\\leq z\\leq3[\/latex] plus the circular top and bottom of the cylinder. Assume that [latex]S[\/latex] is positively oriented.[reveal-answer q=\"934283621\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"934283621\"]Both integrals equal\u00a0[latex]6\\pi[\/latex].[\/hidden-answer]<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the example.<center><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=8250333&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=69sACUrdM4Q&amp;video_target=tpm-plugin-mnpso8ha-69sACUrdM4Q\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/center><center>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP6.65_transcript.html\">transcript for \u201cCP 6.65\u201d here (opens in new window).<\/a><\/center><\/section>\r\n<h2 data-type=\"title\">Using the Divergence Theorem<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">The Divergence Theorem is a powerful computational tool that lets you choose between calculating a complicated flux integral over a closed surface or a (potentially simpler) triple integral of divergence over the interior. This flexibility often transforms intractable problems into manageable calculations.<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>Strategic Thinking<\/strong><\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">When faced with [latex]\\iint_S \\mathbf{F} \\cdot d\\mathbf{S}[\/latex] over a closed surface, ask: Would computing [latex]\\iiint_E \\text{div } \\mathbf{F}\\,dV[\/latex] be easier? For surfaces like cylinders or cubes with multiple pieces, the theorem eliminates the need to break the surface integral into separate pieces\u2014you just compute one triple integral instead.<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>The Zero Divergence Shortcut<\/strong><\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">Here's a remarkable time-saver: if [latex]\\mathbf{F}[\/latex] has the form [latex]\\mathbf{F} = \\langle f(y,z), g(x,z), h(x,y) \\rangle[\/latex] (where each component depends only on two variables, not the \"matching\" variable), then:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\text{div } \\mathbf{F} = 0[\/latex] automatically<\/li>\r\n \t<li class=\"whitespace-normal break-words\">By the Divergence Theorem, flux across any closed surface equals zero<\/li>\r\n<\/ul>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">No calculation needed! This pattern appears when vector fields are rotational in nature\u2014flow entering the region equals flow leaving, so net flux is zero.<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>Justifying Physical Interpretation<\/strong><\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">The Divergence Theorem confirms that divergence measures \"outflow per unit volume.\" Consider a tiny ball [latex]B_r[\/latex] centered at point [latex]P[\/latex]. The flux across its boundary [latex]S_r[\/latex] relates to divergence:<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\" style=\"text-align: center;\">[latex]\\text{div } \\mathbf{F}(P) = \\lim_{r \\to 0} \\frac{1}{V(B_r)} \\iint_{S_r} \\mathbf{F} \\cdot d\\mathbf{S}[\/latex]<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">As the ball shrinks to [latex]P[\/latex], divergence emerges as the net outward flux per unit volume\u2014the fundamental measure of how much the field \"spreads out\" from that point.<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>When to Use It<\/strong><\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">Use the Divergence Theorem when the closed surface is complicated (multiple pieces, difficult parameterization) but computing divergence and integrating over the solid is straightforward.<\/p>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Use the divergence theorem to calculate flux integral [latex]\\displaystyle\\iint_S{\\bf{F}}\\cdot d{\\bf{S}}[\/latex], where [latex]S[\/latex] is the boundary of the box given by [latex]0\\leq x\\leq2[\/latex], [latex]1\\leq y\\leq4[\/latex], [latex]0\\leq z\\leq1[\/latex], and [latex]{\\bf{F}}=\\langle x^2+yz,y-z,2x+2y+2z\\rangle[\/latex] (see the following figure).[caption id=\"attachment_5430\" align=\"aligncenter\" width=\"323\"]<img class=\"size-full wp-image-5430\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/02172609\/6.89b.jpg\" alt=\"This figure is a vector diagram in three dimensions. The box of the figure spans x from 0 to 2; y from 0 to 4; and z from 0 to 1. The vectors point up increasingly with distance from the origin; toward larger x with increasing distance from the origin; and toward smaller y values with increasing height.\" width=\"323\" height=\"302\" \/> Figure 1. The box given by [latex]0\\leq x\\leq2[\/latex], [latex]1\\leq y\\leq4[\/latex], [latex]0\\leq z\\leq1[\/latex], and [latex]{\\bf{F}}=\\langle x^2+yz,y-z,2x+2y+2z\\rangle[\/latex][\/caption][reveal-answer q=\"087582178\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"087582178\"][latex]30[\/latex].[\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">Let [latex]{\\bf{v}}=\\left\\langle\\frac{x}z,\\frac{y}z,0\\right\\rangle[\/latex] be the velocity field of a fluid. Let [latex]C[\/latex] be the solid cube given by [latex]1\\leq x\\leq4[\/latex], [latex]2\\leq y\\leq5[\/latex], [latex]1\\leq z\\leq4[\/latex], and let [latex]S[\/latex] be the boundary of this cube (see the following figure). Find the flow rate of the fluid across [latex]S[\/latex].[caption id=\"attachment_5432\" align=\"aligncenter\" width=\"485\"]<img class=\"size-full wp-image-5432\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/02172827\/6.90a.jpg\" alt=\"This is a figure of a diagram of the given vector field in three dimensions. The x components are x\/z, the y components are y\/z, and the z components are 0.\" width=\"485\" height=\"526\" \/> Figure 3. The solid cube given by [latex]1\\leq x\\leq4[\/latex], [latex]2\\leq y\\leq5[\/latex], [latex]1\\leq z\\leq4[\/latex][\/caption][reveal-answer q=\"390475920\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"390475920\"][latex]9\\ln(16)[\/latex][\/hidden-answer]<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<center><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=8250334&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=wPx5BQt9BY8&amp;video_target=tpm-plugin-fwr205nw-wPx5BQt9BY8\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/center><center>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP6.67_transcript.html\">transcript for \u201cCP 6.67\u201d here (opens in new window).<\/a><\/center><\/section>\r\n<h2 data-type=\"title\">Application to Electrostatic Fields<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">The Divergence Theorem provides a powerful framework for understanding electrostatic fields, culminating in Gauss' Law\u2014one of the fundamental laws of electromagnetism. This connection reveals how abstract mathematical theorems translate directly into physical laws governing electricity.<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>Electrostatic Fields and Inverse-Square Laws<\/strong><\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">A stationary point charge [latex]q[\/latex] at the origin creates an electrostatic field:<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\" style=\"text-align: center;\">[latex]\\mathbf{E} = \\dfrac{q}{4\\pi\\varepsilon_0} \\mathbf{F}_r[\/latex]<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">where [latex]\\mathbf{F}_r = \\frac{1}{r^2}\\left\\langle\\frac{x}{r}, \\frac{y}{r}, \\frac{z}{r}\\right\\rangle[\/latex] is a radial field and [latex]\\varepsilon_0 \\approx 8.854 \\times 10^{-12}[\/latex] F\/m is the electric constant. This field obeys an <strong>inverse-square law<\/strong>: the force at any point is inversely proportional to the square of the distance from the charge. The field points outward (unlike gravity, which points inward).<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>Key Properties of the Radial Field<\/strong><\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">The divergence of [latex]\\mathbf{F}_r[\/latex] is zero everywhere except at the origin. However, the flux of [latex]\\mathbf{F}_r[\/latex] across any closed surface [latex]S[\/latex] has a remarkable property:<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\" style=\"text-align: center;\">[latex]\\iint_S \\mathbf{F}_r \\cdot d\\mathbf{S} = \\begin{cases} 0 &amp; \\text{if origin is outside } S \\\\ 4\\pi &amp; \\text{if origin is inside } S \\end{cases}[\/latex]<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">This flux depends only on whether [latex]S[\/latex] encloses the origin, not on the shape or size of [latex]S[\/latex].<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>Gauss' Law<\/strong><\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">For an electrostatic field [latex]\\mathbf{E}[\/latex] created by charge [latex]q[\/latex], Gauss' Law states:<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\" style=\"text-align: center;\">[latex]\\iint_S \\mathbf{E} \\cdot d\\mathbf{S} = \\frac{Q}{\\varepsilon_0}[\/latex]<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">where [latex]Q[\/latex] is the total charge enclosed by [latex]S[\/latex]. The flux across a closed surface equals the enclosed charge divided by the electric constant. This law extends to multiple charges: simply sum all charges inside [latex]S[\/latex].<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>Why This Matters<\/strong><\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">Gauss' Law lets you calculate flux without explicitly knowing the electric field everywhere\u2014you only need the total enclosed charge. This demonstrates the Divergence Theorem's power in physics: relating local properties (charge distribution) to global properties (total flux).<\/p>\r\n\r\n<\/div>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Understand the divergence theorem<\/li>\n<li>Calculate flux using the divergence theorem<\/li>\n<li>Apply the divergence theorem to physical problems<\/li>\n<\/ul>\n<\/section>\n<h2 data-type=\"title\">Overview of Theorems<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">All the major theorems we&#8217;ve studied\u2014from the Fundamental Theorem of Calculus through Stokes&#8217; Theorem\u2014share a common structure: they relate an integral of a &#8220;derivative&#8221; over a region to an integral over the boundary of that region. Understanding this pattern reveals the deep unity underlying these seemingly different results.<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>The Common Pattern<\/strong><\/p>\n<p class=\"font-claude-response-body whitespace-pre-wrap break-words\">Each theorem follows the template: [latex]\\text{(Integral of derivative over region)} = \\text{(Integral over boundary)}[\/latex]<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>The Hierarchy<\/strong><\/p>\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\"><strong>Fundamental Theorem of Calculus<\/strong>: [latex]\\int_a^b f'(x)\\,dx = f(b) - f(a)[\/latex]\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Derivative: [latex]f'[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Region: line segment [latex][a,b][\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Boundary: endpoints [latex]a[\/latex] and [latex]b[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Fundamental Theorem for Line Integrals<\/strong>: [latex]\\int_C \\nabla f \\cdot d\\mathbf{r} = f(P_1) - f(P_0)[\/latex]\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Derivative: gradient [latex]\\nabla f[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Region: curve [latex]C[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Boundary: endpoints [latex]P_0[\/latex] and [latex]P_1[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Green&#8217;s Theorem (Circulation Form)<\/strong>: [latex]\\iint_D (Q_x - P_y)\\,dA = \\int_C \\mathbf{F} \\cdot d\\mathbf{r}[\/latex]\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Derivative: curl ([latex]Q_x - P_y = \\text{curl } \\mathbf{F} \\cdot \\mathbf{k}[\/latex])<\/li>\n<li class=\"whitespace-normal break-words\">Region: planar region [latex]D[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Boundary: closed curve [latex]C[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Green&#8217;s Theorem (Flux Form)<\/strong>: [latex]\\iint_D (P_x + Q_y)\\,dA = \\int_C \\mathbf{F} \\cdot \\mathbf{N}\\,ds[\/latex]\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Derivative: divergence ([latex]P_x + Q_y = \\text{div } \\mathbf{F}[\/latex])<\/li>\n<li class=\"whitespace-normal break-words\">Region: planar region [latex]D[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Boundary: closed curve [latex]C[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Stokes&#8217; Theorem<\/strong>: [latex]\\iint_S \\text{curl } \\mathbf{F} \\cdot d\\mathbf{S} = \\int_C \\mathbf{F} \\cdot d\\mathbf{r}[\/latex]\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Derivative: curl<\/li>\n<li class=\"whitespace-normal break-words\">Region: surface [latex]S[\/latex] (in space, not necessarily planar)<\/li>\n<li class=\"whitespace-normal break-words\">Boundary: closed curve [latex]C[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>The Big Picture<\/strong><\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">Each theorem generalizes the previous one to higher dimensions, extending the Fundamental Theorem of Calculus from one dimension to curves, surfaces, and eventually three-dimensional regions. The &#8220;derivative&#8221; changes (from [latex]f'[\/latex] to gradient to curl to divergence), but the fundamental relationship between interior and boundary remains constant.<\/p>\n<\/div>\n<h2 data-type=\"title\">Stating the Divergence Theorem<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">The Divergence Theorem completes the pattern of fundamental theorems by relating a triple integral of divergence over a solid region to a flux integral over its boundary surface. It&#8217;s the three-dimensional analog of Green&#8217;s Theorem (flux form), extending the relationship between divergence and flux to solid regions.<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>The Core Formula<\/strong><\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">For a closed surface [latex]S[\/latex] (oriented outward) enclosing solid region [latex]E[\/latex] and vector field [latex]\\mathbf{F}[\/latex]:<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\" style=\"text-align: center;\">[latex]\\iiint_E \\text{div } \\mathbf{F}\\,dV = \\iint_S \\mathbf{F} \\cdot d\\mathbf{S}[\/latex]<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">The left side integrates divergence throughout the interior volume. The right side measures flux across the boundary surface. This remarkable connection means you can calculate either integral by computing the other.<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>Understanding the Proof<\/strong><\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">Imagine dividing solid [latex]E[\/latex] into tiny boxes. For each small box with volume [latex]\\Delta V[\/latex], the flux out of the box is approximately [latex]\\text{div } \\mathbf{F} \\cdot \\Delta V[\/latex]. When you add up fluxes over all boxes, the fluxes across shared interior faces cancel (one box&#8217;s outward flux equals the adjacent box&#8217;s inward flux). Only fluxes across faces on the boundary of [latex]E[\/latex] survive. As box size shrinks to zero, this sum approaches both the triple integral of divergence and the surface integral of flux.<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>Physical Interpretation<\/strong><\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">The Divergence Theorem confirms that divergence measures &#8220;outflow per unit volume.&#8221; For a tiny ball [latex]B_r[\/latex] of radius [latex]r[\/latex] centered at point [latex]P[\/latex]:<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\" style=\"text-align: center;\">[latex]\\text{div } \\mathbf{F}(P) = \\lim_{r \\to 0} \\frac{1}{V(B_r)} \\iint_{S_r} \\mathbf{F} \\cdot d\\mathbf{S}[\/latex]<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">Divergence at [latex]P[\/latex] equals the net outward flux per unit volume in the limit as the region shrinks to [latex]P[\/latex].<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>Strategic Use<\/strong><\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">Computing flux across a complicated closed surface? Use the Divergence Theorem to convert to a (potentially simpler) triple integral of divergence over the interior.<\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Verify the divergence theorem for vector field [latex]{\\bf{F}}(x,y,z)=\\langle x+y+z,y,2x-y\\rangle[\/latex] and surface [latex]S[\/latex] given by the cylinder [latex]x^{2}+y^{2}=1[\/latex], [latex]0\\leq z\\leq3[\/latex] plus the circular top and bottom of the cylinder. Assume that [latex]S[\/latex] is positively oriented.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q934283621\">Show Solution<\/button><\/p>\n<div id=\"q934283621\" class=\"hidden-answer\" style=\"display: none\">Both integrals equal\u00a0[latex]6\\pi[\/latex].<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=8250333&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=69sACUrdM4Q&amp;video_target=tpm-plugin-mnpso8ha-69sACUrdM4Q\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<div style=\"text-align: center;\">You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP6.65_transcript.html\">transcript for \u201cCP 6.65\u201d here (opens in new window).<\/a><\/div>\n<\/section>\n<h2 data-type=\"title\">Using the Divergence Theorem<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">The Divergence Theorem is a powerful computational tool that lets you choose between calculating a complicated flux integral over a closed surface or a (potentially simpler) triple integral of divergence over the interior. This flexibility often transforms intractable problems into manageable calculations.<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>Strategic Thinking<\/strong><\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">When faced with [latex]\\iint_S \\mathbf{F} \\cdot d\\mathbf{S}[\/latex] over a closed surface, ask: Would computing [latex]\\iiint_E \\text{div } \\mathbf{F}\\,dV[\/latex] be easier? For surfaces like cylinders or cubes with multiple pieces, the theorem eliminates the need to break the surface integral into separate pieces\u2014you just compute one triple integral instead.<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>The Zero Divergence Shortcut<\/strong><\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">Here&#8217;s a remarkable time-saver: if [latex]\\mathbf{F}[\/latex] has the form [latex]\\mathbf{F} = \\langle f(y,z), g(x,z), h(x,y) \\rangle[\/latex] (where each component depends only on two variables, not the &#8220;matching&#8221; variable), then:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">[latex]\\text{div } \\mathbf{F} = 0[\/latex] automatically<\/li>\n<li class=\"whitespace-normal break-words\">By the Divergence Theorem, flux across any closed surface equals zero<\/li>\n<\/ul>\n<p class=\"font-claude-response-body whitespace-normal break-words\">No calculation needed! This pattern appears when vector fields are rotational in nature\u2014flow entering the region equals flow leaving, so net flux is zero.<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>Justifying Physical Interpretation<\/strong><\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">The Divergence Theorem confirms that divergence measures &#8220;outflow per unit volume.&#8221; Consider a tiny ball [latex]B_r[\/latex] centered at point [latex]P[\/latex]. The flux across its boundary [latex]S_r[\/latex] relates to divergence:<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\" style=\"text-align: center;\">[latex]\\text{div } \\mathbf{F}(P) = \\lim_{r \\to 0} \\frac{1}{V(B_r)} \\iint_{S_r} \\mathbf{F} \\cdot d\\mathbf{S}[\/latex]<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">As the ball shrinks to [latex]P[\/latex], divergence emerges as the net outward flux per unit volume\u2014the fundamental measure of how much the field &#8220;spreads out&#8221; from that point.<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>When to Use It<\/strong><\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">Use the Divergence Theorem when the closed surface is complicated (multiple pieces, difficult parameterization) but computing divergence and integrating over the solid is straightforward.<\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Use the divergence theorem to calculate flux integral [latex]\\displaystyle\\iint_S{\\bf{F}}\\cdot d{\\bf{S}}[\/latex], where [latex]S[\/latex] is the boundary of the box given by [latex]0\\leq x\\leq2[\/latex], [latex]1\\leq y\\leq4[\/latex], [latex]0\\leq z\\leq1[\/latex], and [latex]{\\bf{F}}=\\langle x^2+yz,y-z,2x+2y+2z\\rangle[\/latex] (see the following figure).<\/p>\n<figure id=\"attachment_5430\" aria-describedby=\"caption-attachment-5430\" style=\"width: 323px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-5430\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/02172609\/6.89b.jpg\" alt=\"This figure is a vector diagram in three dimensions. The box of the figure spans x from 0 to 2; y from 0 to 4; and z from 0 to 1. The vectors point up increasingly with distance from the origin; toward larger x with increasing distance from the origin; and toward smaller y values with increasing height.\" width=\"323\" height=\"302\" \/><figcaption id=\"caption-attachment-5430\" class=\"wp-caption-text\">Figure 1. The box given by [latex]0\\leq x\\leq2[\/latex], [latex]1\\leq y\\leq4[\/latex], [latex]0\\leq z\\leq1[\/latex], and [latex]{\\bf{F}}=\\langle x^2+yz,y-z,2x+2y+2z\\rangle[\/latex]<\/figcaption><\/figure>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q087582178\">Show Solution<\/button><\/p>\n<div id=\"q087582178\" class=\"hidden-answer\" style=\"display: none\">[latex]30[\/latex].<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Let [latex]{\\bf{v}}=\\left\\langle\\frac{x}z,\\frac{y}z,0\\right\\rangle[\/latex] be the velocity field of a fluid. Let [latex]C[\/latex] be the solid cube given by [latex]1\\leq x\\leq4[\/latex], [latex]2\\leq y\\leq5[\/latex], [latex]1\\leq z\\leq4[\/latex], and let [latex]S[\/latex] be the boundary of this cube (see the following figure). Find the flow rate of the fluid across [latex]S[\/latex].<\/p>\n<figure id=\"attachment_5432\" aria-describedby=\"caption-attachment-5432\" style=\"width: 485px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-5432\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/02172827\/6.90a.jpg\" alt=\"This is a figure of a diagram of the given vector field in three dimensions. The x components are x\/z, the y components are y\/z, and the z components are 0.\" width=\"485\" height=\"526\" \/><figcaption id=\"caption-attachment-5432\" class=\"wp-caption-text\">Figure 3. The solid cube given by [latex]1\\leq x\\leq4[\/latex], [latex]2\\leq y\\leq5[\/latex], [latex]1\\leq z\\leq4[\/latex]<\/figcaption><\/figure>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q390475920\">Show Solution<\/button><\/p>\n<div id=\"q390475920\" class=\"hidden-answer\" style=\"display: none\">[latex]9\\ln(16)[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=8250334&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=wPx5BQt9BY8&amp;video_target=tpm-plugin-fwr205nw-wPx5BQt9BY8\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<div style=\"text-align: center;\">You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP6.67_transcript.html\">transcript for \u201cCP 6.67\u201d here (opens in new window).<\/a><\/div>\n<\/section>\n<h2 data-type=\"title\">Application to Electrostatic Fields<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">The Divergence Theorem provides a powerful framework for understanding electrostatic fields, culminating in Gauss&#8217; Law\u2014one of the fundamental laws of electromagnetism. This connection reveals how abstract mathematical theorems translate directly into physical laws governing electricity.<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>Electrostatic Fields and Inverse-Square Laws<\/strong><\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">A stationary point charge [latex]q[\/latex] at the origin creates an electrostatic field:<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\" style=\"text-align: center;\">[latex]\\mathbf{E} = \\dfrac{q}{4\\pi\\varepsilon_0} \\mathbf{F}_r[\/latex]<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">where [latex]\\mathbf{F}_r = \\frac{1}{r^2}\\left\\langle\\frac{x}{r}, \\frac{y}{r}, \\frac{z}{r}\\right\\rangle[\/latex] is a radial field and [latex]\\varepsilon_0 \\approx 8.854 \\times 10^{-12}[\/latex] F\/m is the electric constant. This field obeys an <strong>inverse-square law<\/strong>: the force at any point is inversely proportional to the square of the distance from the charge. The field points outward (unlike gravity, which points inward).<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>Key Properties of the Radial Field<\/strong><\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">The divergence of [latex]\\mathbf{F}_r[\/latex] is zero everywhere except at the origin. However, the flux of [latex]\\mathbf{F}_r[\/latex] across any closed surface [latex]S[\/latex] has a remarkable property:<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\" style=\"text-align: center;\">[latex]\\iint_S \\mathbf{F}_r \\cdot d\\mathbf{S} = \\begin{cases} 0 & \\text{if origin is outside } S \\\\ 4\\pi & \\text{if origin is inside } S \\end{cases}[\/latex]<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">This flux depends only on whether [latex]S[\/latex] encloses the origin, not on the shape or size of [latex]S[\/latex].<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>Gauss&#8217; Law<\/strong><\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">For an electrostatic field [latex]\\mathbf{E}[\/latex] created by charge [latex]q[\/latex], Gauss&#8217; Law states:<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\" style=\"text-align: center;\">[latex]\\iint_S \\mathbf{E} \\cdot d\\mathbf{S} = \\frac{Q}{\\varepsilon_0}[\/latex]<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">where [latex]Q[\/latex] is the total charge enclosed by [latex]S[\/latex]. The flux across a closed surface equals the enclosed charge divided by the electric constant. This law extends to multiple charges: simply sum all charges inside [latex]S[\/latex].<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>Why This Matters<\/strong><\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">Gauss&#8217; Law lets you calculate flux without explicitly knowing the electric field everywhere\u2014you only need the total enclosed charge. This demonstrates the Divergence Theorem&#8217;s power in physics: relating local properties (charge distribution) to global properties (total flux).<\/p>\n<\/div>\n","protected":false},"author":15,"menu_order":27,"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-796","chapter","type-chapter","status-publish","hentry"],"part":438,"module-header":"fresh_take","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\/796","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":13,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/796\/revisions"}],"predecessor-version":[{"id":2258,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/796\/revisions\/2258"}],"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\/796\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=796"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=796"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=796"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=796"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}