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APEX Calculus

Gregory Hartman, Ph.D., Sean Fitzpatrick, Ph.D. (Editor), Alex Jordan, Ph.D. (Editor), Carly Vollet, M.S. (Editor)

Section 15.7 The Divergence Theorem and Stokes’ Theorem

Subsection 15.7.1 The Divergence Theorem

Theorem 15.4.20 gives the Divergence Theorem in the plane, which states that the flux of a vector field across a closed curve equals the sum of the divergences over the region enclosed by the curve. Recall that the flux was measured via a line integral, and the sum of the divergences was measured through a double integral.
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We now consider the three-dimensional version of the Divergence Theorem. It states, in words, that the flux across a closed surface equals the sum of the divergences over the domain enclosed by the surface. Since we are in space (versus the plane), we measure flux via a surface integral, and the sums of divergences will be measured through a triple integral.
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Figure 15.7.1. Introducing the Divergenge Theorem
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Aside

Example 15.7.3. Using the Divergence Theorem in space.

Let \(D\) be the domain in space bounded by the planes \(z=0\) and \(z=2x\text{,}\) along with the cylinder \(x=1-y^2\text{,}\) as graphed in Figure 15.7.4, let \(\surfaceS\) be the boundary of \(D\text{,}\) and let \(\vec F = \langle x+y,y^2, 2z\rangle\text{.}\)
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The \(x\text{,}\) \(y\text{,}\) and \(z\) axes are plotted in three dimensions. The \(z\) axis points up, the \(x\) axis points back and to the left, and the \(y\) axis points toward us and to the left.
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The surface consists of three components:
  • In the \(xy\) plane is the region bounded by the \(y\) axis and the parabola \(x=1-y^2\text{.}\)
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    The parabola has its vertex at \((1,0,0)\) and it meets the \(y\) axis at \((0,1,0)\) and \((0,-1,0)\)
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  • Another parabolic region lies in the plane \(z=2x\text{.}\) This plane intersects the \(y\) axis along the segment \(-1\leq y\leq 1\text{,}\) which is the common bottom of the two parabolic regions; it forms a sharp edge where the two planes meet.
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    The parabolic boundary of this region is the curve given by \(\vec{r}(t) = \la 1-t^2, t, 2-2t^2\ra\text{.}\) This is a parabola in the plane \(z=2x\) that lies directly above the parabola \(x=1-y^2\) in the \(xy\) plane.
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  • The remaining surface is the partion of the parabolic cylinder \(x=1-y^2\) that lies between the planes \(z=0\) and \(z=2x\text{.}\)
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Overall, the surfaces form a wedge shape similar to a segment that might be cut from a tree being chopped down.
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Figure 15.7.4. The surfaces used in Example 15.7.3
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Verify the Divergence Theorem by finding the total outward flux of \(\vec F\) across \(\surfaceS\text{,}\) and show this is equal to \(\iiint_D \divv\vec F\, dV\text{.}\)
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Solution.
The surface \(\surfaceS\) is piecewise smooth, comprising surfaces \(\surfaceS_1\text{,}\) which is part of the plane \(z=2x\text{,}\) surface \(\surfaceS_2\text{,}\) which is part of the cylinder \(x=1-y^2\text{,}\) and surface \(\surfaceS_3\text{,}\) which is part of the plane \(z=0\text{.}\) To find the total outward flux across \(\surfaceS\text{,}\) we need to compute the outward flux across each of these three surfaces.
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We leave it to the reader to confirm that surfaces \(\surfaceS_1\text{,}\) \(\surfaceS_2\) and \(\surfaceS_3\) can be parametrized by \(\vec r_1\text{,}\) \(\vec r_2\) and \(\vec r_3\) respectively as
\begin{align*} \vec r_1(u,v) \amp = \la v(1-u^2), u, 2v(1-u^2)\ra,\\ \vec r_2(u,v) \amp = \la (1-u^2), u, 2v(1-u^2)\ra,\\ \vec r_3(u,v) \amp = \la v(1-u^2), u, 0\ra\text{,} \end{align*}
where \(-1\leq u\leq 1\) and \(0\leq v\leq 1\) for all three functions.
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We compute a unit normal vector \(\vec n\) for each as \(\frac{\vec r_u\times\vec r_v}{\snorm{\vec r_u\times\vec r_v}}\text{,}\) though recall that as we are integrating \(\vec F\cdot \vec n\, dS\text{,}\) we actually only use \(\vec r_u\times\vec r_v\text{.}\) Finally, in previous flux computations, it did not matter which direction \(\vec n\) pointed as long as we made note of its direction. When using the Divergence Theorem, we need \(\vec n\) to point to the outside of the closed surface, so in practice this means we’ll either use \(\vec r_u\times\vec r_v\) or \(\vec r_v\times\vec r_u\text{,}\) depending on which points outside of the closed surface \(\surfaceS\text{.}\)
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We leave it to the reader to confirm the following cross products and integrations are correct.
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For \(\surfaceS_1\text{,}\) we need to use \(\vec r_{1v}\times\vec r_{1u} = \langle 2(u^2-1),0,1-u^2\rangle\text{.}\) (Note the \(z\)-component is nonnegative as \(u\leq 1\text{,}\) therefore this vector always points up, meaning to the outside, of \(\surfaceS\text{.}\)) The flux across \(\surfaceS_1\) is:
\begin{align*} \text{ Flux across }\surfaceS_1:\, \amp = \iint_{\surfaceS_1} \vec F\cdot \vec n_1\, dS\\ \amp = \int_0^1\int_{-1}^1 \vec F\big(\vec r_1(u,v)\big)\cdot \big(\vec r_{1v}\times\vec r_{1u}\big)\, du\, dv\\ \amp = \int_0^1\int_{-1}^1 \la v(1-u^2)+u, u^2,4v(1-u^2)\ra \cdot \la 2(u^2-1),0,1-u^2\ra\, du\, dv\\ \amp = \int_0^1\int_{-1}^1 \big(2u^4v+2u^3-4u^2v-2u+2v\big)\, du\, dv\\ \amp = \frac{16}{15}\text{.} \end{align*}
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For \(\surfaceS_2\text{,}\) we use \(\vec r_{2u}\times\vec r_{2v} = \langle 2(1-u^2), 4u(1-u^2),0\rangle\text{.}\) (Note the \(x\)-component is always nonnegative, meaning this vector points outside \(\surfaceS\text{.}\)) The flux across \(\surfaceS_2\) is:
\begin{align*} \text{ Flux across } \surfaceS_2:\, \amp = \iint_{\surfaceS_2} \vec F\cdot \vec n_2\, dS\\ \amp = \int_0^1\int_{-1}^1 \vec F\big(\vec r_2(u,v)\big)\cdot \big(\vec r_{2u}\times\vec r_{2v}\big)\, du\, dv\\ \amp = \int_0^1\int_{-1}^1 \la 1-u^2+u, u^2, 4v(1-u^2)\ra \cdot \la 2(1-u^2), 4u(1-u^2),0\ra\, du\, dv\\ \amp = \int_0^1\int_{-1}^1 \big(4u^5-2u^4-2u^3+4u^2-2u-2\big)\, du\, dv\\ \amp = \frac{32}{15}\text{.} \end{align*}
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For \(\surfaceS_3\text{,}\) we use \(\vec r_{3u}\times\vec r_{3v} = \langle 0,0,u^2-1\rangle\text{.}\) (Note the \(z\)-component is never positive, meaning this vector points down, outside of \(\surfaceS\text{.}\)) The flux across \(\surfaceS_3\) is:
\begin{align*} \text{ Flux across } \surfaceS_3:\, \amp = \iint_{\surfaceS_3} \vec F\cdot \vec n_3\, dS\\ \amp = \int_0^1\int_{-1}^1 \vec F\big(\vec r_3(u,v)\big)\cdot \big(\vec r_{3u}\times\vec r_{3v}\big)\, du\, dv\\ \amp = \int_0^1\int_{-1}^1 \la v(1-u^2)+u,u^2,0\ra \cdot \la 0,0,u^2-1\ra\, du\, dv\\ \amp = \int_0^1\int_{-1}^1 0\, du\, dv\\ \amp = 0\text{.} \end{align*}
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Thus the total outward flux, measured by surface integrals across all three component surfaces of \(\surfaceS\text{,}\) is \(16/15+32/15+0 = 48/15 = 16/5 = 3.2\text{.}\) We now find the total outward flux by integrating \(\divv \vec F\) over \(D\text{.}\)
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Following the steps outlined in Section 14.6, we see the bounds of \(x\text{,}\) \(y\) and \(z\) can be set as (thinking “surface to surface, curve to curve, point to point”):
\begin{equation*} 0\leq z\leq 2x; 0\leq x\leq 1-y^2; -1\leq y\leq 1\text{.} \end{equation*}
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With \(\divv \vec F = 1+2y+2 = 2y+3\text{,}\) we find the total outward flux of \(\vec F\) over \(\surfaceS\) as:
\begin{equation*} \text{ Flux = } \iiint_D\divv \vec F\, dV = \int_{-1}^1\int_0^{1-y^2}\int_0^{2x}\big(2y+3\big)\, dz\, dx\, dy = 16/5\text{,} \end{equation*}
the same result we obtained previously.
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In Example 15.7.3 we see that the total outward flux of a vector field across a closed surface can be found two different ways because of the Divergence Theorem. One computation took far less work to obtain. In that particular case, since \(\surfaceS\) was comprised of three separate surfaces, it was far simpler to compute one triple integral than three surface integrals (each of which required partial derivatives and a cross product). In practice, if outward flux needs to be measured, one would choose only one method. We will use both methods in this section simply to reinforce the truth of the Divergence Theorem.
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We practice again in the following example.
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Example 15.7.5. Using the Divergence Theorem in space.

Let \(\surfaceS\) be the surface formed by the paraboloid \(z=1-x^2-y^2\text{,}\) \(z\geq 0\text{,}\) and the unit disk centered at the origin in the \(xy\)-plane, graphed in Figure 15.7.6, and let \(\vec F = \langle 0,0,z\rangle\text{.}\) (This surface and vector field were used in Example 15.6.10.)
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This is the same image used in Figure 15.6.11. A circular paraboloid has its vertex at \((0,0,1)\) on the \(z\) axis and opens downward, meeting the \(xy\) plane along the unit circle.
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The interior of the unit circle is a disk that serves as a cap at the bottom of the paraboloid.
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Figure 15.7.6. The surfaces used in Example 15.7.5
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Verify the Divergence Theorem; find the total outward flux across \(\surfaceS\) and evaluate the triple integral of \(\divv \vec F\text{,}\) showing that these two quantities are equal.
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Solution.
We find the flux across \(\surfaceS\) first. As \(\surfaceS\) is piecewise-smooth, we decompose it into smooth components \(\surfaceS_1\text{,}\) the disk, and \(\surfaceS_2\text{,}\) the paraboloid, and find the flux across each.
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In Example 15.6.10, we found the flux across \(\surfaceS_1\) is 0. We also found that the flux across \(\surfaceS_2\) is \(\pi/2\text{.}\) (In that example, the normal vector had a positive \(z\) component hence was an outer normal.) Thus the total outward flux is \(0+\pi/2 = \pi/2\text{.}\)
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We now compute \(\iiint_D \divv \vec F\, dV\text{.}\) We can describe \(D\) as the domain bounded by (think “surface to surface, curve to curve, point to point”):
\begin{equation*} 0\leq z\leq 1-x^2-y^2, -\sqrt{1-x^2}\leq y\leq \sqrt{1-x^2}, -1\leq x\leq 1\text{.} \end{equation*}
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This description of \(D\) is not very easy to integrate. With polar, we can do better. Let \(R\) represent the unit disk, which can be described in polar simply as \(r\text{,}\) where \(0\leq r\leq 1\) and \(0\leq \theta\leq 2\pi\text{.}\) With \(x=r\cos \theta\) and \(y=r\sin\theta\text{,}\) the surface \(\surfaceS_2\) becomes
\begin{equation*} z=1-x^2-y^2 \Rightarrow 1-(r\cos\theta)^2-(r\sin\theta)^2 \Rightarrow 1-r^2\text{.} \end{equation*}
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Thus \(D\) can be described as the domain bounded by:
\begin{equation*} 0\leq z\leq 1-r^2, 0\leq r\leq 1, 0\leq \theta\leq 2\pi\text{.} \end{equation*}
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With \(\divv \vec F = 1\text{,}\) we can integrate, recalling that \(dV = r\, dz\, dr\, d\theta\text{:}\)
\begin{equation*} \iiint_D\divv \vec F\, dV = \int_0^{2\pi}\int_0^1\int_0^{1-r^2} r\, dz\, dr\, d\theta = \frac{\pi}2\text{,} \end{equation*}
which matches our flux computation above.
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Example 15.7.7. A “paradox” of the Divergence Theorem and Gauss’s Law.

The magnitude of many physical quantities (such as light intensity or electromagnetic and gravitational forces) follow an “inverse square law”: the magnitude of the quantity at a point is inversely proportional to the square of the distance to the source of the quantity.
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Let a point light source be placed at the origin and let \(\vec F\) be the vector field which describes the intensity and direction of the emanating light. At a point \((x,y,z)\text{,}\) the unit vector describing the direction of the light passing through that point is \(\langle x,y,z\rangle/\sqrt{x^2+y^2+z^2}\text{.}\) As the intensity of light follows the inverse square law, the magnitude of \(\vec F\) at \((x,y,z)\) is \(k/(x^2+y^2+z^2)\) for some constant \(k\text{.}\) Taken together,
\begin{equation*} \vec F(x,y,z) = \frac{k}{(x^2+y^2+z^2)^{3/2}}\langle x,y,z\rangle\text{.} \end{equation*}
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Consider the cube, centered at the origin, with sides of length \(2a\) for some \(a \gt 0\) (hence corners of the cube lie at \((a,a,a)\text{,}\) \((-a,-a,-a)\text{,}\) etc., as shown in Figure 15.7.8). Find the flux across the six faces of the cube and compare this to \(\iint_D \divv\vec F\, dV\text{.}\)
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A cube is drawn in space, relative to three-dimensional coordinate axes. It is centered at the origin, and its faces are parallel to the coordinate planes.
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One corner of the cube (the nearest one in the perspective used for the image) is labeled with the coordinates \((a,a,a)\text{.}\) The opposite corner is labeled with coordinates \((-a,-a,-a)\text{.}\)
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Figure 15.7.8. The cube used in Example 15.7.7
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Solution.
Let \(\surfaceS_1\) be the “top” face of the cube, which can be parametrized by \(\vec r(u,v) = \langle u,v,a\rangle\) for \(-a\leq u\leq a\text{,}\) \(-a\leq v\leq a\text{.}\) We leave it to the reader to confirm that \(\vec r_u\times \vec r_v = \langle 0,0,1\rangle\text{,}\) which points outside of the cube.
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The flux across this face is:
\begin{align*} \text{ Flux } \amp = \iint_{\surfaceS_1} \vec F\cdot \vec n\, dS\\ \amp = \int_{-a}^a\int_{-a}^a \vec F\big(\vec r(u,v)\big)\cdot \big(\vec r_u\times \vec r_v\big)\, du\, dv\\ \amp = \int_{-a}^a\int_{-a}^a \frac{k\ a}{(u^2+v^2+a^2)^{3/2}}\, du\, dv.\\ \end{align*}
This double integral is not trivial to compute, requiring multiple trigonometric substitutions. This example is not meant to stress integration techniques, so we leave it to the reader to confirm the result is
\begin{align*} \amp = \frac{2k\pi}3\text{.} \end{align*}
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Note how the result is independent of \(a\text{;}\) no matter the size of the cube, the flux through the top surface is always \(2k\pi/3\text{.}\)
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An argument of symmetry shows that the flux through each of the six faces is \(2k\pi/3\text{,}\) thus the total flux through the faces of the cube is \(6\times2k\pi/3 = 4k\pi\text{.}\)
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It takes a bit of algebra, but we can show that \(\divv\vec F = 0\text{.}\) Thus the Divergence Theorem would seem to imply that the total flux through the faces of the cube should be
\begin{equation*} \text{ Flux } =\iiint_D \divv \vec F\, dV = \iiint_D 0\, dV = 0\text{,} \end{equation*}
but clearly this does not match the result from above. What went wrong?
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Revisit the statement of the Divergence Theorem. One of the conditions is that the components of \(\vec F\) must be differentiable on the domain enclosed by the surface. In our case, \(\vec F\) is not differentiable at the origin — it is not even defined! As \(\vec F\) does not satisfy the conditions of the Divergence Theorem, it does not apply, and we cannot expect \(\iint_\surfaceS \vec F\cdot\vec n\, dA = \iiint_D \divv\vec F\, dV\text{.}\)
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Since \(\vec F\) is differentiable everywhere except the origin, the Divergence Theorem does apply over any domain that does not include the origin. Let \(\surfaceS_2\) be any surface that encloses the cube used before, and let \(\hat D\) be the domain between the cube and \(\surfaceS_2\text{;}\) note how \(\hat D\) does not include the origin and so the Divergence Theorem does apply over this domain. The total outward flux over \(\hat D\) is thus \(\iint_{\hat D}\divv \vec F\, dV = 0\text{,}\) which means the amount of flux coming out of \(\surfaceS_2\) is the same as the amount of flux coming out of the cube. The conclusion: the flux across any surface enclosing the origin will be \(4k\pi\text{.}\)
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This has an important consequence in electrodynamics. Let \(q\) be a point charge at the origin. The electric field generated by this point charge is
\begin{equation*} \vec E = \frac{q}{4\pi \epsilon_0}\frac{\la x,y,z\ra}{(x^2+y^2+z^2)^{3/2}}\text{,} \end{equation*}
i.e., it is \(\vec F\) with \(k = q/(4\pi \epsilon_0)\text{,}\) where \(\epsilon_0\) is a physical constant (the “permittivity of free space”). Gauss’s Law states that the outward flux of \(\vec E\) across any surface enclosing the origin is \(q/\epsilon_0\text{.}\)
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Figure 15.7.9. Exploring Gauss’s Law
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Our interest in the Divergence Theorem is twofold. First, its truth alone is interesting: to study the behavior of a vector field across a closed surface, one can examine properties of that field within the surface. Secondly, it offers an alternative way of computing flux. When there are multiple methods of computing a desired quantity, one has power to select the easiest computation as illustrated next.
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Example 15.7.10. Using the Divergence Theorem to compute flux.

Let \(\surfaceS\) be the cube bounded by the planes \(x=\pm 1\text{,}\) \(y=\pm 1\text{,}\) \(z=\pm 1\text{,}\) and let \(\vec F = \langle x^2y,2yz,x^2z^3\rangle\text{.}\) Compute the outward flux of \(\vec F\) over \(\surfaceS\text{.}\)
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Solution.
We compute \(\divv \vec F = 2xy+2z+3x^2z^2\text{.}\) By the Divergence Theorem, the outward flux is the triple integral over the domain \(D\) enclosed by \(\surfaceS\text{:}\)
\begin{equation*} \text{ Outward flux: } \int_{-1}^1\int_{-1}^1\int_{-1}^1(2xy+2z+3x^2z^2)\, dz\, dy\, dx = \frac83\text{.} \end{equation*}
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The direct flux computation requires six surface integrals, one for each face of the cube. The Divergence Theorem offers a much more simple computation.
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Subsection 15.7.2 Stokes’ Theorem

Just as the spatial Divergence Theorem of this section is an extension of the planar Divergence Theorem, Stokes’ Theorem is the spatial extension of Green’s Theorem. Recall that Green’s Theorem states that the circulation of a vector field around a closed curve in the plane is equal to the sum of the curl of the field over the region enclosed by the curve. Stokes’ Theorem effectively makes the same statement: given a closed curve that lies on a surface \(\surfaceS\text{,}\) the circulation of a vector field around that curve is the same as the sum of “the curl of the field” across the enclosed surface. We use quotes around “the curl of the field” to signify that this statement is not quite correct, as we do not sum \(\curl \vec F\text{,}\) but \(\curl \vec F\cdot\vec n\text{,}\) where \(\vec n\) is a unit vector normal to \(\surfaceS\text{.}\) That is, we sum the portion of \(\curl \vec F\) that is orthogonal to \(\surfaceS\) at a point.
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Figure 15.7.11. Introducing Stokes’ Theorem
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Green’s Theorem dictated that the curve was to be traversed counterclockwise when measuring circulation. Stokes’ Theorem will follow a right hand rule: when the thumb of one’s right hand points in the direction of \(\vec n\text{,}\) the path \(C\) will be traversed in the direction of the curling fingers of the hand (this is equivalent to traversing counterclockwise in the plane).
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In general, the best approach to evaluating the surface integral in Stokes’ Theorem is to parametrize the surface \(\surfaceS\) with a function \(\vec r(u,v)\text{.}\) We can find a unit normal vector \(\vec n\) as
\begin{equation*} \vec n = \frac{\vec r_u\times\vec r_v}{\snorm{\vec r_u\times\vec r_v}}\text{.} \end{equation*}
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Since \(dS = \snorm{\vec r_u\times\vec r_v}\, dA\text{,}\) the surface integral in practice is evaluated as
\begin{equation*} \iint_\surfaceS (\curl \vec F)\cdot (\vec r_u\times\vec r_v)\, dA\text{,} \end{equation*}
where \(\vec r_u\times\vec r_v\) may be replaced by \(\vec r_v\times\vec r_u\) to properly match the direction of this vector with the orientation of the parametrization of \(C\text{.}\)
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Example 15.7.13. Verifying Stokes’ Theorem.

Considering the planar surface \(f(x,y) = 7-2x-2y\text{,}\) let \(C\) be the curve in space that lies on this surface above the circle of radius 1 and centered at \((1,1)\) in the \(xy\)-plane, let \(\surfaceS\) be the planar region enclosed by \(C\text{,}\) as illustrated in Figure 15.7.14, and let \(\vec F = \langle x+y,2y, y^2\rangle\text{.}\) Verify Stoke’s Theorem by showing \(\oint_C \vec F\cdot \, d\vec r = \iint_\surfaceS (\curl\vec F)\cdot \vec n\, dS\text{.}\)
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The usual three-dimensional coordinate axes are shown, with the \(x\) axis toward us and to the left, the \(y\) axis toward us and to the right, and the \(z\) axis pointing up.
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A circle is drawn in the \(xy\) plane. It lies in the first quadrant; its center is at \((1,1,0)\text{,}\) and its radius is \(1\text{.}\) It intercepts the \(x\) axis at \((1,0,0)\text{,}\) and the \(y\) axis at \((0,1,0)\text{.}\)
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A plane in space is drawn in the first octant. It is shaped like a rectangle; its highest corner is on the \(z\) axis at \((0,0,7)\text{;}\) the opposite corner is lowest, at \((2,2,-1)\text{.}\)
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On the plane, a curve \(C\) is drawn. The curve is an ellipse, consisting of all points in the plane that lie above the circle in the \(xy\) plane. The surface \(\surfaceS\) for this problem is the region on and inside this ellipse.
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Figure 15.7.14. As given in Example 15.7.13, the surface \(\surfaceS\) is the portion of the plane bounded by the curve
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Solution.
We begin by parametrizing \(C\) and then find the circulation. A unit circle centered at \((1,1)\) can be parametrized with \(x=\cos(t)+1\text{,}\) \(y=\sin(t)+1\) on \(0\leq t\leq 2\pi\text{;}\) to put this curve on the surface \(f\text{,}\) make the \(z\) component equal \(f(x,y)\text{:}\) \(z = 7-2(\cos(t)+1)-2(\sin(t)+1) = 3-2\cos(t) - 2\sin(t)\text{.}\) All together, we parametrize \(C\) with \(\vec r(t) = \la \cos(t)+1, \sin(t)+1, 3-2\cos(t)-2\sin(t)\ra\text{.}\)
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The circulation of \(\vec F\) around \(C\) is
\begin{align*} \oint_C\vec F\cdot \, d\vec r \amp = \int_0^{2\pi}\vec F\big(\vec r(t)\big)\cdot \vrp(t)\, dt\\ \amp = \int_0^{2\pi}\big(2\sin^3t-2\cos(t)\sin^2t+3\sin^2t-3\cos(t)\sin(t)\big)\, dt\\ \amp = 3\pi\text{.} \end{align*}
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We now parametrize \(\surfaceS\text{.}\) (We reuse the letter “r” for our surface as this is our custom.) Based on the parametrization of \(C\) above, we describe \(\surfaceS\) with \(\vec r(u,v) = \la v\cos(u)+1, v\sin(u)+1, 3-2v\cos(u)-2v\sin(u)\ra\text{,}\) where \(0\leq u\leq 2\pi\) and \(0\leq v\leq 1\text{.}\)
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We leave it to the reader to confirm that \(\vec r_u\times \vec r_v = \langle 2v,2v,v\rangle\text{.}\) As \(0\leq v\leq 1\text{,}\) this vector always has a non-negative \(z\)-component, which the right-hand rule requires given the orientation of \(C\) used above. We also leave it to the reader to confirm \(\curl\vec F = \langle 2y,0,-1\rangle\text{.}\)
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The surface integral of Stokes’ Theorem is thus
\begin{align*} \iint_\surfaceS (\curl\vec F)\cdot \vec n\, dS \amp = \iint_\surfaceS (\curl\vec F)\cdot (\vec r_u\times \vec r_v)\, dA\\ \amp = \int_0^1\int_0^{2\pi} \langle 2v\sin(u)+2,0,-1\rangle\cdot\langle 2v,2v,v\rangle\, du\, dv\\ \amp = 3\pi\text{,} \end{align*}
which matches our previous result.
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Figure 15.7.15. An example with Stokes’ Theorem
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One of the interesting results of Stokes’ Theorem is that if two surfaces \(\surfaceS_1\) and \(\surfaceS_2\) share the same boundary, then \(\iint_{\surfaceS_1} (\curl \vec F)\cdot \vec n\, dS = \iint_{\surfaceS_2} (\curl \vec F)\cdot \vec n\, dS\text{.}\) That is, the value of these two surface integrals is somehow independent of the interior of the surface. We demonstrate this principle in the next example.
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Example 15.7.16. Stokes’ Theorem and surfaces that share a boundary.

Let \(C\) be the curve given in Example 15.7.13 and note that it lies on the surface \(z = 6-x^2-y^2\text{.}\) Let \(\surfaceS\) be the region of this surface bounded by \(C\text{,}\) and let \(\vec F = \langle x+y,2y,y^2\rangle\) as in the previous example. Compute \(\iint_\surfaceS (\curl\vec F)\cdot \vec n\, dS\) to show it equals the result found in the previous example.
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This image has elements in common with Figure 15.7.14. We have the same three-dimensional coordinate axes, and the same circle in the \(xy\) plane, centered at \((1,1,0)\text{,}\) of radius \(1\text{.}\)
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The portion of the circular paraboloid \(z=6-x^2-y^2\) that lies in the first octant is plotted. It has its vertex at \((0,0,6)\text{,}\) and opens downward. On the surface is an ellipse. Although the surface is curved, the ellipse is the same ellipse that was drawn in Figure 15.7.14. The surface \(\surfaceS\) is the portion of the paraboloid on and inside the ellipse.
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(a)
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This image shows both the circular paraboloid from Figure 15.7.17.(a) and the plane from Figure 15.7.14.
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The two surfaces are shown intersecting along the same ellipse. This illustrates how the curve \(C\) is the boundary of both surfaces.
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(b)
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Figure 15.7.17. As given in Example 15.7.16, the surface \(\surfaceS\) is the portion of the plane bounded by the curve
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Solution.
We begin by demonstrating that \(C\) lies on the surface \(z=6-x^2-y^2\text{.}\) We can parametrize the \(x\) and \(y\) components of \(C\) with \(x=\cos(t)+1\text{,}\) \(y=\sin(t)+1\) as before. Lifting these components to the surface \(z=6-x^2-y^2\) gives the \(z\) component as \(z = 6-x^2-y^2 = 6-(\cos(t)+1)^2-(\sin(t)+1)^2 = 3-2\cos(t)-2\sin(t)\text{,}\) which is the same \(z\) component as found in Example 15.7.13. Thus the curve \(C\) lies on the surface \(z=6-x^2-y^2\text{,}\) as illustrated in Figure 15.7.17.
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Since \(C\) and \(\vec F\) are the same as in the previous example, we already know that \(\oint_C\vec F\cdot\, d\vec r = 3\pi\text{.}\) We confirm that this is also the value of \(\iint_\surfaceS (\curl\vec F)\cdot \vec n\, dS\text{.}\)
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We parametrize \(\surfaceS\) with
\begin{equation*} \vec r(u,v) = \langle v\cos(u)+1,v\sin(u)+1, 6-(v\cos(u)+1)^2-(v\sin(u)+1)^2\rangle\text{,} \end{equation*}
where \(0\leq u\leq 2\pi\) and \(0\leq v\leq 1\text{,}\) and leave it to the reader to confirm that
\begin{equation*} \vec r_u\times \vec r_v = \la 2v\big(v\cos(u)+1\big), 2v\big(v\sin(u)+1\big),v\ra\text{,} \end{equation*}
which also conforms to the right-hand rule with regard to the orientation of \(C\text{.}\) With \(\curl \vec F = \langle 2y,0,-1\rangle\) as before, we have
\begin{align*} \amp\iint_\surfaceS (\curl\vec F)\cdot \vec n\, dS\\ \quad\quad\amp = \int_0^1\int_0^{2\pi} \la 2v\sin(u)+2,0,-1\ra\cdot \la 2v\big(v\cos(u)+1\big), 2v\big(v\sin(u)+1\big),v\ra\, du\, dv \\ \quad\quad\amp =3\pi\text{.} \end{align*}
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Even though the surfaces used in this example and in Example 15.7.13 are very different, because they share the same boundary, Stokes’ Theorem guarantees they have equal “sum of curls” across their respective surfaces.
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Figure 15.7.18. Another example with Stokes’ Theorem
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Subsection 15.7.3 A Common Thread of Calculus

We have threefold interest in each of the major theorems of this chapter: the Fundamental Theorem of Line Integrals, Green’s, Stokes’ and the Divergence Theorems. First, we find the beauty of their truth interesting. Second, each provides two methods of computing a desired quantity, sometimes offering a simpler method of computation.
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There is yet one more reason of interest in the major theorems of this chapter. These important theorems also all share an important principle with the Fundamental Theorem of Calculus, introduced in Chapter 5.
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Revisit this fundamental theorem, adopting the notation used heavily in this chapter. Let \(I\) be the interval \([a,b]\) and let \(y=F(x)\) be differentiable on \(I\text{,}\) with \(F\,'(x) = f(x)\text{.}\) The Fundamental Theorem of Calculus states that
\begin{equation*} \int_I f(x)\, dx = F(b) - F(a)\text{.} \end{equation*}
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That is, the sum of the rates of change of a function \(F\) over an interval \(I\) can also be calculated with a certain sum of \(F\) itself on the boundary of \(I\) (in this case, at the points \(x=a\) and \(x=b\)).
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Each of the named theorems above can be expressed in similar terms. Consider the Fundamental Theorem of Line Integrals: given a function \(f(x,y)\text{,}\) the gradient \(\nabla f\) is a type of rate of change of \(f\text{.}\) Given a curve \(C\) with initial and terminal points \(A\) and \(B\text{,}\) respectively, this fundamental theorem states that
\begin{equation*} \int_C \nabla f\, ds = f(B) - f(A)\text{,} \end{equation*}
where again the sum of a rate of change of \(f\) along a curve \(C\) can also be evaluated by a certain sum of \(f\) at the boundary of \(C\) (i.e., the points \(A\) and \(B\)).
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Green’s Theorem is essentially a special case of Stokes’ Theorem, so we consider just Stokes’ Theorem here. Recalling that the curl of a vector field \(\vec F\) is a measure of a rate of change of \(\vec F\text{,}\) Stokes’ Theorem states that over a surface \(\surfaceS\) bounded by a closed curve \(C\text{,}\)
\begin{equation*} \iint_\surfaceS \big(\curl \vec F\big)\cdot \vec n\, dS = \oint_C \vec F\cdot d\vec r\text{,} \end{equation*}
i.e., the sum of a rate of change of \(\vec F\) can be calculated with a certain sum of \(\vec F\) itself over the boundary of \(\surfaceS\text{.}\) In this case, the latter sum is also an infinite sum, requiring an integral.
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Finally, the Divergence Theorems state that the sum of divergences of a vector field (another measure of a rate of change of \(\vec F\)) over a region can also be computed with a certain sum of \(\vec F\) over the boundary of that region. When the region is planar, the latter sum of \(\vec F\) is an integral; when the region is spatial, the latter sum of \(\vec F\) is a double integral.
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The common thread among these theorems: the sum of a rate of change of a function over a region can be computed as another sum of the function itself on the boundary of the region. While very general, this is a very powerful and important statement.
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Exercises 15.7.4 Exercises

Terms and Concepts

2.
What property of a vector field does the Divergence Theorem relate to flux?
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3.
What property of a vector field does Stokes’ Theorem relate to circulation?
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4.
Stokes’ Theorem is the spatial version of what other theorem?
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Problems

Exercise Group.
In the following exercises, a closed surface \(\surfaceS\) enclosing a domain \(D\) and a vector field \(\vec F\) are given. Verify the Divergence Theorem on \(\surfaceS\text{;}\) that is, show \(\iint_\surfaceS \vec F\cdot \vec n\, dS = \iiint_D\divv \vec F\, dV\text{.}\)
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5.
\(\surfaceS\) is the surface bounding the domain \(D\) enclosed by the plane \(z=2-x/2-2y/3\) and the coordinate planes in the first octant; \(\vec F = \langle x^2,y^2,x\rangle\text{.}\)
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A surface consisting of the four faces of a tetrahedron is plotted relative to three-dimensional coordinate axes. Each face is a triangle; there is one in each coordinate plane, and the fourth face lies in the plane \(z=2-x/2-2y/3\text{.}\)
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The plane \(z=2-x/2-2y/3\) is plotted in the first octant. It meets the coordinate axes at the points \((4,0,0)\text{,}\) \((0,3,0)\text{,}\) and \((0,0,2)\text{.}\) These are three of the four vertices of the tetrahedron; the remaining vertex is at the origin.
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6.
\(\surfaceS\) is the surface bounding the domain \(D\) enclosed by the cylinder \(x^2+y^2=1\) and the planes \(z=-3\) and \(z=3\text{;}\) \(\vec F = \langle -x,y,z\rangle\text{.}\)
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The surface consists of three parts:
  1. The circular cylinder \(x^2+y^2=1\text{,}\) which is symmetric about the \(z\) axis, from \(z=-3\) to \(z=3\text{.}\)
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  2. The disk bounded by \(x^2+y^2=1\) in the plane \(z=3\text{;}\) this is the top cap of the cylinder.
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  3. The disk bounded by \(x^2+y^2=1\) in the plane \(z=-3\text{;}\) this is the bottom cap of the cylinder.
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7.
\(\surfaceS\) is the surface bounding the domain \(D\) enclosed by \(z=xy(3-x)(3-y)\) and the plane \(z=0\text{;}\) \(\vec F = \langle 3x,4y,5z+1\rangle\text{.}\)
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The graph \(z=xy(3-x)(3-y)\) is plotted relative to three-dimensional coordinate axes. This surface meets the \(xy\) plane along a square, with sides given by the \(x\) and \(y\) axes, and the lines \(x=3\) and \(y=3\) in the \(xy\) plane.
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This square in the plane is part of the surface, and the other part is the graph. The graph is shaped like a steep hill, with a single peak in the center of the square.
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8.
\(\surfaceS\) is the surface composed of \(\surfaceS_1\text{,}\) the paraboloid \(z=4-x^2-y^2\) for \(z\geq 0\text{,}\) and \(\surfaceS_2\text{,}\) the disk of radius 2 centered at the origin; \(\vec F = \langle x,y,z^2\rangle\text{.}\)
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The surface consists of the circular paraboloid \(z=4-x^2-y^2\text{,}\) which opens downward from \((0,0,4)\) and meets the \(xy\) plane along the circle \(x^2+y^2=4\text{,}\) as well as the disk in the \(xy\) plane that lies on and inside the circle of intersection between the plane and the paraboloid.
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Exercise Group.
In the following exercises, a closed curve \(C\) that is the boundary of a surface \(\surfaceS\) is given along with a vector field \(\vec F\text{.}\) Verify Stokes’ Theorem on \(C\text{;}\) that is, show \(\oint_C \vec F\cdot d\vec r = \iint_\surfaceS\big(\curl \vec F\,\big)\cdot\vec n\, dS\text{.}\)
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9.
\(C\) is the curve parametrized by \(\vec r(t) = \langle \cos(t), \sin(t), 1\rangle\) and \(\surfaceS\) is the portion of \(z=x^2+y^2\) enclosed by \(C\text{;}\) \(\vec F = \langle z,-x,y\rangle\text{.}\)
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The surface \(z=x^2+y^2\) is plotted relative to a set of three-dimensional coordinate axes. The domain used for the graph is the unit disk, so that the surface is shaped like a bowl. The boundary of the surface is a circle in the plane \(z=1\text{.}\)
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10.
\(C\) is the curve parametrized by \(\vec r(t) = \langle \cos(t), \sin(t), e^{-1}\rangle\) and \(\surfaceS\) is the portion of \(z=e^{-x^2-y^2}\) enclosed by \(C\text{;}\) \(\vec F = \langle -y,x,1\rangle\text{.}\)
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The graph \(z=e^{-x^2-y^2}\) is plotted relative to a set of three-dimensional coordinate axes, for \(0\leq x^2+y^2\leq 1\text{.}\) The surface is shaped like a bell or a dome, opening downward from a vertext at \((0,0,1)\text{.}\)
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The boundary of the surface is a circle; the circle lies in the horizontal plane \(z=e^{-1}\text{,}\) has radius \(1\text{,}\) and is centered on the \(z\) axis.
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11.
\(C\) is the curve that follows the triangle with vertices at \((0,0,2)\text{,}\) \((4,0,0)\) and \((0,3,0)\text{,}\) traversing the the vertices in that order and returning to \((0,0,2)\text{,}\) and \(\surfaceS\) is the portion of the plane \(z=2-x/2-2y/3\) enclosed by \(C\text{;}\) \(\vec F = \langle y,-z,y\rangle\text{.}\)
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The curve \(C\) is plotted in space relative to three-dimensional coordinate axes. It is a triangle, given by the path from \((0,0,2)\) on the \(z\) axis, to \((4,0,0)\) on the \(x\) axis, to \((0,3,0)\) on the \(y\) axis, and then back to \((0,0,2)\text{.}\)
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The surface \(\surfaceS\) is the portion of the plane in the first octant that lies on and inside this triangle.
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12.
\(C\) is the curve whose \(x\) and \(y\) coordinates follow the parabola \(y=1-x^2\) from \(x=1\) to \(x=-1\text{,}\) then follow the line from \((-1,0)\) back to \((1,0)\text{,}\) where the \(z\) coordinates of \(C\) are determined by \(f(x,y) = 2x^2+y^2\text{,}\) and \(\surfaceS\) is the portion of \(z=2x^2+y^2\) enclosed by \(C\text{;}\) \(\vec F = \langle y^2+z,x,x^2-y\rangle\text{.}\)
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The usual three-dimensional coordinate axes are drawn. The surface is the graph \(z=2x^2+y^2\text{,}\) which is an elliptic paraboloid.
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The domain for the graph is not illustrated, but it is the region between the \(x\) axis and the parabola \(y=1-x^2\text{.}\)
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The surface is shaped like a hammock or a sail. Its boundary consists of two parabolic curves that meet at two cusps. The cusps are the highest points on the surface; they lie in the \(xz\) plane at \((1,0,2)\) and \((-1,0,2)\text{.}\)
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One part of the boundary is a parabola in the \(xz\) plane, given by \(z=2x^2\) for \(-1\leq x\leq 1\text{.}\) The other part is the portion of the elliptic paraboloid \(z=2x^2+y^2\) that lies above the curve \(y=1-x^2\) in the \(xy\) plane.
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Exercise Group.
In the following exercises, a closed surface \(\surfaceS\) and a vector field \(\vec F\) are given. Find the outward flux of \(\vec F\) over \(\surfaceS\) either through direct computation or through the Divergence Theorem.
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13.
\(\surfaceS\) is the surface formed by the intersections of \(z=0\) and \(z=(x^2-1)(y^2-1)\text{;}\) \(\vec F = \langle x^2+1,yz,xz^2\rangle\text{.}\)
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The graph \(z=(x^2-1)(y^2-1)\) is plotted relative to three-dimensional coordinate axes. This surface meets the \(xy\) plane along a square, with sides given by the lines \(x=\pm 1\) and \(y=\pm 1\) in the \(xy\) plane.
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The graph is shaped like a hill, with its peak on the \(z\) axis at \((0,0,1)\text{.}\) The boundary of the graph is the square in the \(xy\) plane.
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The surface consists of this graph and the region in the \(xy\) plane bounded by the square \(-1\leq x,y\leq 1\text{.}\)
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14.
\(\surfaceS\) is the surface formed by the intersections of the planes \(z=\frac12(3-x)\text{,}\) \(x=1\text{,}\) \(y=0\text{,}\) \(y=2\) and \(z=0\text{;}\) \(\vec F = \langle x,y^2,z\rangle\text{.}\)
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The surface \(\surfaceS\) consists of all five faces of a triangular prism. The prism is plotted relative to a set of three-dimensional coordinate axes.
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Two of the faces are triangles. These lie in the planes \(y=0\) and \(y=2\text{.}\) The vertices of the triangles are at \((1,0,0)\text{,}\) \((3,0,0)\text{,}\) and \((1,0,1)\) for the face in the \(xz\) plane, and at \((1,2,0)\text{,}\) \((3,2,0)\text{,}\) and \((1,2,1)\) for the face in the plane \(y=2\text{.}\)
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One face is in the \(xy\) plane; it is a square given by \(1\leq x\leq 3\) and \(0\leq y\leq 2\text{.}\)
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Another face is in the plane \(x=1\text{;}\) it is a rectangle with \(0\leq y\leq 2\) and \(0\leq z\leq 1\text{.}\)
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The final face is in the plane \(z=\frac12(3-x)\text{.}\) This face lies on the hypotenuse of each triangular face. It is a rectangle, with vertices \((3,0,0)\text{,}\) \((3,2,0)\text{,}\) \((1,2,1)\text{,}\) and \((1,0,1)\text{.}\)
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15.
\(\surfaceS\) is the surface formed by the intersections of the planes \(z=2y\text{,}\) \(y=4-x^2\) and \(z=0\text{;}\) \(\vec F = \langle xz,0,xz\rangle\text{.}\)
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This is another wedge-shaped surface, similar to one in Figure 15.7.4.
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There are three components to the surface.
  • The first is the region in the \(xy\) plane bounded by the \(x\) axis and the parabola \(y=4-x^2\text{.}\) It meets the \(x\) axis along the interval \(-2\leq x\leq 2\text{.}\)
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  • The second part lies in the plane \(z=2y\text{.}\) This is a planar surface, bounded below by the segment \(-2\leq x\leq 2\) along the \(x\) axis, and above by a parabola that lies in the plane \(z=2y\text{.}\)
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    This part of the surface lies directly above the part in the \(xy\) plane, and meets it along the \(x\) axis.
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  • The last part is the portion of the parabolic cylinder \(y=4-x^2\) that lies between the planes \(z=0\) and \(z=2y\text{.}\)
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16.
\(\surfaceS\) is the surface formed by the intersections of the cylinder \(z=1-x^2\) and the planes \(y=-2\text{,}\) \(y=2\) and \(z=0\text{;}\) \(\vec F = \langle 0,y^3,0\rangle\text{.}\)
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There are four components to this surface. Overall, the surface looks like the exterior of a barn or greenhouse with an arched roof.
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  • The graph \(z=1-x^2\) is a parabolic cylinder, opening downwards. It is shaped like an arch, with its peak along the line with \(z=1\) directly above the \(y\) axis. The cylinder extends to the \(xy\) plane, and is plotted for \(-2\leq y\leq 2\text{.}\)
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  • The “ends” of the surface lie in the planes \(y=2\) and \(y=-2\text{.}\) They are bounded above by the parabola \(z=1-x^2\text{,}\) and below by the \(xy\) plane, for \(-1\leq x\leq 1\text{.}\)
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  • The bottom of the surface is a rectangle in the \(xy\) plane, given by \(-1\leq x\leq 1\) and \(-2\leq y\leq 2\text{.}\) This is where the parabolic cylinder, and the two ends, meet the \(xy\) plane.
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Exercise Group.
In the following exercises, a closed curve \(C\) that is the boundary of a surface \(\surfaceS\) is given along with a vector field \(\vec F\text{.}\) Find the circulation of \(\vec F\) around \(C\) either through direct computation or through Stokes’ Theorem.
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17.
\(C\) is the curve whose \(x\)- and \(y\)-values are determined by the three sides of a triangle with vertices at \((-1,0)\text{,}\) \((1,0)\) and \((0,1)\text{,}\) traversed in that order, and the \(z\)-values are determined by the function \(z=xy\text{;}\) \(\vec F = \langle z-y^2,x,z\rangle\text{.}\)
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The graph \(z=xy\) is a hyperbolic paraboloid; it is plotted here relative to the usual three-dimensional coordinate axes, over a triangular domain.
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The surface has a twisted, wavy shape. It has one edge that lies along the \(x\) axis, for \(-1\leq x\leq 1\text{.}\) For \(0\leq x\leq 1\text{,}\) the surface bends upward, with its highest points above the line \(y=x\text{.}\) It then bends back down to meet the \(y\) axis.
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The portion for \(-1\leq x\leq 0\) is like a mirror image of the other side, but it lies below the \(xy\) plane. The lowest points lie below the line \(y=-x\text{,}\) and the surface dips down from the \(x\) and \(y\) axes to meet these points.
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The other part of the boundary begins on the \(x\) axis at \((1,0,0)\text{.}\) It curves upward in a parabolic arch, meeting the \(y\) axis at \((0,1,0)\text{.}\) The boundary then dips downward in another parabolic curve that rises up to meet the \(x\) axis again at \((-1,0,0)\text{.}\)
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18.
\(C\) is the curve whose \(x\)- and \(y\)-values are given by \(\vec r(t) = \langle 2\cos(t),2\sin(t)\rangle\) and the \(z\)-values are determined by the function \(z=x^2+y^3-3y+1\text{;}\) \(\vec F = \langle -y,x,z\rangle\text{.}\)
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This might be one of the more complicated surfaces in the book. It is plotted relative to the usual set of coordinate axes in three dimensions. There is a local minimum that lies along the \(y\) axis. The surface is bowl-shaped near the minimum.
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The boundary of the surface is wavy. There is a peak on the boundary at the point \((0,2,6)\text{.}\) If we travel along the boundary in either direction from this point, the surface dips down when \(y\) is close to \(1\text{,}\) and then climbs back up again.
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Above the \(y\) axis, the surface climbs from the minimum to a saddle point when \(y=-1\text{.}\) The surface climbs up if we move away from the saddle point in the \(x\) direction, and it dips back down if we move in the \(y\) direction.
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If the surface were flipped over, it would look like a saddle with a back rest, and a hump in the front.
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19.
\(C\) is the curve whose \(x\)- and \(y\)-values are given by \(\vec r(t) = \langle \cos(t),3\sin(t)\rangle\) and the \(z\)-values are determined by the function \(z=5-2x-y\text{;}\) \(\vec F = \langle -\frac13y,3x,\frac23y-3x\rangle\text{.}\)
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The surface is planar; it is bounded by an ellipse that lies in the plane \(z=5-2x-y\text{.}\) The ellipse is centered on the \(z\) axis. The surface consists of all points in the plane that lie on or inside this ellipse.
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20.
\(C\) is the curve whose \(x\)- and \(y\)-values are sides of the square with vertices at \((1,1)\text{,}\) \((-1,1)\text{,}\) \((-1,-1)\) and \((1,-1)\text{,}\) traversed in that order, and the \(z\)-values are determined by the function \(z=10-5x-2y\text{;}\) \(\vec F = \langle 5y^2,2y^2,y^2\rangle\text{.}\)
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The image shows a rectangle in space, viewed in perspective, relative to a set of three-dimensional coordinate axes.
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The sides of the rectangle lie above the square given by \(-1\leq x\leq 1\) and \(-1\leq y\leq 1\text{.}\) The \(z\) coordinates of the points on the rectangle are given by the plane equation \(z=10-5x-2y\text{.}\)
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Exercise Group.
The following exercises are designed to challenge your understanding and require no computation.
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21.
Let \(\surfaceS\) be any closed surface enclosing a domain \(D\text{.}\) Consider \(\vec F_1 = \langle x,0,0\rangle\) and \(\vec F_2=\langle y,y^2,z-2yz\rangle\text{.}\)
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These fields are clearly very different. Why is it that the total outward flux of each field across \(\surfaceS\) is the same?
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22.
(a)
Green’s Theorem can be used to find the area of a region enclosed by a curve by evaluating a line integral with the appropriate choice of vector field \(\vec F\text{.}\) What condition on \(\vec F\) makes this possible?
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(b)
Likewise, Stokes’ Theorem can be used to find the surface area of a region enclosed by a curve in space by evaluating a line integral with the appropriate choice of vector field \(\vec F\text{.}\) What condition on \(\vec F\) makes this possible?
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23.
The Divergence Theorem establishes equality between a particular double integral and a particular triple integral. What types of circumstances would lead one to choose to evaluate the triple integral over the double integral?
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24.
Stokes’ Theorem establishes equality between a particular line integral and a particular double integral. What types of circumstances would lead one to choose to evaluate the double integral over the line integral?
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