Overview of Theorems

Before examining the divergence theorem, it is helpful to begin with an overview of the versions of the Fundamental Theorem of Calculus we have discussed:

1. The Fundamental Theorem of Calculus:
   
   $${\displaystyle {\int }_a^b{f}^{\prime }\left(x\right)dx=f\left(b\right)-f\left(a\right)}.$$
   
   This theorem relates the integral of derivative $f^{\prime }$ over line segment $\left[a,b\right]$ along the x-axis to a difference of $f$ evaluated on the boundary.
2. The Fundamental Theorem for Line Integrals:
   
   $${\displaystyle {\int }_C\nabla f·d\text{r}=f\left(P_1\right)-f\left(P_0\right)},$$
   
   where $P_0$ is the initial point of C and $P_1$ is the terminal point of C. The Fundamental Theorem for Line Integrals allows path C to be a path in a plane or in space, not just a line segment on the x-axis. If we think of the gradient as a derivative, then this theorem relates an integral of derivative $\nabla f$ over path C to a difference of $f$ evaluated on the boundary of C.
3. Green’s theorem, circulation form:
   
   $${\displaystyle {\iint }_D(Q_x-P_y)dA}={\displaystyle {\int }_C\text{F}·d\text{r}}.$$
   
   Since $Q_x-P_y=\text{curl}\text{F}·\text{k}$ and curl is a derivative of sorts, Green’s theorem relates the integral of derivative curlF over planar region D to an integral of F over the boundary of D.
4. Green’s theorem, flux form:
   
   $${\displaystyle {\iint }_D(P_x+Q_y)dA}={\displaystyle {\int }_C\text{F}·\text{N}ds}.$$
   
   Since $P_x+Q_y=\text{div}\text{F}$ and divergence is a derivative of sorts, the flux form of Green’s theorem relates the integral of derivative divF over planar region D to an integral of F over the boundary of D.
5. Stokes’ theorem:
   
   $${\displaystyle {\iint }_S\text{curl}\text{F}·d\text{S}}={\displaystyle {\int }_C\text{F}·d\text{r}}.$$
   
   If we think of the curl as a derivative of sorts, then Stokes’ theorem relates the integral of derivative curlF over surface S (not necessarily planar) to an integral of F over the boundary of S.

Stating the Divergence Theorem

The divergence theorem follows the general pattern of these other theorems. If we think of divergence as a derivative of sorts, then the divergence theorem relates a triple integral of derivative divF over a solid to a flux integral of F over the boundary of the solid. More specifically, the divergence theorem relates a flux integral of vector field F over a closed surface S to a triple integral of the divergence of F over the solid enclosed by S.

Figure 6.87 The divergence theorem relates a flux integral across a closed surface S to a triple integral over solid E enclosed by the surface.

Recall that the flux form of Green’s theorem states that ${\displaystyle {\iint }_D\text{div}\text{F}dA={\displaystyle {\int }_C\text{F}·\text{N}ds}}.$ Therefore, the divergence theorem is a version of Green’s theorem in one higher dimension.

The proof of the divergence theorem is beyond the scope of this text. However, we look at an informal proof that gives a general feel for why the theorem is true, but does not prove the theorem with full rigor. This explanation follows the informal explanation given for why Stokes’ theorem is true.

Proof

Let B be a small box with sides parallel to the coordinate planes inside E (Figure 6.88). Let the center of B have coordinates $\left(x,y,z\right)$ and suppose the edge lengths are $\text{Δ}x,\text{Δ}y,$ and $\text{Δ}z$ (Figure 6.88(b)). The normal vector out of the top of the box is k and the normal vector out of the bottom of the box is $\text{−}\text{k}.$ The dot product of $\text{F}=\langle P,Q,R\rangle$ with k is R and the dot product with $\text{−}\text{k}$ is $\text{−}R.$ The area of the top of the box (and the bottom of the box) $\text{Δ}S$ is $\text{Δ}x\text{Δ}y.$

Figure 6.88 (a) A small box B inside surface E has sides parallel to the coordinate planes. (b) Box B has side lengths $\text{Δ}x,\text{Δ}y,$ and $\text{Δ}z$ (c) If we look at the side view of B, we see that, since $(x,y,z)$ is the center of the box, to get to the top of the box we must travel a vertical distance of $\text{Δ}z\text{/}2$ up from $(x,y,z).$ Similarly, to get to the bottom of the box we must travel a distance $\text{Δ}z\text{/}2$ down from $(x,y,z).$

The flux out of the top of the box can be approximated by $R\left(x,y,z+\frac{\text{Δ}z}{2}\right)\text{Δ}x\text{Δ}y$ (Figure 6.88(c)) and the flux out of the bottom of the box is $\text{−}R\left(x,y,z-\frac{\text{Δ}z}{2}\right)\text{Δ}x\text{Δ}y.$ If we denote the difference between these values as $\text{Δ}R,$ then the net flux in the vertical direction can be approximated by $\text{Δ}R\text{Δ}x\text{Δ}y.$ However,

$$\text{Δ}R\text{Δ}x\text{Δ}y=\left(\frac{\text{Δ}R}{\text{Δ}z}\right)\text{Δ}x\text{Δ}y\text{Δ}z\approx \left(\frac{\partial R}{\partial z}\right)\text{Δ}V.$$

Therefore, the net flux in the vertical direction can be approximated by $\left(\frac{\partial R}{\partial z}\right)\text{Δ}V.$ Similarly, the net flux in the x-direction can be approximated by $\left(\frac{\partial P}{\partial x}\right)\text{Δ}V$ and the net flux in the y-direction can be approximated by $\left(\frac{\partial Q}{\partial y}\right)\text{Δ}V.$ Adding the fluxes in all three directions gives an approximation of the total flux out of the box:

$$\text{Total flux}\approx \left(\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z}\right)\text{Δ}V=\text{div}\text{F}\text{Δ}V.$$

This approximation becomes arbitrarily close to the value of the total flux as the volume of the box shrinks to zero.

The sum of $\text{div}\text{F}\text{Δ}V$ over all the small boxes approximating E is approximately ${\displaystyle {\iiint }_E\text{div}\text{F}dV}.$ On the other hand, the sum of $\text{div}\text{F}\text{Δ}V$ over all the small boxes approximating E is the sum of the fluxes over all these boxes. Just as in the informal proof of Stokes’ theorem, adding these fluxes over all the boxes results in the cancelation of a lot of the terms. If an approximating box shares a face with another approximating box, then the flux over one face is the negative of the flux over the shared face of the adjacent box. These two integrals cancel out. When adding up all the fluxes, the only flux integrals that survive are the integrals over the faces approximating the boundary of E. As the volumes of the approximating boxes shrink to zero, this approximation becomes arbitrarily close to the flux over S.

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Example 6.77

Verifying the Divergence Theorem

Verify the divergence theorem for vector field $\text{F}=\langle x-y,x+z,z-y\rangle$ and surface S that consists of cone $x^2+y^2=z^2,0\le z\le 1,$ and the circular top of the cone (see the following figure). Assume this surface is oriented outward.

Solution

Let E be the solid cone enclosed by S. To verify the theorem for this example, we show that ${\displaystyle {\iiint }_E\text{div}\text{F}dV=}{\displaystyle \underset{S}{\iint }\text{F}·d\text{S}}$ by calculating each integral separately.

To compute the triple integral, note that $\text{div}\text{F}=P_x+Q_y+R_z=2,$ and therefore the triple integral is

$$\begin{array}{cc}\hfill {\displaystyle {\iiint }_E\text{div}\text{F}dV}& =2{\displaystyle {\iiint }_{E}dV}\hfill \\ & =2\left(\text{volume of}E\right).\hfill \end{array}$$

The volume of a right circular cone is given by $\pi r^2\frac{h}{3}.$ In this case, $h=r=1.$ Therefore,

$${\displaystyle {\iiint }_E\text{div}\text{F}dV}=2\left(\text{volume of}E\right)=\frac{2\pi }{3}.$$

To compute the flux integral, first note that S is piecewise smooth; S can be written as a union of smooth surfaces. Therefore, we break the flux integral into two pieces: one flux integral across the circular top of the cone and one flux integral across the remaining portion of the cone. Call the circular top $S_1$ and the portion under the top $S_2.$ We start by calculating the flux across the circular top of the cone. Notice that $S_1$ has parameterization

$$\text{r}\left(u,v\right)=\langle u\text{cos}v,u\text{sin}v,1\rangle ,0\le u\le 1,0\le v\le 2\pi .$$

Then, the tangent vectors are ${\text{t}}_u=\langle \text{cos}v,\text{sin}v,0\rangle$ and ${\text{t}}_v=\langle \text{−}u\text{cos}v,u\text{sin}v,0\rangle .$ Therefore, the flux across $S_1$ is

$$\begin{array}{cc}\hfill {\displaystyle {\iint }_{S_1}\text{F}·d\text{S}}& ={\displaystyle {\int }_0^1{\displaystyle {\int }_0^{2\pi }\text{F}\left(\text{r}\left(u,v\right)\right)}}·\left({\text{t}}_u\times {\text{t}}_v\right)dA\hfill \\ & ={\displaystyle {\int }_0^1{\displaystyle {\int }_0^{2\pi }\langle u\text{cos}v-u\text{sin}v,u\text{cos}v+1,1-u\text{sin}v\rangle ·\langle 0,0,u\rangle }}dvdu\hfill \\ & ={\displaystyle {\int }_0^1{\displaystyle {\int }_0^{2\pi }u-u^2\text{sin}v}}dvdu=\pi .\hfill \end{array}$$

We now calculate the flux over $S_2.$ A parameterization of this surface is

$$\text{r}\left(u,v\right)=\langle u\text{cos}v,u\text{sin}v,u\rangle ,0\le u\le 1,0\le v\le 2\pi .$$

The tangent vectors are ${\text{t}}_u=\langle \text{cos}v,\text{sin}v,1\rangle$ and ${\text{t}}_v=\langle \text{−}u\text{sin}v,u\text{cos}v,0\rangle ,$ so the cross product is

$${\text{t}}_u\times {\text{t}}_v=\langle \text{−}u\text{cos}v,\text{−}u\text{sin}v,u\rangle .$$

Notice that the negative signs on the x and y components induce the inward orientation of the cone. Since the surface is oriented outward, we use vector ${\text{t}}_v\times {\text{t}}_u=\langle u\text{cos}v,u\text{sin}v,\text{−}u\rangle$ in the flux integral. The flux across $S_2$ is then

$$\begin{array}{cc}\hfill {\displaystyle {\iint }_{S_2}\text{F}·d\text{S}}& ={\displaystyle {\int }_0^1{\displaystyle {\int }_0^{2\pi }\text{F}\left(\text{r}\left(u,v\right)\right)}}·\left({\text{t}}_v\times {\text{t}}_u\right)dA\hfill \\ & ={\displaystyle {\int }_0^1{\displaystyle {\int }_0^{2\pi }\langle u\text{cos}v-u\text{sin}v,u\text{cos}v+u,u-\text{sin}v\rangle ·\langle u\text{cos}v,u\text{sin}v,\text{−}u\rangle }}\hfill \\ & ={\displaystyle {\int }_0^1{\displaystyle {\int }_0^{2\pi }{u}^2}}{\text{cos}}^{2}v+2u^2\text{sin}v-u^{2}dvdu=-\frac{\pi }{3}.\hfill \end{array}$$

The total flux across S is

$${\displaystyle {\iint }_{S_2}\text{F}·d\text{S}}={\displaystyle {\iint }_{S_1}\text{F}·d\text{S}}+{\displaystyle {\iint }_{S_2}\text{F}·d\text{S}}=\frac{2\pi }{3}={\displaystyle {\iiint }_E\text{div}}\text{F}dV,$$

and we have verified the divergence theorem for this example.

Checkpoint 6.65

Verify the divergence theorem for vector field $\text{F}(x,y,z)=\langle x+y+z,y,2x-y\rangle$ and surface S given by the cylinder $x^2+y^2=1,0\le z\le 3$ plus the circular top and bottom of the cylinder. Assume that S is oriented outward.

Recall that the divergence of continuous field F at point P is a measure of the “outflowing-ness” of the field at P. If F represents the velocity field of a fluid, then the divergence can be thought of as the rate per unit volume of the fluid flowing out less the rate per unit volume flowing in. The divergence theorem confirms this interpretation. To see this, let P be a point and let $B_r$ be a ball of small radius r centered at P (Figure 6.89). Let $S_r$ be the boundary sphere of $B_r.$ Since the radius is small and F is continuous, $\text{div}\text{F}\left(Q\right)\approx \text{div}\text{F}\left(P\right)$ for all other points Q in the ball. Therefore, the flux across $S_r$ can be approximated using the divergence theorem:

$${\displaystyle {\iint }_{S_r}\text{F}·d\text{S}}={\displaystyle {\iiint }_{B_r}\text{div}}\text{F}dV\approx {\displaystyle {\iiint }_{B_r}\text{div}}\text{F}\left(P\right)dV.$$

Since $\text{div}\text{F}\left(P\right)$ is a constant,

$${\displaystyle {\iiint }_{B_r}\text{div}}\text{F}\left(P\right)dV=\text{div}\text{F}\left(P\right)V\left(B_r\right).$$

Therefore, flux ${\displaystyle {\iint }_{S_r}\text{F}·d\text{S}}$ can be approximated by $\text{div}\text{F}\left(P\right)V\left(B_r\right).$ This approximation gets better as the radius shrinks to zero, and therefore

$$\text{div}\text{F}\left(P\right)=\underset{r\to 0}{\text{lim}}\frac{1}{V\left(B_r\right)}{\displaystyle {\iint }_{S_r}\text{F}·d\text{S}}.$$

This equation says that the divergence at P is the net rate of outward flux of the fluid per unit volume.

Figure 6.89 Ball $B_r$ of small radius r centered at P.

Using the Divergence Theorem

The divergence theorem translates between the flux integral of closed surface S and a triple integral over the solid enclosed by S. Therefore, the theorem allows us to compute flux integrals or triple integrals that would ordinarily be difficult to compute by translating the flux integral into a triple integral and vice versa.

Checkpoint 6.66

Use the divergence theorem to calculate flux integral ${\displaystyle {\iint }_S\text{F}·d\text{S}},$ where S is the boundary of the box given by $0\le x\le 2,1\le y\le 4,0\le z\le 1,$ and $\text{F}=\langle x^2+yz,y-z,2x+2y+2z\rangle$ (see the following figure).

Example 6.79

Applying the Divergence Theorem

Let $\text{v}=\langle -\frac{y}{z},\frac{x}{z},0\rangle$ be the velocity field of a fluid. Let C be the solid cube given by $1\le x\le 4,2\le y\le 5,1\le z\le 4,$ and let S be the boundary of this cube (see the following figure). Find the flow rate of the fluid across S.

Figure 6.90 Vector field $\text{v}=\langle -\frac{y}{z},\frac{x}{z},0\rangle .$

Solution

The flow rate of the fluid across S is ${\displaystyle {\iint }_S\text{v}·d\text{S}}.$ Before calculating this flux integral, let’s discuss what the value of the integral should be. Based on Figure 6.90, we see that if we place this cube in the fluid (as long as the cube doesn’t encompass the origin), then the rate of fluid entering the cube is the same as the rate of fluid exiting the cube. The field is rotational in nature and, for a given circle parallel to the xy-plane that has a center on the z-axis, the vectors along that circle are all the same magnitude. That is how we can see that the flow rate is the same entering and exiting the cube. The flow into the cube cancels with the flow out of the cube, and therefore the flow rate of the fluid across the cube should be zero.

To verify this intuition, we need to calculate the flux integral. Calculating the flux integral directly requires breaking the flux integral into six separate flux integrals, one for each face of the cube. We also need to find tangent vectors, compute their cross product, and use Equation 6.19. However, using the divergence theorem makes this calculation go much more quickly:

$$\begin{array}{cc}\hfill {\displaystyle {\iint }_S\text{v}·d\text{S}}& ={\displaystyle {\iiint }_C\text{div}\left(\text{v}\right)}dV\hfill \\ & ={\displaystyle {\iiint }_{C}0}dV=0.\hfill \end{array}$$

Therefore the flux is zero, as expected.

Checkpoint 6.67

Let $\text{v}=\langle \frac{x}{z},\frac{y}{z},0\rangle$ be the velocity field of a fluid. Let C be the solid cube given by $1\le x\le 4,2\le y\le 5,1\le z\le 4,$ and let S be the boundary of this cube (see the following figure). Find the flow rate of the fluid across S.

Example 6.79 illustrates a remarkable consequence of the divergence theorem. Let S be a piecewise, smooth closed surface and let F be a vector field defined on an open region containing the surface enclosed by S. If F has the form $\text{F}=\langle f\left(y,z\right),g\left(x,z\right),h\left(x,y\right)\rangle ,$ then the divergence of F is zero. By the divergence theorem, the flux of F across S is also zero. This makes certain flux integrals incredibly easy to calculate. For example, suppose we wanted to calculate the flux integral ${\displaystyle {\iint }_S\text{F}·d\text{S}}$ where S is a cube and

Calculating the flux integral directly would be difficult, if not impossible, using techniques we studied previously. At the very least, we would have to break the flux integral into six integrals, one for each face of the cube. But, because the divergence of this field is zero, the divergence theorem immediately shows that the flux integral is zero.

We can now use the divergence theorem to justify the physical interpretation of divergence that we discussed earlier. Recall that if F is a continuous three-dimensional vector field and P is a point in the domain of F, then the divergence of F at P is a measure of the “outflowing-ness” of F at P. If F represents the velocity field of a fluid, then the divergence of F at P is a measure of the net flow rate out of point P (the flow of fluid out of P less the flow of fluid in to P). To see how the divergence theorem justifies this interpretation, let $B_r$ be a ball of very small radius r with center P, and assume that $B_r$ is in the domain of F. Furthermore, assume that $B_r$ has a positive, outward orientation. Since the radius of $B_r$ is small and F is continuous, the divergence of F is approximately constant on $B_r.$ That is, if $P^{\prime }$ is any point in $B_r,$ then $\text{div}\text{F}(P)\approx \text{div}\text{F}(P^{\prime }).$ Let $S_r$ denote the boundary sphere of $B_r.$ We can approximate the flux across $S_r$ using the divergence theorem as follows:

As we shrink the radius r to zero via a limit, the quantity $\text{div}\text{F}(P)V(B_r)$ gets arbitrarily close to the flux. Therefore,

and we can consider the divergence at P as measuring the net rate of outward flux per unit volume at P. Since “outflowing-ness” is an informal term for the net rate of outward flux per unit volume, we have justified the physical interpretation of divergence we discussed earlier, and we have used the divergence theorem to give this justification.

Application to Electrostatic Fields

The divergence theorem has many applications in physics and engineering. It allows us to write many physical laws in both an integral form and a differential form (in much the same way that Stokes’ theorem allowed us to translate between an integral and differential form of Faraday’s law). Areas of study such as fluid dynamics, electromagnetism, and quantum mechanics have equations that describe the conservation of mass, momentum, or energy, and the divergence theorem allows us to give these equations in both integral and differential forms.

One of the most common applications of the divergence theorem is to electrostatic fields. An important result in this subject is Gauss’ law. This law states that if S is a closed surface in electrostatic field E, then the flux of E across S is the total charge enclosed by S (divided by an electric constant). We now use the divergence theorem to justify the special case of this law in which the electrostatic field is generated by a stationary point charge at the origin.

If $(x,y,z)$ is a point in space, then the distance from the point to the origin is $r=\sqrt{x^2+y^2+z^2}.$ Let ${\text{F}}_r$ denote radial vector field ${\text{F}}_r=\frac{1}{r^2}\langle \frac{x}{r},\frac{y}{r},\frac{z}{r}\rangle .$ The vector at a given position in space points in the direction of unit radial vector $\langle \frac{x}{r},\frac{y}{r},\frac{z}{r}\rangle$ and is scaled by the quantity $1\text{/}{r}^2.$ Therefore, the magnitude of a vector at a given point is inversely proportional to the square of the vector’s distance from the origin. Suppose we have a stationary charge of q Coulombs at the origin, existing in a vacuum. The charge generates electrostatic field E given by

where the approximation ${\epsilon }_0=8.854\times {10}^{\mathrm{-12}}$ farad (F)/m is an electric constant. (The constant ${\epsilon }_0$ is a measure of the resistance encountered when forming an electric field in a vacuum.) Notice that E is a radial vector field similar to the gravitational field described in Example 6.6. The difference is that this field points outward whereas the gravitational field points inward. Because

we say that electrostatic fields obey an inverse-square law. That is, the electrostatic force at a given point is inversely proportional to the square of the distance from the source of the charge (which in this case is at the origin). Given this vector field, we show that the flux across closed surface S is zero if the charge is outside of S, and that the flux is $q\text{/}{\epsilon }_0$ if the charge is inside of S. In other words, the flux across S is the charge inside the surface divided by constant ${\epsilon }_0.$ This is a special case of Gauss’ law, and here we use the divergence theorem to justify this special case.

To show that the flux across S is the charge inside the surface divided by constant ${\epsilon }_0,$ we need two intermediate steps. First we show that the divergence of ${\text{F}}_r$ is zero and then we show that the flux of ${\text{F}}_r$ across any smooth surface S is either zero or $4\pi .$ We can then justify this special case of Gauss’ law.

Notice that since the divergence of ${\text{F}}_r$ is zero and E is ${\text{F}}_r$ scaled by a constant, the divergence of electrostatic field E is also zero (except at the origin).

In other words, this theorem says that the flux of ${\text{F}}_r$ across any piecewise smooth closed surface S depends only on whether the origin is inside of S.