The Divergence Theorem: Learn It 2

Giselle Miller

The Divergence Theorem: Learn It 2

Stating the Divergence Theorem

The divergence theorem follows the general pattern of the other theorems we’ve studied. If we think of divergence as a derivative of sorts, then the divergence theorem relates a triple integral of derivative div [latex]\mathbf{F}[/latex] over a solid to a flux integral of [latex]\mathbf{F}[/latex] over the boundary of the solid.

More specifically, the divergence theorem relates a flux integral of vector field [latex]\mathbf{F}[/latex] over a closed surface [latex]S[/latex] to a triple integral of the divergence of [latex]\mathbf{F}[/latex] over the solid enclosed by [latex]S[/latex].

the divergence theorem

Let [latex]S[/latex] be a piecewise, smooth closed surface that encloses solid [latex]E[/latex] in space. Assume that [latex]S[/latex] is oriented outward, and let [latex]{\bf{F}}[/latex] be a vector field with continuous partial derivatives on an open region containing [latex]E[/latex] (Figure 1). Then:

[latex]\displaystyle\iiint_e\text{div }{\bf{F}}dV=\displaystyle\int_C{\bf{F}}\cdot d{\bf{S}}[/latex].

A diagram of a closed surface S, vector field, and solid E enclosed by the surface in three dimensions. The surface is a roughly rectangular prism with curved sides. The normal vectors stretch out and away from the surface. The arrows have negative x components and positive y and z components. — Figure 1. The divergence theorem relates a flux integral across a closed surface [latex]S[/latex] to a triple integral over solid [latex]E[/latex] enclosed by the surface.

Recall that the flux form of Green’s theorem states that [latex]\displaystyle\iint_D\text{div }{\bf{F}}dA=\displaystyle\int_C{\bf{F}}\cdot{\bf{N}}ds[/latex]. 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 [latex]B[/latex] be a small box with sides parallel to the coordinate planes inside [latex]E[/latex] (Figure 2). Let the center of [latex]B[/latex] have coordinates [latex](x, y, z)[/latex] and suppose the edge lengths are [latex]\Delta x[/latex], [latex]\Delta y[/latex], and [latex]\Delta z[/latex] (Figure 2(b)). The normal vector out of the top of the box is [latex]{\bf{k}}[/latex] and the normal vector out of the bottom of the box is [latex]-{\bf{k}}[/latex]. The dot product of [latex]{\bf{F}}=\langle P,Q,R\rangle[/latex] with [latex]{\bf{k}}[/latex] is [latex]R[/latex] and the dot product with [latex]-{\bf{k}}[/latex] is [latex]-R[/latex]. The area of the top of the box (and the bottom of the box) [latex]\Delta S[/latex] is [latex]\Delta x\Delta y[/latex].

This figure has three diagrams. The first is a surface E in three dimensions with a small box B inside it. The second just has box B. The height is labeled as delta z, the width is labeled as delta x, and the width is labeled as delta y. An arrow perpendicular to the top points up and away from the box and is labeled k. An arrow perpendicular to the bottom points down and away from the box and is labeled –k. The third diagram is a side views of box B. The center is (x, y, z), the midpoint of the side below it is (x, y, z – delta z / 2), and the midpoint of the size above it is (x, y, z + delta z / 2). The height is delta z. — Figure 2. (a) A small box [latex]B[/latex] inside surface [latex]E[/latex] has sides parallel to the coordinate planes. (b) Box [latex]B[/latex] has side lengths [latex]{\Delta x}[/latex],[latex]{\Delta y}[/latex], and [latex]{\Delta z}[/latex] (c) If we look at the side view of [latex]B[/latex], we see that, since [latex](x, y, z)[/latex] is the center of the box, to get to the top of the box we must travel a vertical distance of [latex]{\Delta z}/2[/latex] up from [latex](x, y, z)[/latex]. Similarly, to get to the bottom of the box we must travel a distance [latex]{\Delta z}/2[/latex] down from [latex](x, y, z)[/latex].

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

[latex]\large{\Delta R\Delta x\Delta y=\left(\frac{\Delta R}{\Delta z}\right)\Delta x\Delta y\Delta z\approx\left(\frac{\partial R}{\partial z}\right)\Delta V}[/latex].

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

Total flux [latex]\approx\left(\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z}\right)\Delta V=\text{div }{\bf{F}}\Delta V[/latex].

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

The sum of [latex]\text{div }{\bf{F}}\Delta V[/latex] over all the small boxes approximating [latex]E[/latex] is approximately [latex]\displaystyle\iiint_E\text{div }{\bf{F}}dV[/latex]. On the other hand, the sum of [latex]\text{div }{\bf{F}}\Delta V[/latex] over all the small boxes approximating [latex]E[/latex] 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 [latex]E[/latex]. As the volumes of the approximating boxes shrink to zero, this approximation becomes arbitrarily close to the flux over [latex]S[/latex].

[latex]_\blacksquare[/latex]

Verify the divergence theorem for vector field [latex]{\bf{F}}=\langle x-y,x+z, z-y\rangle[/latex] and surface [latex]S[/latex] that consists of cone [latex]x^{2}+y^{2}=z^{2}[/latex], [latex]0\leq z\leq1[/latex], and the circular top of the cone (see the following figure). Assume this surface is positively oriented.

This figure is a vector diagram in three dimensions. The cone x^2 + y^2 = z^2 is shown. Its point is at the origin, and it opens up. There is a cover across the top. The arrows seem to be following the shape of the cone. — Figure 3. The vector field [latex]{\bf{F}}=\langle x-y,x+z, z-y\rangle[/latex] and surface [latex]S[/latex] that consists of cone [latex]x^{2}+y^{2}=z^{2}[/latex], [latex]0\leq z\leq1[/latex], and the circular top of the cone

Let [latex]E[/latex] be the solid cone enclosed by [latex]S[/latex]. To verify the theorem for this example, we show that [latex]\displaystyle\iiint_E\text{div }{\bf{F}}dV=\displaystyle\iint_s{\bf{F}}\cdot d{\bf{S}}[/latex] by calculating each integral separately.

To compute the triple integral, note that [latex]\text{div }{\bf{F}}=P_x+Q_y+R_z=2[/latex], and therefore the triple integral is

[latex]\large{\begin{aligned} \displaystyle\iiint_E\text{div }{\bf{F}}dV&=2\displaystyle\iiint_EdV \\ &=2\text{(volume of E)} \end{aligned}}[/latex].

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

[latex]\large{\displaystyle\iiint_E\text{div }{\bf{F}}dV=2}\text{(volume of E)}=\frac{2\pi}{3}[/latex].

To compute the flux integral, first note that [latex]S[/latex] is piecewise smooth; [latex]S[/latex] 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 [latex]S_1[/latex] and the portion under the top [latex]S_2[/latex]. We start by calculating the flux across the circular top of the cone. Notice that [latex]S_1[/latex] has parameterization

[latex]\large{{\bf{r}}(u,v)=\langle u,\cos v,u\sin v,1\rangle, \ 0\leq u\leq1, \ 0\leq v\leq2\pi}[/latex].

Then, the tangent vectors are [latex]{\bf{t}}_u\langle\cos v,\sin v, 0\rangle[/latex] and [latex]{\bf{t}}_v=\langle-u\cos v,u\sin v,0\rangle[/latex]. Therefore, the flux across [latex]S_1[/latex] is

[latex]\begin{aligned} \displaystyle\iint_{S_1}{\bf{F}}\cdot d{\bf{S}}&=\displaystyle\int_0^1\displaystyle\int_0^{2\pi}{\bf{F}}({\bf{r}}(u,v))\cdot({\bf{t}}_u\times{\bf{t}}_v)dA \\ &=\displaystyle\int_0^1\displaystyle\int_0^{2\pi}\langle u\cos v-u\sin v,u\cos v+1,1-u\sin v\rangle\cdot\langle0,0,u\rangle \ dvdu \\ &=\displaystyle\int_0^1\displaystyle\int_0^{2\pi}u-u^2\sin v \ dvdu=\pi \end{aligned}[/latex].

We now calculate the flux over [latex]S_2[/latex]. A parameterization of this surface is

[latex]\large{{\bf{r}}(u,v)=\langle u,\cos v,u\sin v,1\rangle, \ 0\leq u\leq1, \ 0\leq v\leq2\pi}[/latex].

The tangent vectors are [latex]{\bf{t}}_u\langle\cos v,\sin v, 1\rangle[/latex] and [latex]{\bf{t}}_v\langle-u\sin v,u\cos v, 0\rangle[/latex], so the cross product is

[latex]\large{{\bf{t}}_u\times{\bf{t}}_v=\langle-u\cos v, -\sin v, u\rangle}[/latex].

Notice that the negative signs on the [latex]x[/latex] and [latex]y[/latex] components induce the negative (or inward) orientation of the cone. Since the surface is positively oriented, we use vector [latex]{\bf{t}}_u\times{\bf{t}}_v=\langle u\cos v,u\sin v,-u\rangle[/latex] in the flux integral. The flux across [latex]S_2[/latex] is then

[latex]\begin{aligned} \displaystyle\iint_{S_2}{\bf{F}}\cdot d{\bf{S}}&=\displaystyle\int_0^1\displaystyle\int_0^{2\pi}{\bf{F}}({\bf{r}}(u,v))\cdot({\bf{t}}_u\times{\bf{t}}_v)dA \\ &=\displaystyle\int_0^1\displaystyle\int_0^{2\pi}\langle u\cos v-u\sin v,u\cos v+u,u-u\sin v\rangle\cdot\langle u\cos v,u\sin v,-u\rangle \\ &=\displaystyle\int_0^1\displaystyle\int_0^{2\pi}u^2\cos^2v+2u^2\sin v-u^2 \ dvdu=-\frac{\pi}3 \end{aligned}[/latex].

The total flux across [latex]S[/latex] is

[latex]\large{\displaystyle\iint_{S_2}{\bf{F}}\cdot d{\bf{S}}=\displaystyle\iint_{S_1}{\bf{F}}\cdot d{\bf{S}}+\displaystyle\iint_{S_2}{\bf{F}}\cdot d{\bf{S}}=\frac{2\pi}3=\displaystyle\iiint_e\text{div }{\bf{F}}dV}[/latex],

and we have verified the divergence theorem for this example.

Physical Interpretation of Divergence

Recall that the divergence of continuous field [latex]\mathbf{F}[/latex] at point [latex]P[/latex] is a measure of the “outflowing-ness” of the field at [latex]P[/latex]. If [latex]\mathbf{F}[/latex] 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 [latex]P[/latex] be a point and let [latex]B_r[/latex] be a ball of small radius [latex]r[/latex] centered at [latex]P[/latex] (Figure 4). Let [latex]S_r[/latex] be the boundary sphere of [latex]B_r[/latex].

Since the radius is small and [latex]{\bf{F}}[/latex] is continuous, [latex]\text{div }{\bf{F}}(Q)\approx\text{div }{\bf{F}}(P)[/latex] for all other points [latex]Q[/latex] in the ball. Therefore, the flux across [latex]S_r[/latex] can be approximated using the divergence theorem:

[latex]\displaystyle\iint_{S_r}{\bf{F}}\cdot dS=\displaystyle\iiint_{B_r}\text{div }{\bf{F}}dV\approx\displaystyle\iiint_{B_r}\text{div }{\bf{F}}(P)dV[/latex].

Since [latex]\text{div }{\bf{F}}(P)[/latex] is a constant,

[latex]\displaystyle\iiint_{B_r}\text{div }{\bf{F}}(P)dV=\text{div }{\bf{F}}(P)V(B_r)[/latex].

Therefore, flux [latex]\displaystyle\iint_{S_r}{\bf{F}}\cdot d{\bf{S}}[/latex] can be approximated by [latex]\text{div }{\bf{F}}(P)V(B_r)[/latex]. This approximation gets better as the radius shrinks to zero, and therefore:

[latex]\text{div }{\bf{F}}(P)=\displaystyle\lim_{r\to0}\frac{1}{V(B_r)}\displaystyle\iint_{S_r}{\bf{F}}\cdot d{\bf{S}}[/latex].

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

This figure is a diagram of ball B_r, with small radius r centered at P. Arrows are drawn pointing up and to the right across the ball. — Figure 4. Ball [latex]B_r[/latex] of small radius [latex]r[/latex] centered at [latex]P[/latex].