M344: Calculus III
Section 16.9
The Divergence Theorem¶
In the last section we learned about Stokes' theorem, which may be regarded as an extension of the curl-form of Green's theorem (Section 16.5) to surfaces in $\mathbb{R}^3$. Recall that in Section 16.5 we also reframed Green's theorem in terms of the divergence of a vector field $\mathbf{F}$. Under the assumptions, we showed that
$$
\int_{\partial D} \mathbf{F}\cdot d\mathbf{n} = \int\!\!\!\!\int_D \text{div}\, \mathbf{F}\, dA.
$$
This version of Green's theorem extends to $3$ dimensions in the form of The Divergence Theorem.
Let $E$ be a simple solid region and let $S = \partial E$ be the boundary surface of $E$, given with positive outward orientation. Let $\mathbf{F}$ be a vector field whose component functions have continuous partial derivatives on an open region that contains $E$. Then
$$ \int\!\!\!\!\int_{\partial E} \mathbf{F}\cdot d\mathbf{S} = \int\!\!\!\!\int\!\!\!\!\int_E \text{div}\,\mathbf{F}\, dV. $$
The phrase "simple solid region" in the statement refers to a region that may be regarded simultaneously as type I, type II, and type III. This assumption can be relaxed slightly, but we will not do so in this course.
Video proof
A video goes here.
Examples¶
We now work through some examples to solidify our understanding of the Divergence Theorem. Please try all of these yourself before watching the video solution. If necessary, start the video to give you a hint as to how you should proceed, then "pause and ponder" before watching the end of the video.
1. Example¶
Compute both integrals to verify that the conclusion of the Divergence theorem holds for the vector field $\mathbf{F}(x,y,z) = \left\langle y^2z^3, 2yz, 4z^2 \right\rangle$ on the solid region $E$ that is enclosed by the paraboloid $z = x^2 + y^2$ and the plane $z = 9$.
Video solution
A video goes here.
2. Example¶
Use the Divergence theorem to compute the flux of $\mathbf{F}$ across $S$, where
$$
\mathbf{F}(x,y,z) = xe^y\,\mathbf{i} + \big(z - e^y\big)\,\mathbf{j} - xy\,\mathbf{k},
$$
and $S$ is the ellipsoid $x^2 + 2y^2 + 3z^2 = 4$.
Video solution
A video goes here.
3. Example¶
Suppose that $E$ and $S = \partial E$ satisfy the hypotheses of the Divergence theorem and $f$ is a (scalar) function with continuous partial derivatives on $E$. Show that
$$
\int\!\!\!\!\int\!\!\!\!\int_E \nabla f\; dV = \int\!\!\!\!\int_{\partial E} f\,\mathbf{n}\; dS,
$$
where each of these integrals of vector functions are defined by computing the integrals on each component function.
Video solution
A video goes here.
4. Example¶
Suppose that $E$ and $S = \partial E$ satisfy the hypotheses of the Divergence theorem, and $f$ is a function with continuous second-order partial derivatives on a region containing $E$. Recall that the normal derivative of $f$ along $S$ is defined to be the directional derivative,
$$
\frac{\partial f}{\partial \mathbf{n}} = \nabla f\cdot\mathbf{n}.
$$
Show that
$$
\int\!\!\!\!\int\!\!\!\!\int_E \Delta f\; dV = \int\!\!\!\!\int_{\partial E} \frac{\partial f}{\partial \mathbf{n}}\; dS.
$$
Video solution
A video goes here.
5. Example¶
Suppose that $E$ and $S = \partial E$ satisfy the hypotheses of the Divergence theorem, and consider the vector field
$$
\mathbf{F}(x,y,z) = \big\langle x,y,z \big\rangle.
$$
Show that the volume of $E$ may be computed via a surface integral of $\mathbf{F}$ around the boundary. In particular,
$$
\text{vol}(E) = \frac{1}{3}\int\!\!\!\!\int_{\partial E} \mathbf{F}\cdot d\mathbf{S}.
$$
Video solution
A video goes here.
Discussion¶
Questions? You can ask them here.