Math2111: Chapter 5: Integral theorems. Section 1: Divergence theorem

In this blog entry you can find lecture notes for Math2111, several variable calculus. See also the table of contents for this course. This blog entry printed to pdf is available here.

The divergence theorem is a generalisation of Green’s theorem (or more precisely of Green’s theorem in normal form). This theorem states that the flux of a vector field out of a closed surface equals to the integral of the divergence of that vector field over the volume enclosed by the surface (recall that Green’s theorem states that the flux out of a simple closed curve equals to the integral of the divergence over the region enclosed by the curve).

Gauss’ Divergence theorem
Let ${}S \subset \mathbb{R}^3$ be a closed smooth surface and let $\boldsymbol{F}:\mathbb{R}^3\to\mathbb{R}^3$ be continuously differentiable. Let ${}T$ be the region enclosed by ${}S$. Then

$\displaystyle \iiint_T (\nabla \cdot \boldsymbol{F}) \,\mathrm{d} V = \iint_S \boldsymbol{F}\cdot \mathrm{d} \boldsymbol{\mathcal{S}},$

where the surface is oriented such that the normal vector points outward.

Example
Find the flux of the vector field $\boldsymbol{F}(x,y,z)= -y \widehat{\boldsymbol{i}} + x \widehat{\boldsymbol{j}} + z \widehat{\boldsymbol{k}}$ out of the cylinder $x^2+y^2\le 1$ and $0\le z \le 1.$
$\Box$

Example
Evaluate $\iint_S \boldsymbol{F}\cdot\mathrm{d} \boldsymbol{\mathcal{S}}$ where $\boldsymbol{F}(x,y,z)= xy \widehat{\boldsymbol{i}} + (z+\mathrm{e}^{\cos xz}) \widehat{\boldsymbol{j}} + \sin x \mathrm{e}^y \widehat{\boldsymbol{k}}$ and ${}S$ is the surface of the region ${}T$ bounded by $z=1-x^2$ and the planes $z=0,$ $y=0$ and $y+z=2.$
$\Box$