Section 6.10 Divergence Theorem

6.10 Divergence Theorem

In this section, we study Divergence Theorem. Recall that the flux form of Green’s theorem states that 

$$\int {\int }_D\text{div}\overrightarrow{F}dA={\int }_C\overrightarrow{F}\cdot \overrightarrow{N}ds.$$ The divergence theorem is a version of Green’s theorem in one higher dimension. If we think of divergence as a derivative of sorts, then the divergence theorem relates a triple integral of derivative div$\overrightarrow{F}$ over a solid to a flux integral of $\overrightarrow{F}$ over the boundary of the solid. More specifically, the divergence theorem relates a flux integral of vector field $\overrightarrow{F}$ over a closed surface $S$ to a triple integral of the divergence of $\overrightarrow{F}$ over the solid enclosed by $S$.

Theorem: The Divergence Theorem

Let $S$ be a piecewise, smooth closed surface that encloses solid $E$ in space. Assume that $S$ is oriented outward, and let $\overrightarrow{F}$ be a vector field with continuous partial derivatives on an open region containing $E$ . Then 

$$\int \int {\int }_E\text{div}\overrightarrow{F}dV=\int {\int }_S\overrightarrow{F}\cdot dS$$

$$=\int {\int }_S\overrightarrow{F}\cdot \overrightarrow{N}dS$$

$$=\int {\int }_D\overrightarrow{F}\cdot (\overrightarrow{s_u}\times \overrightarrow{s_v})dA.$$

Example 1: Find $\int {\int }_S\overrightarrow{F}\cdot dS$, where $\overrightarrow{F}=<z{e}^y,xy,z^2>$ and $S$ is the surface, oriented outward, bounded by $x^2+z^2=4$, $y=1$ and $y=-1$.

Exercise 1: Find $\int {\int }_S\overrightarrow{F}\cdot dS$, where $\overrightarrow{F}=<yz+\text{cos}(z),e^{x+z}+y,z^2>$ and $S$ is the surface, oriented outward, bounded by $x^2+y^2=4$, $z=0$ and $z=1$.

Example 2: Find $\int {\int }_S\overrightarrow{F}\cdot dS$, where $\overrightarrow{F}=<x^3+z^3,x^3+y^3,z^3+x^3>$ and $S$ is the surface of sphere radius $3$, oriented outward.

Exercise 2: Find $\int {\int }_S\overrightarrow{F}\cdot dS$, where $\overrightarrow{F}=<x^3+y^3,z^3+y^3,z^3+y^3>$ and $S$ is the surface of sphere radius $2$, oriented outward.

Example 3: Find $\int {\int }_S\overrightarrow{F}\cdot dS$, where $\overrightarrow{F}=<xz,x^2+z^2,z^2>$ and $S$ is the surface, oriented outward, bounded by $x^2+z^2=9$, $y=0$ and $y=z+1$.

Exercise 3: Find $\int {\int }_S\overrightarrow{F}\cdot dS$, where $\overrightarrow{F}=<yz,x^2+y^2,yz>$ and $S$ is the surface, oriented outward, bounded by $x^2+y^2=4$, $z=0$ and $z=x-2$.

Example 4: Find $\int {\int }_S\overrightarrow{F}\cdot dS$ where $\overrightarrow{F}=<\frac{1}{3}{x}^3+y^3,\frac{1}{3}\text{sin}(z{)}^3+x^2,\frac{1}{3}{z}^3+y^4>$ and $S$ is the surface bounded by $y=4-x^2-z^2$ with $y\ge 0$, oriented outward.

Group work:

1. Find $\int {\int }_S\overrightarrow{F}\cdot dS$ where $\overrightarrow{F}=<x{e}^y,z-e^y,-xy>$ and $S$ is the ellipsoid $x^2+4y^2+5z^2=20$ oriented outward.

2. Find $\int {\int }_S\overrightarrow{F}\cdot dS$ where $\overrightarrow{F}=<2x^3+y^3,\frac{1}{3}\text{tan}(z{)}^3+y^3,3y^{2}z>$ and $S$ is the surface bounded by $z=9-x^2-y^2$ with $z\ge 0$, oriented outward.

3. Find $\int {\int }_S\overrightarrow{F}\cdot dS$, where $\overrightarrow{F}=<xy,-y^2+z^2,3yz>$ and $S$ is the surface, oriented outward, bounded by $x^2+y^2=25$, $z=0$ and $z=y+5$.