16.9 Videos Guide

16.9a

Theorem (statement and proof):

• The Divergence Theorem (the 3-D analog of Green’s Theorem):
  $\iint^{\ }_{S}{\mathbf{F}\cdot d\mathbf{S}=\iiint^{\ }_{E}{\mbox{div}\ \mathbf{F}\ dV}}$, where $S$ is the boundary surface of $E$, a solid region whose surfaces are continuous, with outward orientation

Exercises:

16.9b

• Verify that the Divergence Theorem is true for the vector field $\mathbf{F}$ on the region $E$.
  $\mathbf{F}(x,\ y,\ z)=\left\langle x^{2},\ -y,\ z\right\rangle$,
  $E$ is the solid cylinder $y^{2}+z^{2}\leq9$, $0\leq x\leq2$

16.9c

• Use the Divergence Theorem to calculate the surface integral $\iint^{\ }_{S}{\mathbf{F}\cdot d\mathbf{S}}$; that is, calculate the flux of $\mathbf{F}$ across $S$.

  • $\mathbf{F}(x,\ y,\ z)=\left(x^{3}+y^{3}\right)\mathbf{\ i}+\left(y^{3}+z^{3}\right)\ \mathbf{j}+(z^{3}+x^{3})\ \mathbf{k}$,
    $S$ is the sphere of radius 2 with center $(0,\ 0,\ 0)$

16.9d

• $\mathbf{F}(x,\ y,\ z)=(xy+2xz)\mathbf{i}+\left(x^{2}+y^{2}\right)\ \mathbf{j}+\left(xy-z^{2}\right)\ \mathbf{k}$,
  $S$ is the surface of the solid bounded by the cylinder $x^{2}+y^{2}=4$ and the planes $z=y-2$ and $z=0$