16.7 Videos Guide

16.7a

• Surface integral of a scalar field $f(x,\ y,\ z)$

  • $\iint^{\ }_{S}{f(x,\ y,\ z)\ dS}=\iint^{\ }_{D}{f\left(\mathbf{r}(u,\ v)\right)\left|\mathbf{r}_{u}\times\mathbf{r}_{v}\right|\ dA}$

Note that $dS=\left|\mathbf{r}_{u}\times\mathbf{r}_{v}\right|\ dA$.

Exercises:

• Evaluate the surface integral.

  • $\iint^{\ }_{S}{xyz\ dS}$,
    $S$ is the cone with parametric equations $x=u\cos v$, $y=u\sin v$, $z=u$,
    $0\leq u\leq1$, $0\leq v\leq\frac{\pi}{2}$

16.7b

• $\iint^{\ }_{S}{xy\ dS}$,
  $S$ is the part of the plane $2x+2y+z=4$ that lies in the first octant

16.7c

• If $S$ consists of multiple surfaces $S_{i}$, then
  $\iint^{\ }_{S}{f(x,\ y,\ z)\ dS}=\iint^{\ }_{S_{1}}{f(x,\ y,\ z)\ dS}+\iint^{\ }_{S_{2}}{f(x,\ y,\ z)\ dS}+\cdots\iint^{\ }_{S_{n}}{f(x,\ y,\ z)\ dS}$

Exercise:

• Evaluate the surface integral.
  $\iint^{\ }_{S}{\left(x^{2}+y^{2}+z^{2}\right)\ dS}$,
  $S$ is the part of the cylinder $x^{2}+y^{2}=9$ between the planes $z=0$ and $z=2$, together with its top and bottom disks

16.7d

• Surface integral of a vector field $\mathbf{F}(x,\ y,\ z)$

  • Flux is $\iint^{\ }_{S}{\mathbf{F}\cdot d\mathbf{S}}=\iint^{\ }_{S}{\mathbf{F}\cdot\mathbf{n}\ dS}=\iint^{\ }_{D}{\mathbf{F}\cdot\left(\mathbf{r}_{u}\times\mathbf{r}_{v}\right)dA}$
    Note that $d\mathbf{S}=\mathbf{n}dS=\left(\mathbf{r}_{u}\times\mathbf{r}_{v}\right)\ dA$, where $\mathbf{n}$ is a unit normal vector and $\mathbf{r}_{u}\times\mathbf{r}_{v}$ is simply a normal vector to the surface $S$.

• If $x$ and $y$ are the parameters, we have $\iint^{\ }_{S}{\mathbf{F}\cdot d\mathbf{S}}=\iint^ {}_{D}{\left(-P\frac{\partial g}{\partial x}-Q\frac{\partial g}{\partial y}+R\right)dA}$, for upward orientation. The signs of the integrand change for downward orientation.

Exercises:

16.7e

• Evaluate the surface integral $\iint^{\ }_{S}{\mathbf{F}\cdot d\mathbf{S}}$ for the given vector field $\mathbf{F}$ and the oriented surface $S$. In other words, find the flux of $\mathbf{F}$ across $S$. For closed surfaces, use the positive (outward) orientation.

  • $\mathbf{F}(x,\ y,\ z)=-x\ \mathbf{i}-y\ \mathbf{j}+z^{3}\ \mathbf{k}$,
    $S$ is the part of the cone $z=\sqrt{x^{2}+y^{2}}$ between the planes $z=1$ and $z=3$ with downward orientation

16.7f

• $\mathbf{F}(x,\ y,\ z)=x\ \mathbf{i}+y\ \mathbf{j}+5\ \mathbf{k}$,
  $S$ is the boundary of the region enclosed by the cylinder $x^{2}+z^{2}=1$ and the planes $y=0$ and $x+y=2$