16.8 Videos Guide

16.8a

Theorem (statement):

• Stokes’ Theorem: $\int^{\ }_{C}{\mathbf{F}\cdot d\mathbf{r}=\iint^{\ }_{S}{\mbox{curl}\ \mathbf{F}}\cdot d\mathbf{S}}$, where $C$ is the positively oriented piecewise-smooth boundary curve of $S$, an oriented piecewise-smooth surface

• Green’s Theorem as a special case of Stokes’ Theorem via a vector form of Green’s Theorem
  $$\int^{\ }_{C}{\mathbf{F}\cdot d\mathbf{r}}=\iint^{\ }_{S}{\mbox{curl}\ \mathbf{F}}\cdot d\mathbf{S=}\iint^{\ }_{D}{\left(\mbox{curl}\ \mathbf{F}\right)\cdot\mathbf{k}\ dA}$$

• 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 $S$

Exercises:

16.8b

• Verify that Stokes’ Theorem is true for the given vector field $\mathbf{F}$ and surface $S$.
  $\mathbf{F}(x,\ y,\ z)=-2yz\ \mathbf{i}+y\ \mathbf{j}+3x\ \mathbf{k}$,
  $S$ is the part of the paraboloid $z=5-x^{2}-y^{2}$ that lies above the plane $z=1$, oriented upward

16.8c

• Use Stokes’ Theorem to evaluate $\iint^{\ }_{S}{\mbox{curl}\ \mathbf{F}}\cdot d\mathbf{S}$.
  $\mathbf{F}(x,\ y,\ z)=x^{2}\sin z\ \mathbf{i}+y^{2}\mathbf{\ j}+xy\ \mathbf{k}$,
  $S$ is the part of the paraboloid $z=1-x^{2}-y^{2}$ that lies above the $xy$-plane, oriented upward

16.8d

• Use Stokes’ Theorem to evaluate $\int^{\ }_{C}{\mathbf{F}\cdot d\mathbf{r}}$. In each case $C$ is oriented counterclockwise as viewed from above.

  • $\mathbf{F}(x,\ y,\ z)=\mathbf{i}+(x+yz)\ \mathbf{j}+\left(xy-\sqrt{z}\right)\ \mathbf{k}$,
    $C$ is the boundary of the part of the plane $3x+2y+z=1$ in the first octant

16.8e

• $\mathbf{F}(x,\ y,\ z)=2y\ \mathbf{i}+xz\ \mathbf{j}+(x+y)\ \mathbf{k}$,
  $C$ is the curve of intersection of the plane $z=y+2$ and the cylinder
  $$x^{2}+y^{2}=1$$