In-class work for 24 July 2012
1. Use Stokes' theorem to evaluate $\displaystyle\int\int_S curl \vec{F} \cdot d\vec{S}$.
    a.) $\vec{F}(x,y,z) = < 2y \cos z, e^x \sin z, xe^y >$ where $S$ is the hemisphere $x^2+y^2+z^2=9$, $z \geq 0$, oriented upward
    b.) $\vec{F}(x,y,z) = < x^2y^3z, \sin(xyz), xyz >$, $S$ is the part of the cone $y^2=x^2+z^2$ that lies between the planes $y=0$ and $y=3$, oriented in the direction of the positive $y$-axis
    c.) $\vec{F}(x,y,z) = < e^{xy} \cos z, x^2 z, xy >$; $S$ is the hemisphere $x = \sqrt{1-y^2-z^2}$, oriented in the direction of the positive $x$-axis

2. Use Stokes' theorem to evaluate $\displaystyle\int_C \vec{F} \cdot d \vec{r}$. In each case, $C$ is oriented counterclockwise as viewed from above.
    a.) $\vec{F}(x,y,z) = < e^{-x}, e^x, e^z >$; $C$ is the boundary of the part of the plane $2x+y+2z=2$ in the first octant
    b.) $\vec{F}(x,y,z) = < xy, 2z, 3y >$; $C$ is the curve of intersection of the plane $x+z=5$ and the cylinder $x^2+y^2=9$.
    c.) $\vec{F}(x,y,z) = < x^2z, xy^2, z^2 >$; $C$ is the curve of intersection of the plane $x+y+z=1$ and the cylinder $x^2+y^2=9$ oriented counterclockwise as viewed from above

3. A particle moves along line segments from the origin to the points $(1,0,0)$, $(1,2,1)$, $(0,2,1)$, and back to the origin under the influence of the force field $\vec{F}(x,y,z) = < z^2, 2xy, 4y^2 >$. Find the work done.

4. Verify that Stokes' Theorem is not true for the given $\vec{F}$ and the surface $S$.
    a.) $\vec{F}(x,y,z) = < y^2, x, z^2 >$, $S$ is the part of the paraboloid $z=x^2+y^2$ that lies below the plane $z=1$, oridented upwards
    b.) $\vec{F}(x,y,z) = < x, y, xyz >$; $S$ is the part of the plane $2x+y+z=2$ that lies in the first octant, oriented upward