In-class work for 25 July 2012 (last day review)
1. Determine whether or not $\vec{F}$ is a conservative vector field. If it is, find its potential function.
    a.) $\vec{F}(x,y) = < xe^y, ye^x >$
    b.) $\vec{F}(x,y) = < 6x+5y, 5x+4y >$

2. Find the gradient vector field of $f$
    a.) $f(x,y,z) = \sin(x) + e^y \cos x$
    b.) $f(x,y) = 4x^4y^2+3x^3y + 2yx + 1$

3. Evaluate the line integral.
    a.) $\displaystyle\int_C xy^4 ds$; $C$ is the right half of the circle $x^2+y^2=16$
    b.) $\displaystyle\int_C < yz, xz, xy+2z > \cdot d \hspace{1pt} \vec{r}$; $C$ is the line segment from $(1,0,-2)$ to $(4,6,3)$.
    c.) $\displaystyle\int_C < x^2y^3,-y\sqrt{x} > \cdot d \hspace{1pt} \vec{r}$; $\vec{r}(t) = < t^2, -t^3 >, 0 \leq t \leq 1$
    d.) $\displaystyle\int_C xe^y dx$; $C$ is the arc of the curve $x=e^y$ from $(1,0)$ to $(e,1)$
    e.) $\displaystyle\int_C < z, y, -x > \cdot d \hspace{1pt} \vec{r}$; $\vec{r}(t) = < t^3, -t^2, t >, 0 \leq 1 \leq 1$

4. Use Green's theorem to evaluate.
    a.) $\displaystyle\int_C xy^2 dx + x^3 dy$; $C$ is the rectangle with vertices $(0,0), (2,0), (2,3),$ and $(0,3)$.
    b.) $\displaystyle\int_C (y+e^{\sqrt{x}}) dx + (2x+ \cos y^2) dy$; $C$ is the boundary of the region enclosed by the parabolas $y=x^2$ and $x=y^2$

5. Compute
    a.) $\mathrm{curl} < \log(x), \log(xy), \log(xyz) >$
    b.) $\mathrm{div} < xe^{-y}, xz, ze^y >$

6. Find a parametric representation of...
    a.) the plane that passes through the point $(1,2,-3)$ and contains the vectors $<1,1,-1>$ and $<1,-1,1>$.
    b.) the part of the sphwere $x^2+y^2+z^2=4$ that lies above the cone $z = \sqrt{x^2+y^2}$

7. Compute
    a.) the surface area of the part of the hyperbolic paraboloid $z=y^2-x^2$ that lies between the cylinders $x^2+y^2=1$ and $x^2+y^2=4$
    b.) an equation of the tangent plane to the parametric surface defined by parametrization $\left\{ \begin{array}{ll} x = u + v \\ y = 3u^2 \\ z = u - v \end{array} \right\}$ at the point $(2,3,0)$.

8. Compute the surface integral
    a.) $\displaystyle\int\int_S yz dS$, $S$ is the part of the plane $x+y+z=1$ that lies in the first octant
    b.) $\displaystyle\int\int_S < x, -z, y >$; $S$ is the part of the sphere $x^2+y^2+z^2=4$ in the first octant with orientation toward the origin
    c.) $\displaystyle\int\int_S xy dS$; $S$ is the boundary of the region enclosed by the cylinder $x^2+z^2=4$
    d.) $\displaystyle\int\int_S < 0, y, -z >$; $S$ consists of hte paraboloid $y=x^2+z^2, 0 \leq y \leq 1$ and the disk $x^2+z^2 \leq 1, y = 1$

9. Use Stokes' theorem to evaluate
    a.) $\displaystyle\int\int_S \mathrm{curl} < xyz, xy, x^2yz > \cdot d \vec{S}$; $S$ consists of the top and the four sides (but not the bottom) of the cube with vertices $(\pm 1, \pm 1, \pm 1)$ oriented outward
    b.) $\displaystyle\int_C < x+y^2, y+z^2, z+x^2 > \cdot d \vec{r}$;$C$ is the triangle with vertices $(1,0,0), (0,1,0),$ and $(0,0,1)$.

10. Use the divergence theorem to evaluate
    a.)$\displaystyle\int\int_S < x^2z^3, 2xyz^3, xz^4 > \cdot d \vec{S}$; $S$ is the surface of the box with vertices $( \pm 1, \pm 2, \pm 3)$
    b.) $\displaystyle\int\int_S < e^x \sin y, e^x \cos y, yz^2 >$; $S$ is the surface of the box bounded by the planes $x=0, x=1, y=0, y=1, z=0,$ and $z=2$.