In-class work for 13 July 2012
1. Determine whether $\vec{F}$ is a conservative vector field. If it is, find a function $f$ such that $\nabla f = \vec{F}$.
    a.) $\vec{F}(x,y) = < e^x \cos y, e^x \sin y >$
    b.) $\vec{F}(x,y) = < 3x^2-2y^2, 4xy+3 >$

2. Consider $\vec{F}$ and $C$ below. Use them to compute $\displaystyle\int_C \nabla f \cdot d \vec{r}$.
    a.) $\vec{F}(x,y) = < x^2, y^2 >$; $C$ is the arc of the parabola $y=2x^2$ from $(-1,2)$ to $(2,8)$
    b.) $\vec{F}(x,y) = \left< \dfrac{y^2}{1+x^2}, 2y \arctan(x) \right>$; $C \colon \vec{r}(t) = < t^2, 2t >$; $0 \leq t \leq 1$
    c.) $\vec{F}(x,y,z) = < 2xz+y^2, 2xy, x^2+3z^2 >$; $C \colon x=t^2, y=t+1, z=2t-1; 0 \leq t \leq 1$
    d.) $\vec{F}(x,y,z) = < e^y, xe^y, (z+1)e^z >$; $C \colon \vec{r}(t) = < t, t^2, t^3 >$; $0 \leq t \leq 1$

3. Find the work done by the force field $\vec{F}=< e^{-y}, -xe^{-y} >$ moving an object from $P=(1,1)$ to $Q=(2,4)$.

4. Determine whether the set is (i) open, (ii) connected, and/or (iii) simply-connected.
    a.) $\{(x,y) \colon 1 < x^2+y^2 < 4 \}$
    b.) $\{(x,y) \colon x^2+y^2 \leq 1$ or $4 \leq x^2 + y^2 \leq 9 \}$
    c.) $\{(x,y) \colon x \neq 0 \}$