In-class work for 18 July 2012
1. Find the curl and divergence of the vector field
a.) $\vec{F}(x,y,z) = < 0, \cos xz, -\sin xy >$
b.) $\vec{F}(x,y,z) = <0, e^{xy}\sin z, y \arctan(x/z) >$
2. Let $f$ be a scalar field and $\vec{F}$ a vector field. State whether each expression is meaningful. If not, explain why. If so, state whether it is a scalar field or a vector field.
a.) $div \vec{F}$
b.) $grad \vec{F}$
c.) $div(grad f)$
3. Determine whether or not the vector field is conservative. If it is, find a function $f$ such that $\vec{F} = \nabla f$.
a.) $\vec{F}(x,y,z) = < xyz^2, x^2yz^2, x^2y^2z >$
b.) $\vec{F}(x,y,z) = < e^z, 1, xe^z >$
c.) $\vec{F}(x,y,z) = < yee^{-x}, e^{-x}, 2z >$
d.) $\vec{F}(x,y,z) = < y \cos xy, x \cos xy, -\sin z >$
4. Identify the surface with the given vector equation.
a.) $\vec{r}(u,v) = <2 \sin u, 3 \cos u, v >$; $0 \leq v \leq 2$
b.) $\vec{r}(s,t) = < s \sin 2t, s^2, s \cos 2t >$
5. Find a parametric representation for the surface.
a.) Lower half of the ellipsoid $2x^2+4y^2+z^2=1$
b.) The part of the hyperbolid $x^2+y^2-z^2=1$ that lies to the right of the $xz$-plane.