Math 215 - Multivariate Calculus - Schedule

Rapid Recall

Draw the surface $\vec r(t,v) = (3\cos t, 3\sin t, v)$ for $0\leq t\leq 2\pi$ and $1\leq v\leq 4$.

Solution

It's a right circular cylinder of radius 3, whose center lies along the $z$-axis for $1\leq z\leq 4$.

Solution

It's a helix, radius is 3, spiraling counterclockwise (when viewed from above) at it wraps around the $z$ axis for $0\leq t\leq 6\pi$.

For the vector field $\vec F = (xyz, 3x+4y+5z, xy+z^2)$, compute $D\vec F$ and then compute $\vec \nabla \cdot \vec F$ and $\vec \nabla \times \vec F$. Recall $\vec \nabla = \left(\frac{\partial}{\partial x},\frac{\partial}{\partial y},\frac{\partial}{\partial z}\right)$.

Solution

We compute

$D\vec F = \begin{bmatrix}\begin{matrix}yz\\3\\y\end{matrix}&\begin{matrix}xz\\4\\x\end{matrix}&\begin{matrix}xy\\5\\2z\end{matrix}\end{bmatrix}$,
$\vec \nabla \cdot \vec F = yz+4+2z$, and
$\vec \nabla \times \vec F = (x-5, xy-y, 3-xz)$.

Do you notice how every single entry from the derivative of $\vec F$ shows up in exactly one spot in one of the latter quantities?

Draw each curve or surface given below.
1. $\vec r(u,v) = (4\cos u,v, 3\sin u)$ for $0\leq u\leq \pi$ and $0\leq v\leq 7$.
2. $\vec r(t) = (3\cos t,3\sin t,t)$ for $0\leq t\leq 6\pi$.
3. $\vec r(u,v) = (u\cos v,u\sin v,v)$ for $0\leq v\leq 6\pi$ and $2\leq u\leq 4$.
4. $\vec r(t) = (0,t,9-t^2)$ for $0\leq t\leq 3$.
5. $\vec r(x,y) = (x,y,9-x^2-y^2)$ for for $0\leq x \leq 3$ and $-3\leq y\leq 3$.
6. $\vec r(u,v) = (u\cos v,u\sin v,9-u^2)$ for $0\leq u\leq 3$ and $0\leq v\leq 2\pi$.
For the vector field $\vec F = (x^2,3z+x,2y+4z)$, compute both $\vec \nabla \cdot F$ and $\vec \nabla \times F$.
For the same vector field compute each of the following, or explain why they cannot be computed.
- $\vec \nabla\times(\vec \nabla \cdot \vec F)$
- $\vec \nabla\cdot (\vec \nabla \times \vec F)$
- $\vec \nabla\cdot (\vec \nabla \cdot \vec F)$
- $\vec \nabla\times(\vec \nabla \times \vec F)$
For the vector field $\vec F = (M,N,P)$, compute each of the above that can be computed.