Rapid Recall
- 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 the curve $\vec r(t) = (3\cos t,3\sin t,t)$ for $0\leq t\leq 6\pi$.
Solution
It's a helix, radius is 3, spiraling counterclockwise (when viewed from above) at it wraps around the $z$ axis for $0\leq t\leq 6\pi$.
- 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 $1\leq z\leq 4$.
Group problems
- Suppose $f=x^2y+3z$.
- Compute $df$.
- Compute $\int_{ (1,2,3) }^{ (0,-1,2) } df$.
- Let $\vec F = (5y,5x)$.
- Find a potential for $\vec F$, or explain why none exists.
- Compute the work done by $\vec F$ to go once counter-clockwise along the circle $\vec r(t) = (3\cos t, 3\sin t)$.
- Let $\vec F = (-5y,5x)$.
- Find a potential for $\vec F$, or explain why none exists.
- Compute the work done by $\vec F$ to go once counter-clockwise along the circle $\vec r(t) = (3\cos t, 3\sin t)$.
- Draw each curve or surface given below.
- $\vec r(t) = (3\cos t,3\sin t)$ for $0\leq t\leq 2\pi$.
- $\vec r(u,v) = (3\cos u,3\sin u,v)$ for $0\leq u\leq 2\pi$ and $0\leq v\leq 5$.
- $\vec r(u,v) = (4\cos u,v, 3\sin u)$ for $0\leq u\leq \pi$ and $0\leq v\leq 7$.
- $\vec r(t) = (3\cos t,3\sin t,4t)$ for $0\leq t\leq 6\pi$.
- $\vec r(u,v) = (u\cos v,u\sin v,v)$ for $0\leq v\leq 6\pi$ and $2\leq u\leq 4$.
- $\vec r(t) = (0,t,9-t^2)$ for $0\leq t\leq 3$.
- $\vec r(u,v) = (u\cos v,u\sin v,9-u^2)$ for $0\leq v\leq 2\pi$ and $0\leq u\leq 3$.
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