Rapid Recall
Solution
Solution
Solution
Group problems
- Consider the parametric surface $\vec r(u,v) = (u\cos v, u\sin v, u^2)$ for $u\in [1,2]$ and $v\in [0,2\pi]$.
- Compute a normal vector to the surface (so $\vec n = \vec r_u\times \vec r_v$).
- Give an equation of the tangent plane to the surface at $(u,v)=(3/2,\pi/2)$.
- Set up an integral to compute the surface area of the surface.
- Consider the surface parametrized by $\vec r(u,v) = (u, v, u^2+v^2)$ for $-3\leq u\leq 3$ and $0\leq v\leq 3$.
- Compute $d\sigma = \left |\dfrac{\partial \vec r}{\partial u}\times\dfrac{\partial \vec r}{\partial v}\right|dudv$.
- Set up an integral formula to compute $\bar z$ for this surface.
- Give an equation of the tangent plane to the surface at $(u,v)=(2,1)$.
- Consider the parametric surface $\vec r(u,v) = (u^2\cos v, u, u^2\sin v)$ for $u\in [1,2]$ and $v\in [0,2\pi]$.
- Compute a normal vector to the surface (so $\vec n = \vec r_u\times \vec r_v$).
- Give an equation of the tangent plane to the surface at $(u,v)=(3/2,\pi/2)$.
- Set up an integral to compute the surface area of the surface.
We'll end today with a a discussion of flux and the divergence theorem. Thanks for all your hard work. It's been a very enjoyable semester. Let $\vec F = (3x,4y,2z)$. Let $S$ be the surface parametrized by $\vec r(u,v) = (u,v,9-u^2-v^2)$.
- Compute $d\sigma$.
- State a unit normal vector $\hat n$ to the surface that points upwards at all points along the surface.
- Compute $\vec F\cdot\hat n d\sigma$. What does this quantity represent?
- Change the surface to the 6 faces of a rectangular box with $x\in [0,2], y\in [0,3], z\in [0.5] $. For each surface, state a normal vector and $\vec F\cdot\hat n d\sigma$.
- Compute the outward flux across the surface.
- Let's apply the divergence theorem.
- What requirements does the divergence theorem have?
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