Rapid Recall
- For the vector field $\vec F = (2x+3y^2, 6xy+4y)$, compute the work done along the curve $\vec r(t) = (3\cos t, 3\sin t)$ for $t\in [0,\pi] $.
Solution
We'll do this on the board.
- For the vector field $\vec F = (-4y, 4x)$, compute the work done along the curve $\vec r(t) = (3\cos t, 3\sin t)$ for $t\in [0,\pi] $.
Solution
We'll do this on the board.
- For the vector field $\vec F = (-4y, 6x)$, compute the work done along the curve $\vec r(t) = (3\cos t, 3\sin t)$ for $t\in [0,2\pi] $.
Solution
We'll do this on the board.
- Consider the parametric surface $\vec r(u,v) = (2u, 3v, u^2+v^2)$ for $u\in [0,3]$ and $v\in [0,3]$. Start by computing the normal vector $\vec n = \vec r_u\times \vec r_v$, and then give an equation of the tangent plane to this surface at $\vec r(-1,1)$.
Solution
We'll do this on the board.
Discussion
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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