9:00 AM Jamboard Links
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12:45 PM Jamboard Links
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Presenters
9AM
Thanks for sharing things in Perusall. Here are the presenters for today. Remember, I start picking presenters at 8am. If you upload something after that time, I may not see it.
- 6.17 -
12:45PM
Thanks for sharing things in Perusall. Here are the presenters for today. Remember, I start picking presenters at 12 noon. If you upload something after that time, I may not see it.
- 6.17 -
Learning Reminders
- We are in the 13th week of the semester. If you are on track for an A, then ideally you're finishing your SDL project for the 5th unit, and proposed something for the 6th unit.
- There are only 2 weeks left, which means you can submit at most 2 more SDL projects.
- The final SDL project (6th) can be over any topic from the entire semester. You can use it to expand what we do in the 6th unit, or you may choose to revisit something from a prior unit that you would like to spend more time with.
Brain Gains (Recall/Generation)
- Consider the vector field $\vec F(x,y,z) = (M,N,P)$, where each of $M$, $N$, and $P$ are functions of $x$, $y$, and $z$. Compute $D\vec F(x,y,z)$ symbolically, filling the 3 by 3 matrix with appropriate partials of $M$, $N$, and $P$.
Solution
The solution is $$D\vec F = \begin{bmatrix} \begin{matrix}M_x\\N_x\\P_x\end{matrix} &\begin{matrix}M_y\\N_y\\P_y\end{matrix} &\begin{matrix}M_z\\N_z\\P_z\end{matrix} \end{bmatrix}$$
- Consider the surface $\vec r(u,v) = (2u+3v,4u+5v,6u+7v)$. The differential is $$d\vec r=\begin{pmatrix}2\\4\\6\end{pmatrix}du+\begin{pmatrix}3\\5\\7\end{pmatrix}dv.$$ Find the area of the parallelogram with edges $(2,4,6)du$ and $(3,5,7)dv$.
Solution
The magnitude of the cross product of these two vectors is $$\begin{align} |(2,4,6)du\times(3,5,7)dv| &=\left|(2,4,6)\times(3,5,7)\right|dudv\\ &=\left|(4(7)-5(6),3(6)-2(7),2(5)-3(4))\right|dudv\\ &=\left|(-2,4,-2)\right|dudv\\ &=\sqrt{4+16+4}dudv\\ &=\sqrt{24}dudv \end{align}$$
- For a vector field $\vec F(x,y)=(M,N)$ that is continuously differentiable everywhere, Green's theorem states $$\oint_C Mdx+Ndy = \iint_R N_x-M_ydA,$$ where $R$ is region inside a simple closed piecewise smooth curve $C$ that traverses around the boundary of $R$ in a counterclockwise fashion. Let $\vec F = (3x+4y,6x+7y)$ and let $C$ be the curve which starts at $(1,-3)$, travels right to $(5,-3)$, up to $(5,7)$, left to $(1,7)$, and then back down to $(1,-3)$. Find the work done by $\vec F$ along $C$.
Solution
Note that $N_x-M_y=6-4=2$. By Green's theorem, the requested integral is the same as $\iint_R2dA=2A$, twice the area of the rectangle $R$ inside $C$. The width of this rectangle is 4, and the height is 10, so the answer is $W=2A = 2(40)=80$.
- For the surface $\vec r(u,v) = (u,v,9-u^2-v^2)$, compute the normal vector $\vec n = \frac{\partial \vec r}{\partial u}\times \frac{\partial \vec r}{\partial v}$.
Solution
The partial derivatives are $$\frac{\partial \vec r}{\partial u} = (1,0,-2u),\quad \frac{\partial \vec r}{\partial v} = (0,1,-2v), $$ The normal vector $\vec n$ is the cross product of these two, so $$\vec n = (2u, 2v,1).$$
- Set up an iterated double integral to compute the surface area $\ds \sigma = \iint_S |\vec r_u\times \vec r_v|dudv$ of the portion of the surface $\vec r(u,v) = (u,v,9-u^2-v^2)$ for $0\leq u\leq 3$ and $-3\leq v\leq 3$.
Solution
The area of the parallelogram formed by $$\frac{\partial \vec r}{\partial u}du = (1,0,-2u)du\quad \text{and}\quad \frac{\partial \vec r}{\partial v}dv = (0,1,-2v)dv$$ is the magnitude of their cross product. The cross product of these two vectors is $$\vec n = (2u, 2v,1)dudv.$$ The magnitude is $$d\sigma = \sqrt{4u^2+4v^2+1}dudv.$$ The surface area is $$\sigma = \iint_S d\sigma = \int_{0}^{3}\int_{-3}^{3}\sqrt{4u^2+4v^2+1}dvdu.$$
Group problems
- Compute the work done by $\vec F = (-3y,3x)$ to move an object counterclockwise once along the circle $\vec r(t) = (5\cos t, 5\sin t).$
- Compute this work using $ \oint_{C} M dx+Ndy$. [Check $150\pi$.]
- Compute this work using $\iint_R N_x-M_y dA$. (You can set up the integral in rectangular, or polar, or just use facts about area.)
- Compute the work done by $\vec F = (2x-y,2x+4y)$ to move an object counterclockwise once along the triangle with corners $(0,0)$, $(2,0)$, and $(0,3)$.
- Set up the single double integral $\iint_R N_x-M_y dA$.
- Compute the integral (use use facts about area).
- Consider the surface parametrized by $\vec r(u,v) = (u\cos v, u\sin v, u^2)$ for $0\leq u\leq 3$ and $0\leq v\leq 2\pi$.
- Draw the surface.
- Compute $d\sigma = \left |\dfrac{\partial \vec r}{\partial u}\times\dfrac{\partial \vec r}{\partial u}\right|dudv$.
- Set up an integral formula to compute $\bar z$ for this surface.
- Draw each curve or surface given below.
- $\vec r(u,v) = (u\cos v,u\sin v,u)$ for $0\leq v\leq 2\pi$ and $0\leq u\leq 4$. (Check: Cone)
- $\vec r(u,v) = (u\cos v,u\sin v,v)$ for $0\leq v\leq 6\pi$ and $2\leq u\leq 4$. (Check: Spiral stair case)
- $\vec r(t) = (t,9-t^2,0)$ for $0\leq t\leq 3$.
- $\vec r(u,v) = (9-u^2,u\cos v,u\sin v)$ for $0\leq v\leq 2\pi$ and $0\leq u\leq 3$.
- 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 $\vec r(t) = (3\cos t, 3\sin t)$.
- Compute the derivative of each vector field $\vec F$ below (obtaining a square matrix). Then find a potential for $\vec F$ or explain why the vector field has no potential.
- $\vec F = (2x-y,-x+4y)$
- $\vec F = (y^2+2x,2xy)$
- $\vec F = (x+yz,xz+4yz,xy+2y^2)$
- $\vec F = (x+yz,4yz,xy+2y^2)$
- $\vec F = (x+yz,xz+4yz,xy)$
- $\vec F = (yz,xz+4yz,xy+2y^2)$
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