Presentations
6.? - 6.?Pacing Tracker
- The quizzes have included questions for 28 objectives. How many have you passed? What are you plans to master those that you haven't mastered yet?
- Once you've passed 26 objectives, you have hit 80% of the total objectives (needed to pass the class with a grade of A/B/C).
- We've finished units 1 through 5. Have you started your self-directed learning project for each unit?
- The 6th project can be over any topic from the entire semester. Feel free to get started on this one as soon as you have an idea.
- Remember you can submit only one SDL project per week. Plan ahead and don't let yourself get behind.
Finals week plans
- Our last class is Monday next week.
- Final Exam information:
- It's optional.
- Any score above 70% on the final exam will boost your grade a half letter step.
- The final exam has a time limit of 4 hours. Once you click start, the exam will appear for download. You then have 4 hours to complete the exam and upload your work.
- The final will be available in I-Learn. You can take the final at any time. No submissions will be accepted after Wed at midnight.
- For those who have not yet reached 80% mastery on the quizzes, there will be three more attempts for you to reach this benchmark.
- Quiz 13 will have objectives from units 3, 4, 5, and 6.
- Quiz 14A will only have objectives from units 5 and 6. This quiz is due Tues at 5pm, with solutions being released at 5pm. This quiz opens Mon after class.
- Quiz 14B will have all objectives on it. It is due on Wed at midnight. This quiz opens Monday Dec after class.
Brain Gains
- 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 $d\sigma = \left |\dfrac{\partial \vec r}{\partial u}\times\dfrac{\partial \vec r}{\partial v}\right|dudv$.
- 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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