• I-Learn, Problem Set, Class Pictures

Finals week plans

• Our last class will be today. We won't have class Monday, so that you can use the time for quizzes, SDLs, or the final exam. If you have completed 80% mastery on the quizzes, then the only thing left to complete is the final exam.
  • Any score above 70% on the final exam will boost your grade a half letter step, while a score less than 30% will drop your grade a half letter step.
  • The final exam will not be graded in mastery fashion, rather will be graded in a more traditional way (to make comparisons with other courses).
  • 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 appear tonight at 5 in I-Learn. You can take the final at any time up through Wednesday next week at midnight.
• For those who have not yet reached 80% mastery on the quizzes, there will be two more attempts for you to reach this benchmark.
  • Quiz 14A will only have objectives from units 5 and 6. This quiz is due Tuesday at 5pm, with solutions being released at 6pm. This quiz opens Monday morning at 12am.
  • Quiz 14B will have all objectives on it. It is due on Wednesday at midnight. This quiz opens Monday at 5pm (I need time to finish making it. I have to create a quiz with 32 objectives, whereas hopefully you only need to pass a few.)

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)

When I see a problem involving work, I always go through this three step process.

1. Is the curve a simple closed curve (does it form the boundary of some region R)? If so, use Green's Theorem $\int_CMdx+Ndy = \iint_R N_x-M_ydA$.
2. Does the vector field have a potential? Use the fundamental theorem of line integral $\int_C\vec\nabla f \cdot d\vec r = f(B)-f(A)$, work done is the difference in potential.
3. If the curve is not a simple closed curve, and there is no potential, then return to unit 2 and directly use the long formula $\int_CMdx+Ndy$ (and cry a little and know I'll probably make lots of mistakes).

• 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] $.

• 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] $.

• 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] $.

• 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 $\vec r(-1,1)$.

Group problems

1. Consider the surface parametrized by $\vec r(u,v) = (u\cos v, u\sin v, u^2)$ for $0\leq u\leq 2$ and $0\leq v\leq 2\pi$.
   1. Draw the surface.
   2. Compute $d\sigma = \left |\dfrac{\partial \vec r}{\partial u}\times\dfrac{\partial \vec r}{\partial v}\right|dudv$.
   3. Set up an integral formula to compute $\bar z$ for this surface.
2. Consider the parametric surface $\vec r(u,v) = (u, u\cos v,u\sin v)$ for $0\leq v\leq \pi$ and $0\leq u\leq 4$.
   1. Draw the surface.
   2. Compute $\dfrac{\partial \vec r}{\partial u}\times\dfrac{\partial \vec r}{\partial v}$.
   3. Give an equation of the tangent plane to the surface at $(u,v) = (1,\pi/2)$.
   4. Set up an integral formula to compute the surface area $\sigma$.
   5. Set up an integral formula to compute $\bar y$ for this surface.
3. Let $\vec F = (5y,5x)$.
   1. Find a potential for $\vec F$, or explain why none exists.
   2. Compute the work done by $\vec F$ to get from $(3,0)$ to $(0,5)$ along a straight line path.
   3. Compute the work done by $\vec F$ to go once counter-clockwise along the circle $\vec r(t) = (3\cos t, 3\sin t)$.
4. Let $\vec F = (-5y,5x)$.
   1. Find a potential for $\vec F$, or explain why none exists.
   2. Compute the work done by $\vec F$ to get from $(3,0)$ to $(0,5)$ along a straight line path.
   3. Compute the work done by $\vec F$ to go once counter-clockwise along the circle $\vec r(t) = (3\cos t, 3\sin t)$. [Use Green's Theorem.]
5. 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.
   1. $\vec F = (2x-y,-x+4y)$
   2. $\vec F = (y^2+2x,2xy)$
   3. $\vec F = (x+yz,xz+4yz,xy+2y^2)$
   4. $\vec F = (x+yz,4yz,xy+2y^2)$
   5. $\vec F = (x+yz,xz+4yz,xy)$
   6. $\vec F = (yz,xz+4yz,xy+2y^2)$
6. Compute the divergence and curl of each vector field.
   1. $\vec F = (x+yz,xz+4yz,xy+2y^2)$
   2. $\vec F = (x+yz,4yz,xy+2y^2)$
   3. $\vec F = (x+yz,xz+4yz,xy)$
   4. $\vec F = (yz,xz+4yz,xy+2y^2)$
   5. If a vector field has a potential, then what is the curl of that vector field?

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Problem Set
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