Presentations

6.2+

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.

Brain Gains

Group problems

  1. Let $\vec F = (2x+3y, 4x+5y)$.
    1. Compute $D\vec F$.
    2. Why it is impossible to find a function $f$ so that $\vec F = \vec \nabla f$.
  2. Given a vector field $\vec F$, what condition must be true about $D\vec F$ for there to be a function $f$ such that $\vec\nabla f = \vec F$? We call such function $f$ a potential for $\vec F$. When $\vec F$ has a potential, we say that $\vec F$ is a gradient field.
  3. 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. $\vec F = (2x,3y)$ [Check: $D\vec F = \begin{bmatrix}2&0\\0&3\end{bmatrix}$ and $f = x^2+\frac{3}{2}y^2$ yields $\vec \nabla f = (2x,3y)$. We can quickly verify that $\vec\nabla f = \vec F$ by a direct computation. ]
    2. $\vec F = (2y,3x)$
    3. $\vec F = (3y,3x)$
    4. $\vec F = (4x,5y,6z)$
    5. $\vec F = (4x,5z,6y)$
    6. $\vec F = (4x,5z,5y)$
    7. $\vec F = (2x-y,-x+4y)$
    8. $\vec F = (y^2+2x,2xy)$
    9. $\vec F = (x+yz,xz+4yz,xy+2y^2)$
    10. $\vec F = (x+yz,4yz,xy+2y^2)$
    11. $\vec F = (x+yz,xz+4yz,xy)$
    12. $\vec F = (yz,xz+4yz,xy+2y^2)$
  4. Draw each curve or surface given below.
    1. $\vec r(t) = (3\cos t,3\sin t,4t)$ for $0\leq t\leq 6\pi$. (Check: Helix)
    2. $\vec r(u,v) = (u\cos v,u\sin v,u)$ for $0\leq v\leq 2\pi$ and $0\leq u\leq 4$. (Check: Cone)
    3. $\vec r(u,v) = (u\cos v,u\sin v,v)$ for $0\leq v\leq 6\pi$ and $2\leq u\leq 4$. (Check: Spiral stair case)
    4. $\vec r(t) = (0,t,9-t^2)$ for $0\leq t\leq 3$.
    5. $\vec r(u,v) = (u\cos v,u\sin v,9-u^2)$ for $0\leq v\leq 2\pi$ and $0\leq u\leq 3$.
  5. 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 go once counter-clockwise along the circle $\vec r(t) = (3\cos t, 3\sin t)$.
  6. 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 go once counter-clockwise along the circle $\vec r(t) = (3\cos t, 3\sin t)$.

Problem Set
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