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
- Draw the curve $\vec r(t) = (3\cos t,3\sin t,t)$ for $0\leq t\leq 6\pi$.
Solution
It's a helix, radius is 3, spiraling counterclockwise (when viewed from above) as it wraps around the $z$-axis for $0\leq t\leq 6\pi$.
- Draw the surface $\vec r(t,v) = (3\cos t, 3\sin t, v)$ for $0\leq t\leq 2\pi$ and $1\leq v\leq 4$.
Solution
It's a right circular cylinder of radius 3, whose center lies along the $z$-axis for $1\leq z\leq 4$.
- Draw the surface $\vec r(t,v) = (v\cos t, v\sin t, 3)$ for $0\leq t\leq \pi$ and $1\leq v\leq 4$.
Solution
It's half a washer from radius 1 to 4, whose center lies at $(0,0,3)$.
- For the vector field $\vec F = (x+2y+3z, 4x+5y+6z, 7x+8y+9z)$, compute
- $D\vec F$,
- $\vec \nabla \cdot \vec F$, and
- $\vec \nabla \times \vec F$.
Solution
We'll do this together in class.
- Compute the work done by the vector field $\vec F = (4x+2xy,x^2+2y)$ along the curve $C$ parametrized by $\vec r(t) = (3t-1,-5t+2)$ for $0\leq t\leq 1$. [Hint: First find a potential.]
Solution
The vector field has a potential as the derivative $D\vec F =\begin{bmatrix}- &2x \\2x &-\end{bmatrix}$ is symmetric.
- A potential for the vector field is $f(x,y) = 2x^2+x^2y+y^2$ (note $\int 4x+2xy dx = 2x^2+x^2y +C(y)$ and $\int x^2+2y dy = x^2y+y^2+D(x)$).
- The start point is $\vec r(0) = (-1,2)$ and the end point is $\vec r(1) = (2,-3)$.
The work done by $\vec F$ is the difference in potential, which gives $$\int_C\vec F\cdot d\vec r = f(2,-3) - f(-1,2)=(8-12+9)-(2+2+4) = 5-8 = 3.$$
The following two problems are essential problem 7 and 8 from the problem set. I'll leave the solutions here for those interested in doing these problems.
- Find a potential for the vector field $\vec G = \frac{(-x,-y,-z)}{(x^2+y^2+z^2)^{3/2}}$. Hint, start by computing $\int \frac{-x}{(x^2+y^2+z^2)^{3/2}}dx$.
Solution
Using the substitution $u=x^2+y^2+z^2$, we have $$\int \frac{-x}{(x^2+y^2+z^2)^{3/2}}dx = \frac{-1}{2}\int (u)^{-3/2}du =u^{-1/2} = (x^2+y^2+z^2)^{-1/2}.$$ A similar computation yields $$\int \frac{-y}{(x^2+y^2+z^2)^{3/2}}dy = (x^2+y^2+z^2)^{-1/2}\quad\text{and}\quad \int \frac{-z}{(x^2+y^2+z^2)^{3/2}}dz = (x^2+y^2+z^2)^{-1/2}.$$ A potential for $\vec G$ is $$g(x,y,z) = (x^2+y^2+z^2)^{-1/2} = \frac{1}{\sqrt{x^2+y^2+z^2}}.$$
- Find the work done by $\vec G = \frac{(-x,-y,-z)}{(x^2+y^2+z^2)^{3/2}}$ on an object as it moves from $(1,2,2)$ to $(0,3,4)$.
Solution
A potential for $\vec G$ is $$g(x,y,z) = (x^2+y^2+z^2)^{-1/2} = \frac{1}{\sqrt{x^2+y^2+z^2}}.$$ Work done is the difference in potential, which means $$W = g(0,3,4)-g(1,2,2) = \frac{1}{5}-\frac{1}{3} = -\frac{2}{15}.$$ The object moved from 3 units away from the origin to 5 units away from the origin, and negative work was done.
As a side note, the gravitational vector field is $\vec F = \frac{Gm_1m_2(-x,-y,-z)}{(x^2+y^2+z^2)^{3/2}}$, just a constant multiple of the one we worked with above. Electrostatics also uses a very similar vector field.
Group problems
- Recall $\vec \nabla = \left(\frac{\partial}{\partial x},\frac{\partial}{\partial y},\frac{\partial}{\partial z}\right)$. For the vector field $\vec F(x,y,z) = (3xy,2x+4y+5z,y+xz^2)$ do the following.
- Compute the derivative $D \vec F$ (you'll get a 3 by 3 matrix).
- Compute the divergence $\vec \nabla \cdot \vec F$.
- Compute the curl $\vec \nabla \times \vec F$. [Check: $\left(-4, -z^2, 2 - 3x\right)$.]
- Draw each curve or surface given below.
- $\vec r(t) = (3\cos t,3\sin t)$ for $0\leq t\leq 2\pi$.
- $\vec r(u,v) = (3\cos u,3\sin u,v)$ for $0\leq u\leq 2\pi$ and $0\leq v\leq 5$.
- $\vec r(u,v) = (4\cos u,v, 3\sin u)$ for $0\leq u\leq \pi$ and $0\leq v\leq 7$.
- $\vec r(t) = (3\cos t,3\sin t,4t)$ for $0\leq t\leq 6\pi$. (Check: Helix)
- $\vec r(u,v) = (u\cos v,u\sin v,u)$ 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)
- $\vec r(t) = (0,t,9-t^2)$ for $0\leq t\leq 3$.
- $\vec r(u,v) = (u\cos v,u\sin v,9-u^2)$ for $0\leq v\leq 2\pi$ and $0\leq u\leq 3$.
- 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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