Math 215 - Multivariate Calculus - Schedule

Presentations

6.10+

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 occurs on Friday of week 13.
- Quiz 14A occurs on the last day of classes, which is Tuesday April 7, 2026.
- Quiz 14B occurs during our course's scheduled final exam time, which is Wed April 8 at 10:30 AM – 1:45 PM.

The quizzes have included questions for 28 objectives. How many have you passed? What are your 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.

Consider the surface $\vec r(u,v) = (u\cos v,u\sin v,9-u^2)$ for $0\leq v\leq 2\pi$ and $0\leq u\leq 3$.
1. Draw the surface.
2. Compute the differential $d\vec r$.
3. The vector $\frac{\partial \vec r}{\partial u}du$ is a short (assuming $du$ is small) tangent vector to the surface. Compute $\ds \left|\frac{\partial \vec r}{\partial u}du \times \frac{\partial \vec r}{\partial v}dv\right|$.
4. Give a physical interpretation of $\left|\vec r_u \times \vec r_v\right|dudv$.
5. Set up an iterated integral to give the surface area of the surface above.
6. Set up an integral to give the $z$-coordinate of the centroid of the surface.
Let $\vec F = (-5y,5x)$.
1. Set up a single integral to give the work done by $\vec F$ to go once counter-clockwise along the circle $\vec r(t) = (3\cos t, 3\sin t)$.
2. Does $\vec F$ have a potential?
3. 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. Set up the double integral that gives the same result as the single integral we already set up.

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. $\vec r(u,v) = (u\cos v,u\sin v,v)$ for $0\leq v\leq 6\pi$ and $2\leq u\leq 4$. (Check: Spiral stair case)
3. $\vec r(u,v) = (u\cos v,u\sin v,u)$ for $0\leq v\leq 2\pi$ and $0\leq u\leq 4$. (Check: Cone)
4. For the cone in part c., compute $\left|\vec r_u \times \vec r_v\right|$ and then set up an iterated integral that would give the surface area of the cone.
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 $\vec r(t) = (3\cos t, 3\sin t)$.
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. $\vec F = (y^2+2x,2xy)$
8. $\vec F = (x+yz,xz+4yz,xy+2y^2)$
9. $\vec F = (x+yz,4yz,xy+2y^2)$
10. $\vec F = (x+yz,xz+4yz,xy)$
11. $\vec F = (yz,xz+4yz,xy+2y^2)$