Class Meets MWF 1:30–2:20

Room Keller Hall 303

See here for the syllabus.

## Announcements

• (12/6) The final exam will be Monday, December 10 from 2:15pm to 4:15pm. The final will be held in the usual classroom. See here for the full final schedule.

• (12/4) Here is the formula sheet you will have for the final. Remember that no notes are permitted!

• (11/27) Here are a bunch of practice questions for the final exam, taken from your textbook:

• Chapter 11: (p. 650) 9–12, 17, 18, 27, 28, 37, 38, 39, 44, 53

• Chapter 14: (p. 836) 8–10; (p. 838) 1–18, 21, 23–28, 32–37; (p. 840) 6, 8, 9

• Chapter 15: (p. 917) 1, 4, 5, 9, 10, 18, 19, 21, 22, 25, 26, 27, 29–34, 37b, 39, 40, 49–51, 53, 54

• (11/20) I will be letting you make up two of the problems on the midterm, namely numbers 3 and 6. The problems are here. You can see the grades for these problems, given by your student ID, here. If you wish to make up the problems, turn in your solutions on Monday, November 25 and I will grade them, replacing your current scores for the problems. However, for the make up problems you will only be able to get up to 80% of the maximum partial scores for them, namely 12 for number 3 and 8 for number 6.

• (11/14) Here is the formula sheet I will be giving you for the in-class portion of the second midterm. (If there is something I told you that you didn’t need to memorize which isn’t on the formula sheet, then maybe it’s not on the in-class portion…)

• (11/7) The study guide for the second midterm can be found here. Answer key:

1. $\mathrm{curl}\,\vec F = (2-ye^{yz})\vec\imath - x \vec k$; $\mathrm{div}\,\vec F = y + ze^{yz} + 1$

2. $128/15$

3. $9\pi$ (Misprint in the problem. The cone should be $\phi = \pi/3$.)

4. $2\pi/3$

5. $2\sqrt 6 + 2\log(\sqrt 2 + \sqrt 3)$

6. $1/8$ (Misprint in the problem. The transform should be $x = u - v$ and $y = v/2$.)

7. $-5/6$

8. $0$

9. Calculate the partial derivatives of the components of $\vec F$.

10. $-\pi$

• (11/5) There will be no class next Monday (November 12), as the university is closed for (the day after) Armistice Day.

• (11/2) The second midterm will be on Monday, November 19, with the take-home portion of the exam due Wednesday, November 21. The final exam will be Monday, December 10 from 2:15pm to 4:15pm. The final will be held in the usual classroom. For the full final schedule, see here.

• (10/24) As I mentioned at the beginning of class, I’m doing an informal mid-semester evaluation to give you an opportunity to give feedback on my teaching. You can find it here. Participation is optional, but please respond by next Monday. As a reminder, the Google form is set up to not record your email address. Mahalo!

• (10/24) Math majors: The mandatory group advising session is the coming Monday, from 5 to 6:30pm. See this flyer for more information.

• (10/8) The Intergovernmental Panel on Climate Change recently released a report which shows that there is a high risk of climate change-related crisis as early as 2040. For a summary of the report, see this New York Times article.

• (9/28) See the answer key for the first midterm.

• (9/17) I uploaded a study guide for the first midterm. Here are the answers: (1) The answer was in your heart all along. (2) We did this in class, and I don’t recall the answer I got. (3) Many possible answers. I got $-9x - 8y + 6z = 5$. (4) $e^\pi - 1$. (5) $81\pi/2$. (6) $(8e^3 - 4)/9$. [This is wrong, but I don’t remember the answer we calculated in class. A much uglier expression.] (7) $7$. (8) Just set up the integrals. (9) $9/10$. If you find a mistake in these answers, please let me know so I can correct it.

• (9/6) The first midterm will be Monday, September 24. It will cover material from sections 11.5, 11.6, and 14.1–14.4, corresponding to the material up to the homework due September 19.

• (9/3) Here are some online resources for graphing functions of two variables, which can be useful in visualizing what’s going on. I highly recommend referencing them while doing your homework.

• WolframAlpha. You can easily plot graphs.

• You can plot graphs right in google’s search bar. The syntax is “plot f(x,y)”, where f(x,y) is the function you want to plot. See a sample plot.

• If you have access to any of them, Matlab, Maple, and Mathematica all have built-in 3d plotting capabilities.

• (8/22) The university is closed tomorrow and Friday, so we will not meet on Friday.

• (8/22) The tentative dates for the midterms will be:

• Midterm 1: Friday, September 21 Monday, September 24.

• Midterm 2: Friday, November 16.

• (8/20) Please note that I pushed the homework for the first week to be due next week.

## Homework

I will grade among the bolded problems.

• Week 1: No homework.

• Week 2: (Due Wednesday Aug 29) Section 11.5: 1, 4, 8, 23, 28, 33, 42, 48

• Week 3: Canceled! Material will be worked into later homeworks. (Due Wednesday Sept 5) tentatively Section 11.6: 1–12 (evens), 45; Section 14.1: 1, 5, 10, 12, 14, 17, 19.

• Week 4: (Due Wednesday Sept 12) Section 11.6: 45; Section 14.1: 1–6, 9, 10, 15, 16, 17, 21, 27.

• Week 5: (Due Wednesday Sept 19) Section 14.2: 1–4, 13, 15–20, 25, 54; Section 14.3: 2, 7, 9, 11, 15, 16; Section 14.4: 1–3, 6, 8, 20, 23, 33

• Week 6: No homework due to exam.

• Week 7: (Due Wednesday Oct 3) Section 14.5: 9, 17, 22, 23, 30, 31; Section 14.6: 1, 10, 25.

• Bolded Problem You have a weighted die, conveniently represented by the unit cube $0 \le x \le 1$, $0 \le y \le 1$, $0 \le z \le 1$. The density at a point $(x,y,z)$ in the die is given by the function $\delta(x,y,z) = 1 + z$. Find the center of mass of the die.

• Bolded Problem You have another weighted die, but instead of being a cube it is an (eight-sided) octahedron. This solid is given by the inequality $\lvert x\rvert + \lvert y\rvert + \lvert z\rvert \le 3$. The density at a point $(x,y,z)$ in the die is given by $\delta(x,y,z) = 3 + x + y + z$. Sketch a picture of the octaderal die and set up the integral to calculate its mass. (Hint: break the die up into four regions and find the limits of integration for each region separately.) Also set up the integral to calculate one of its moments (your choice which). Without calculating those integrals, which octant do you think the center of mass is in? Explain your reasoning.

• Week 8: (Due Wednesday Oct 10) Section 14.7: 4, 7, 9, 14, 17, 21, 24, 28, 32, 33, 38, 39.

• Week 9: (Due Wednesday Oct 17) Section 14.8: 1, 3, 6, 7, 9, 17, 20; Section 15.1: 9, 13, 14, 19, 21.

• Week 10: (Due Wednesday Oct 24) Section 15.2: 2, 3, 13, 14, 23, 30, 38.

• Week 11: (Due Wednesday Oct 31) Section 15.3: 2, 4, 10, 17, 22, 34.

• Based upon what you know about how to check whether a vector field over $\mathbb R^2$ or $\mathbb R^3$ is conservative, how do you think you can check whether a vector field over $\mathbb R^4$ is conservative? That is, let $\vec F(x,y,z,w) = (M,N,P,Q)$ be a vector field whose inputs and outputs are length 4 vectors of real numbers. Write a condition for $\vec F$ being conservative in terms of the partial derivatives of $M,N,P,Q$.

• (Bolded problem) Calculate the following, where $f(x,y,z) = x\sin(yz)$ and $\vec G = (xyz,0,x+y+z)$:

• $\operatorname{div} \vec G$

• $\operatorname{grad} f$

• $\operatorname{curl} \vec G$

• $\operatorname{grad} \operatorname{div} \vec G$

• $\operatorname{div} \operatorname{grad} f$

• $\operatorname{div} \operatorname{curl} \vec G$

• It makes sense to take the divergence of the curl of a vector field, because $\operatorname{curl} \vec F$ is a vector field and divergence is applied to vector fields. But it doesn’t make sense to take the curl of the divergence of a vector field, because $\operatorname{div} \vec F$ is a scalar function and curl is applied to vector fields. Which of the following make sense?

• $\operatorname{grad} \operatorname{curl} \vec F$

• $\operatorname{curl} \operatorname{grad} f$

• $\operatorname{grad} \operatorname{div} \operatorname{curl} \vec F$

• $\operatorname{curl} \operatorname{div} \operatorname{grad} \operatorname{curl} \operatorname{curl} \vec F$

• $\operatorname{curl} \operatorname{curl} \operatorname{curl} \operatorname{grad} \operatorname{div} \operatorname{curl} \vec F$

• (It may help to think of curl, div, and grad as spam, bacon, and eggs.

• Week 12: (Due Wednesday Nov 7) Section 15.4: 5, 6, 7, 12, 17, 20, 24; Section 15.5: 2, 13, 18, 31b (use the parameterization from 31a).

• (bolded problem) Show that if $f(x,y,z)$ has continious second partial derivatives then $\operatorname{curl} \operatorname{grad} f = (0,0,0)$. (Hint: use Schwarz’s theorem.)
• Week 13: (Due Wednesday Nov 14) Section 15.6: 3, 5, 19, 22, 32; Section 15.7: 2, 7, 10, 19.

• Week 14: No homework due to exam.

• Week 15: (Due Wednesday Nov 28) Section 15.8: 5, 8, 10, 17b, 26, 27, 31.