MAT 223: Multivariable Calculus

Fall 2018

Basic Information

• Instructor: Jed Yang, CC221, 651-638-6405,
• Office hours: Tuesday and Friday 13:30–14:30; or by appointment (instructions)
• Lectures: Mod F (MWF 14:50–15:40) in RC229
• Course website: https://www.mathcs.bethel.edu/yang/mat223.18f/

Course Information

• Official course description: Differential calculus of real functions on Rⁿ: limits, continuity, partial and directional derivatives, mean value theorem, implicit functions, Taylor's Theorem, and optimization techniques (including Lagrange multipliers). Multiple integral theory: change of variables, iterated integrals, and line integration (Green's Theorem).
• Prerequisites: MAT 125: Calculus 2
• Textbook: Jon Rogawski, Multivariable Calculus, 3rd edition, 2015, ISBN: 9781464171758.
• Materials: Student versions of Mathematica will be given to you. Graphing calculators are recommended but not required. Electronic devices such as computers, calculators, and mobile phones are not to be used during exams.

Topics

Multivariable Calculus is a continuation of what you learned in first-year Calculus. We will take the concepts of functions, differentiation, and integration and will extend them to higher dimensions. That is, we will analyze what happens when there is more than one independent and/or dependent variable.

• Mathematical tools: parametric equations and curves, polar coordinates, conic sections, vectors, dot products, cross products, quadric surfaces, cylindrical and spherical coordinates
• Modeling motion: position, velocity, and acceleration vectors, curvature, Kepler's Laws
• Differentation: partial derivatives, gradients, optimization of real-valued functions of more than one variable
• Integration: of functions of more than one variable, double and triple integrals, change of coordinates
• Vector analysis: vector fields, line integrals, Green's theorem

Objectives

I will guide you in learning to:

• Engage in the process of doing and learning mathematics.
• Think and understand mathematics as opposed to practice memorized procedures.
• Make connections between different mathematical representations: in data (numerical), symbolic (algebraic), visual (graphical), and descriptive (written and verbal) forms.
• Collaborate with peers both inside and outside of class.
• Use technology in mathematical investigations through solving problems with authentic, real-world (sometimes "messy") contexts.
• See mathematical principles at work in the natural world and in everyday life.
• Understand and apply important ideas and results in Calculus.
• Employ symbolic manipulation skills in the context of Calculus.

Grading

Your grade will be determined by a weighted arithmetic mean of various components with weights listed in the table on the right. The total score will be converted to a letter grade whose lower bounds are: 93% A, 90% A-, 87% B+, 83% B, 80% B-, 76% C+, 70% C, 66% C-, 63% D+, 58% D, 0% F.

Note that there is no preset curve of how many of each letter grade will be given. If you all do A-level work, you will each get an A. As such, you are encouraged to help each other in the pursuit of perfection.

In many courses I intentionally make one exam harder than others, which gives me information (in a mathematical sense) in separating an A performance from an A- performance. Typically, I will let you know and adjust that exam scores upward. What this means is that you should NOT care about how hard an exam is. If you do A-level work, you will get an A, regardless of the raw numerical score prior to adjustment.

Besides possibly adjusting scores upward for difficult exams, I also reserve the right to lower the grade cutoffs. Both of these help you. I will not hurt you by adjusting your exam scores downward or increasing the grade cutoffs.

Requirements

Attendance and participation. I expect you to attend class. You may not notice me taking attendance during class meetings, but I will notice if you are not in class. Occasional absences will not impact your grade because what I look for is not mere attendance, but engagement and participation.

Indeed, coming to class is not just about showing up; it is also about being fully engaged in the learning experience. If you have a question, others in the class may also be wondering the same thing. So, please speak up and ask questions anytime you need to. Not only will you be helping yourself, but also you will be helping your peers. Attending office hours is another great opportunity to ask questions.

Be mindful of others. Refrain from using mobile phones or laptops for activities unrelated to the learning process. If you prefer to use laptops to take notes, please kindly sit in the back, as the screen may distract others. There is research that suggests taking notes by hand is better for long-term retention (P. A. Mueller and D. M. Oppenheimer, The pen is mightier than the keyboard, Psychological Science 25 (2014), 1159–1168).

Silence and put away mobile phones and do not use laptops for anything other than class-related activities.

It is my sincere hope that every one of you get all the points for attendance and participation.

Reading. Read the book! You should prepare for class by looking over the sections we will cover. Your aim is not to understand every detail, but to get a sense of where we are headed. Even a few minutes of pre-reading can help with class time. We will not have time to cover every single detail in class. As such, after class, read the sections carefully again to fill in the gaps. Keep up with the reading: reading large sections right before an exam is less effective!

Homework. Homework will be assigned most days. The goal of the homework is to give you an opportunity to continuously engage directly with the material. Some of the homework questions are meant to be challenging and to stretch you; simply put, I believe that the homework is where you will do the vast majority of your learning in this class. Grapple with the questions; talk to classmates about solution strategies if you are feeling stuck; do the homework.

Please staple your homework before coming to class and write your name, PO number, and homework number in the top right corner.

Homework is due at the beginning of the next class after it was assigned, unless otherwise stated. In general, late work is not accepted. If there are special circumstances, talk to the instructor. To alleviate your anxiety from accidentally forgetting to bring your homework to class, illness, emergencies, or other situations beyond your control, the lowest three (3) assignments will be dropped.

Because communicating results to others is an important skill, showing your work is as important as getting an answer. In many instances, credit will only be given if your work accompanies your answer. Some of the points will be given for completing the assignment; most will be awarded for showing work and correctness. You are encouraged to collaborate, but what you turn in must be your own work. See "Learning integrity" and the collaboration policy below.

Exams. There are three in-class midterm exams (see calendar below for a tentative schedule), weighted equally. Subsequent exams will mainly focus on the material covered since the previous exam, but can include previous material too. There will be a final exam during the official final exam period covering the entire course.

There are no make-up exams except in circumstances recognized by the instructor as beyond the control of the student. To receive this consideration, the instructor must be notified of the problem before the exam unless this is impossible, in which case as soon as possible.

Time outside of class. I expect a typical student to spend about two hours outside of class for each hour in class. Some students need to spend a bit more than that (which is okay). If you are spending more than 10 hours per week on this course outside of class time, please come talk to me so we can find ways to help you learn the material without spending so much time.

Illness. You should make every effort to attend class when you are healthy. If you become ill, for your well-being and the well-being of the rest of the class, you should not come to class. (Nor should you show up in my office with your germs!) Yes, this sounds like common sense, but it is tempting to try and power through as normal so as not to fall behind. If you become ill, or know that you will need to miss class for some reason, please contact me as soon as you are able, and we will work together to plan how you will keep up and/or make up any missed work.

Policies

Learning integrity.

Search me, O God, and know my heart;
Try me, and know my anxieties;
And see if there is any wicked way in me,
And lead me in the way everlasting.

- Psalm 139:23–24 NKJV

Collaborative work is an integral part of many successful ventures. As such, I expect that you should collaborate with your classmates a lot during your time in this course. However, it is important to understand that there is a big difference between thinking about and solving a problem as part of a group (which is good, both educationally and morally) and copying an answer or letting someone else copy your answer (which is bad, educationally and morally, and has punitive consequences).

In short, I trust you to maintain the utmost level of academic integrity in this course. Please do not break this trust; if you do, there will be repercussions. The formal policy below lays this out explicitly, and supplements Bethel's academic honesty policy.

Collaboration policy.

• You may collaborate on the homework assignments to the extent of formulating ideas as a group, but you may not collaborate in the actual writing of solutions (unless explicitly allowed in the instructions).
• In particular, you may not work from notes taken during collaborative sessions.
• You may not consult any materials from any previous offerings of this course or from any other similar course offered elsewhere unless explicitly permitted.
• You are required to completely understand any solution that you submit and, in case of any doubt, you must be prepared to orally explain your solution to me. If you have submitted a solution that you cannot verbally explain to me, then you have violated this policy.

Accommodation policy. Disability-related accommodations are determined by the Office of Disability Resources and Services (DRS). Students are responsible to contact the Office of Disability Resources and Services. Once DRS determines that accommodations are to be made, they will notify the student and the instructor via e-mail. Students choosing to use the disability-related accommodations must contact the instructor no later than five business days before accommodations are needed. The instructor will provide accommodations, but the student is required to initiate the process for the accommodations.

Concerns and appeals. If you have any concerns regarding the course, your grades, or the instructor, see the instructor first. If needed, see Bethel's academic appeals policy.

Getting Help

If you need help there are multitude of resources you can use:

• Yourself. If you're stuck on a problem or struggling with a concept from class, take a break and think about something else (e.g., your Hebrew assignment, the economics of Star Trek) for a few hours and then try a fresh start.
• Your classmates. You are each other's best resource: talking through the course material with someone else who is also trying to master it is a great way for you both to learn. (And don't discount the learning that you will do while trying to explain to a classmate an idea covered during class that you think you understand; I can't count the number of times that I've discovered that I didn't really understand something until I tried to teach it to someone.) The homework assignments are meant to challenge you, and figuring some of them out together is a great approach.
• Math Lab. The Math Department offers support for students enrolled in math classes by providing a Math Lab four days per week in HC 113 and 114. If you are having any difficulty with your homework in this class, please seek help from the tutors in Math Lab. The Math Lab is not only a great place to get help from tutors, but also is the perfect place to meet other students from class, do homework, and check your work. Plan Math Lab hours into your weekly schedule and develop this habit early on in the course.
• The instructor. Come to my office hours or email to make an appointment. To make an appointment, please email me several times you are available and try to give me at least 24 hours of lead time. Each afternoon, I will look at your availabilities and try to schedule as many people as I can fit for the next day. I will consistently reserve Thursdays for research, and I do not schedule office hours or make appointments for that day. I have this scheduled "research day" so that I can work on my research projects in an uninterrupted block of time. Without reserving a large block, I won't have time for any research. Tuesdays tend to be good days for appointments.

Calendar

Daily/weekly schedule to be updated throughout the term; topics and exam dates are tentative and subject to change.

Before class, please read the textbook section(s) to be covered. After class, start doing the homework assigned that day as soon as possible. Unless otherwise stated, homework will be due at the beginning of next class.

Date	Agenda	Homework
Week 1: Chapter 11 mathematical tools
1. 08/27 M	Introduction; 11.1 parametric equations	hw01: Getting started
2. 08/29 W	11.1	hw02: 11.1 # 8, 16, 20, 21*, 24, 28, 30, 32, 40, 46*, 58, 61–64, 76*. [note] 21: For odd-numbered questions, check answers in the back of the book; obviously, show your work to receive credit. 46: The vertical axis in one of the blue graphs should be labelled x instead of y. Which one? 76: A proof is a convincing argument. I and the TAs will NOT be convinced simply by a bunch of calculations. Use full sentences in English to explain your calculations. (This is good practice to follow for all problems, but it is especially important for proofs.)
3. 08/31 F	11.2 arc length	hw03: 11.2 # 1, 5, 8, 11, 18, 21, 33.
Week 2
4. 09/05 W	11.3 polar coordinates	hw04: 11.3 # 2, 4, 6, 7, 16, 20, 23, 24, 46.
5. 09/07 F	11.4 area and arc length in polar coordinates	hw05: 11.4 # 4, 7, 8, 13, 25, 41.
Week 3: Chapter 12 vector geometry
6. 09/10 M	11.5 conic sections	hw06: 11.5 # 3, 8, 12, 16, 26, 37, 50, 54, 58.
7. 09/12 W	12.1 vectors in 2D	hw07: 12.1 # 1, 16, 18, 21, 27, 37–40, 42, 44, 62.
8. 09/14 F	12.2 vectors in 3D	hw08: 12.2 # 3*, 4, 9–12*, 22, 24, 30, 40, 44, 46, 52. * 3: sketch; 10–11: no need to sketch.
Week 4
9. 09/17 M	12.3 dot products	hw09: 12.3 # 1, 11, 13, 23, 32–34, 38, 41, 46, 49, 53, 57, 64, 67, 71.
10. 09/19 W	12.4 cross products	hw10: 12.4 # 7, 11, 16, 22, 24, 27, 28, 30, 36, 37, 54, 70. Due Monday.
11. 09/21 F	Exam 1 (topics and tips)
Week 5
12. 09/24 M	12.5 planes	hw12: 12.5 # 2, 11, 12, 13, 19, 22, 27, 29, 39, 57, 62, 63, 66, 69.
13. 09/26 W	12.6 quadric surfaces	hw13: 12.6 # 2, 4, 6, 8, 10, 12, 41, 42, 46, 47*. * 47: Consider writing $c$ and $d$ instead of $\alpha$ and $\beta$.
14. 09/28 F	12.7 cylindrical and spherical coordinates	hw14: 12.7 # 2, 8, 14, 24, 30, 34, 40, 52, 62, 68, 83.
Week 6: Chapter 13 vector-valued functions
15. 10/01 M	13.1 vector-valued functions	hw15: 13.1 # 1, 5, 8, 11–13, 21a, 23–25, 32, 34.
16. 10/03 W	13.2 calculus of vector-valued functions	hw16: 13.2 # 4, 6, 10, 16, 20, 22, 28, 34, 36, 37, 43, 54, 57.
17. 10/05 F	13.3 arc length	hw17: 13.3 # 3, 9, 15, 18, 19, 32, 36.
Week 7
18. 10/08 M	13.4 curvature	hw18: 13.4 # 1, 7, 9, 15, 23–26, 28, 29.
19. 10/10 W	13.4 normal vector, osculation	hw19: 13.4 # 43, 46, 47, 56, 58, 60, 61.
Week 8: Chapter 14 differentiation in several variables
20. 10/15 M	13.5 motion	hw20: 13.5 # 4, 16, 19, 20, 21, 29, 30, 35, 37.
21. 10/17 W	14.1 multivariable functions	hw21: 14.1 # 6, 14, 19, 20, 32*, 34*, 37, 38, 39, 44–47. * You may use Mathematica and attach the printout. Due Monday.
22. 10/19 F	Exam 2 (topics and tips)
Week 9
23. 10/22 M	14.2 limits and continuity	hw23: 14.2 # 6–18*, 22–36*. * Even problems only.
24. 10/24 W	14.3 partial derivatives	hw24: 14.3 # 1, 7–12, 16, 20, 24, 34, 43, 49, 64, 65.
25. 10/26 F	14.4 differentiability	hw25: 14.4 # 3, 6, 11, 13, 15, 22, 28, 35, 39.
Week 10
26. 10/29 M	14.5 gradient	hw26: 14.5 # 2–4, 8, 13, 21, 27, 33, 37, 38, 40, 41, 43, 46, 52, 54, 55.
27. 11/02 F	14.6 chain rule	hw27: 14.6 # 7, 13, 15, 18, 20, 21, 22, 24, 26, 28.
Week 11: Chapter 15 multiple integration
28. 11/05 M	14.7 optimization	hw28: 14.7 # 4, 6, 11, 15, 24, 28*, 30, 31, 33, 34. * 28: This exercise is asking 12 separate questions, 2 for each part. Of course, you may organize your answers in an efficient display.
29. 11/07 W	14.8 Lagrange multipliers	hw29: 14.7 # 35, 40. 14.8 # 1, 3, 7, 11, 17, 49.
30. 11/09 F	15.1 double integrals	hw30: 14.7 # 47*. * 47: Do this twice: first without using Lagrange multipliers and then again with Lagrange multipliers. 15.1 # 17, 19, 20, 22, 23, 24, 34, 37, 43, 47.
Week 12
31. 11/12 M	15.2 more general double integrals	hw31: 14.7 # 50*. * 50: Do this once: either using Lagrange multipliers or not; up to you! 15.2 # 3, 9, 17, 21, 28, 30, 34.
32. 11/14 W	15.3 triple integrals	hw32: 15.2 # 42, 46, 52, 61. 15.3 # 9, 15, 21, 28, 39. Due Monday.
33. 11/16 F	Exam 3 (topics and tips)
Week 13: giving thanks
34. 11/19 M	15.5 applications	hw34: 15.5 # 1, 5, 11, 23, 49, 50, 53, 54, 60, 61*, 62*. * 61, 62: no integration needed!
Week 14: Chapter 16 line integrals
35. 11/26 M	16.1 vector fields	hw35: 16.1 # 2, 6, 8, 12, 13–20, 26, 30, 38, 42, 44, 50.
36. 11/28 W	16.2 line integrals	hw36: 16.2 # 5, 7, 9, 13, 23, 27, 37, 38, 41, 42, 43, 45, 53, 55.
37. 11/30 F	16.3 conservative vector fields	hw37: 16.3 # 3, 10–12, 17, 19, 21–23.
Week 15: Chapter 17 fundamental theorem
38. 12/03 M	17.1 Green's theorem (planimeter video)	hw38: 17.1 # 3, 7, 13, 16, 24, 29, 30, 32, 33.
39. 12/05 W	15.4 integrating in polar coordinates	hw39: 15.4 # 4, 10, 22, 28. 15.5 # 4, 13, 21. 17.1 # 5.
40. 12/07 F	15.6 change of variables	hw40: None.
Final Exam: 12/10 Monday 14:45–16:45 (topics and tips)