MAT 223: Multivariable Calculus

Spring 2021

Basic Information

• Instructor: Jed Yang,
• Office hours: Tuesday and Friday 13:30–14:30; or by appointment (instructions)
• Lectures: Mod E (MWF 13:50–14:40) in Zoom
• Course website: https://www.mathcs.bethel.edu/yang/mat223.21s/

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.

Week	Monday	Wednesday	Friday
Unit 1: calculus of vector-valued functions
1	1. 02/01 M Introduction hw02: Getting started	2. 02/03 W 9.2 parametric equations hw03: 9.2 # 5*, 17*, 20, 21, 24, 31, 32, 39, 43, 50, 51. * 5: For odd-numbered questions, check answers in the back of the book; obviously, show your work to receive credit. * 17: Use Mathematica; export as PDF and upload.	3. 02/05 F 9.3 calculus thereof hw04: 9.3 # 6*, 7, 14, 15, 33, 35, 36, 41, 42. * Find an equation $y = f(x)$ for the parametric curve and compute $dy/dx$ in two ways: using Key Idea 9.3.1 and by differentiating $f(x)$.
2	4. 02/08 M 9.4, 9.5 polar coordinates hw05: 9.4 # 6, 8, 10, 17*, 31, 46*. * 17: Use Mathematica. * 46: In the form $r = f(\theta)$. 9.5 # 4, 12, 27, 31.	5. 02/10 W 10.1, 10.2 vectors hw06: 9.5 # 21, 26, 30. 10.1 # 7, 10. 10.2 # 8, 11, 13, 14, 17, 18, 22, 24.	6. 02/12 F 10.3 dot products hw07: 10.3 # 5, 7, 11, 13*, 15*, 18, 21, 26, 27, 32. * Radians only.
3	7. 02/15 M 10.4 cross products hw08: 10.4 # 7, 17, 21, 22, 23, 27–35*. * Odd problems only.	8. 02/17 W 10.5, 10.6 lines and planes hw09: 10.5 # 6, 8, 17, 20. 10.6 # 4, 5, 8, 9, 11*, 13*, 15, 17, 20. * Think through, but do not turn in.	9. 02/19 F 11.1 vector-valued functions hw10: 10.5 # 9, 12, 18, 19. 11.1 # 11*, 15*, 18, 22, 26, 28, 31, 32. * Use Mathematica with ParametricPlot and ParametricPlot3D.
4	10. 02/22 M 11.2 calculus thereof	11. 02/24 W exam1 (topics and tips) hw12: 11.2 # 6, 8, 10, 12, 14, 19*, 20*, 23–26, 29, 32. * To sketch $\vec r(t)$, you may first use Mathematica to plot it; print out or copy down by hand. Do everything else by hand. Also, write one (short) sentence summarizing the main takeaway of these pair of exercises.	12. 02/26 F 11.2, 11.3, 11.5 arc length hw13: 11.2 # 34, 35, 37, 40, 41. 11.3 # 8, 12*, 25, 31, 35. * May use Mathematica.
5	13. 03/01 M 11.4, 11.5 curvature hw14: 11.3 # 15, 17, 36. 11.4 # 7, 8, 11, 12. 11.5 # 7, 10.	14. 03/03 W 11.4, 11.5 motion hw15: 11.4 # 17, 21*, 27, 28. * Also find $\mathbf{B}=\mathbf{T}\times\mathbf{N}$ and confirm that it is a unit vector. 11.5 # 13*, 17*, 19*, 22*. * Sketch by hand or with Mathematica.	15. 03/05 F 12.1 multivariable functions hw16: 11.5 # 23, 25, 29, 31. 12.1 # 8, 10, 14, 17*, 20*, 22*, 24, 26, 30. * Use ContourPlot on Mathematica first (no need to specify $c$ values). Then, find the equations of the level curves by hand. Finally, plot those level curves with Plot (not ContourPlot) on Mathematica. Upload PDF of both ContourPlot and Plot output.
Unit 2: calculus of multivariable functions
6	16. 03/08 M 12.2 limits and continuity hw17: AQ # 1–6*. * Find these Additional Questions posted on Moodle.	17. 03/10 W 12.3 partial derivatives hw18: 12.3 # 6, 10–18*, 28, 32. * Even problems only.	18. 03/12 F 12.4, 12.7 differentiability hw19: 12.4 # 6, 12, 16, 18, 20, 22. 12.7 # 18, 20. AQ # 7, 8.
7	(Spring break)	(Spring break)	(Spring break)
8	19. 03/22 M 12.6 gradient and directional derivatives	20. 03/24 W exam2 (topics and tips) hw21: 12.6 # 8–28 evens. 12.7 # 22, 24.	21. 03/26 F 12.5 chain rule hw22: 12.5 # 8, 10, 14, 16, 20–30 evens.
9	22. 03/29 M 12.8 optimization hw23: 12.2 # 7–10, 12, 13. 12.8 # 6–14 evens. AQ # 9, 10.	23. 03/31 W Lagrange multipliers hw24: 12.8 # 16, 18. Exercises # 2, 3*, 5*, 7*, 8, 10. AQ # 11, 12. * Check answers.	(Good Friday)
10	(Easter Monday)	24. 04/07 W 13.1 iterated integrals hw25: 13.1 # 5, 7, 9, 12, 14, 16. AQ # 13, 14.	25. 04/09 F 13.2 double integrals hw26: 13.2 # 5, 8, 14, 18, 20, 22, 23, 26.
11	26. 04/12 M 13.6 triple integrals	27. 04/14 W exam3 (topics and tips) hw28: 13.6 # 6, 14, 18–20.	28. 04/16 F 13.4 center of mass hw29: 13.4 # 11, 12, 14, 19, 20. 13.6 # 24.
Unit 3: calculus of vector fields
12	29. 04/19 M 14.1 scalar line integrals hw30: 14.1 # 5, 7, 9, 12*, 14*, 15, 17, 19. * Set up by hand, evaluate using Mma or calculator, write the answer; no need to print/attach Mma output.	30. 04/21 W 14.2 vector fields hw31: 14.2 # 6–18 evens.	31. 04/23 F 14.3 vector line integrals hw32: 14.3 # 7, 8, 11, 12, 15, 16, 18, 20.
13	32. 04/26 M 14.4 Green's theorem - planimeter video hw33: 14.4 # 13–15, 17, 18.	33. 04/28 W 13.3, 13.7 polar, cylindrical, and spherical coordinates	34. 04/30 F exam4 (topics and tips) hw35: 13.3 # 6, 10*, 12*, 14*. 13.7 # 6, 26, 32. * Set up by hand, evaluate using Mma or calculator, write the answer; no need to print/attach Mma output.
14	35. 05/03 M 13.7 change of variables hw36: Handout on Moodle.	36. 05/05 W 14.5, 14.6 scalar surface integrals hw37: 14.5 # 4, 6, 16*, 17, 19, 22*, 24*. * 16: parametrize both bounding surfaces. * 22, 24: Set up by hand, evaluate using Mma or calculator, write the answer; no need to print/attach Mma output.	37. 05/07 F 14.6 vector surface integrals hw38: 14.6 # 5*–13 odds. * 5: Density function should be $\delta(x,y,z)=z+10$.
15	38. 05/10 M 14.7 divergence theorem hw39: 14.7 # 5–7. worksheet # 4.	39. 05/12 W 14.7 Stokes' theorem	40. 05/14 F exam5 (topics and tips)
Final Exam: 05/18 Tuesday 08:15–10:15 (cancelled)

Course Information

• Official course description: Differential calculus of real functions on Rⁿ: limits, continuity, partial and directional derivatives, implicit functions, and optimization techniques (including Lagrange multipliers). Multiple integral theory: iterated integrals, change of variables. Vector analysis: dot and cross products, curl, divergence, line integrals, and fundamental theorems.
• Prerequisites: MAT 125: Calculus 2
• Textbook: Gregory Hartman, APEX Calculus, version 4.0, volume 3, 2018, ISBN: 9781719263665.

  The textbook can be downloaded for free or purchased at-cost from Amazon. It is also available online; see supplemental resources at the bottom of this page.

• Materials: Student versions of Mathematica will be given to you. Graphing calculators are not required. Calculator functionality (including on computers 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 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, vectors, dot products, cross products, polar, cylindrical, and spherical coordinates
• Modeling motion: position, velocity, and acceleration vectors, curvature
• Differentiation: 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, surface integrals, Stokes' theorem, divergence theorem (if time allows)

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.
• 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.

Structure and Adjustments

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-, 77% C+, 72% C, 69% C-, 66% D+, 60% 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

Whatever you do, work at it with all your heart, as working for the Lord, not for human masters, since you know that you will receive an inheritance from the Lord as a reward. It is the Lord Christ you are serving.

- Colossians 3:23–24 NIV

I will be trying to make these verses true for me as I work with you throughout this course, and I hope that you will, too.

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. 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).

In general, you are expected to keep your video on most of the time to make your engagement more clear. Let me know if there are legitimate reasons why you need to be off-camera frequently.

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.

Each homework must be scanned and uploaded to Moodle. Instructions for scanning will be given to you.

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 submit homework before class, illness, emergencies, or other situations beyond your control, the lowest four (4) 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. 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 several in-class midterm exams (see calendar for a tentative schedule). 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 to three 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, you should not come to class. 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 may not look up solutions in any form, including from solution manuals or online repositories.
• 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 Accessibility Resources and Services (OARS). Students are responsible to contact the Office of Accessibility Resources and Services. Once OARS 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.
• The book. Reread the sections we covered in class to fill in the gaps.
• 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. Hours, as well as logistics for attending (in person or remotely) will be provided on Moodle. 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 (remote) office hours or email to make an appointment. Please read and follow instructions.
• Supplemental resources.
  • UND has a slightly modified online version of our textbook. The chapter numbers are off by 1; the section numbers seem to be the same. There seems to be a pretty good (and short!) video embedded in each section of UND's online textbook. I think the placement of the video in each section is deliberate. I suggest that you watch the video after first reading what comes before the video (read their version online or from the corresponding section in your textbook). Then, after watching the video, read the rest of the section.
  • More video examples: mathispower4u
  • If you find good resources, let me know and I can post them here for everyone.