# Math 318 - Spring 2019

Instructor: Dr. Silas Johnson
Office: Cupples I, room 107A
Email: silas@wustl.edu
Office hours: Tu 10:30-11:30, Tu 4:15-5:15, Th 2:30-4:00

Lectures (with Dr. Johnson):

• Section 1: 2-3 MWF, Cupples I 113
• Section 2: 3-4 MWF, Cupples I 113

## Course Outline

This course is a more in-depth and theoretical treatment of multivariable calculus and related topics than Math 233. It assumes knowledge of basic computations (partial derivatives, double integrals, line integrals) from Math 233, as well as linear algebra on the level of Math 309. The course will synthesize these ideas, with a result that is significantly more than the sum of its parts. In addition, we will approach mathematical structures like vector spaces from a more advanced perspective, which will provide useful background for students who take courses like Math 4111, 4171, and 429 later on (although this course is not a prerequisite for any of those).

Learning Objectives:

• Gain fluency with vector spaces and the basic language of point-set topology (open and closed sets, etc.)
• Analyze vector-valued functions of vector variables as single objects, and understand the meaning of the derivative of such a function and its relationship to partial derivatives
• Compute derivatives of vector-valued functions, including through the use of the Chain Rule
• Understand and use the Inverse and Implicit Function Theorems
• Define and visualize manifolds, and analyze them algebraically
• Integrate functions over manifolds and interpret the results
• Analyze form fields algebraically
• Integrate form fields over manifolds
• Relate form-field-over-manifold integration to classical theorems of vector calculus
In addition, you should be building a number of general skills throughout the course, such as:
• Solving problems you have never seen before
• Working with others to solve problems
• Communicating your own work to others in a way they can understand
• Comparing and contrasting different problems or topics
• Understanding how different ideas depend on each other

## Teaching and Learning Philosophy

The best (and possibly only) way to learn math is to do math. It is crucial to your success that you engage with the material both in and outside of class. We will try to spend a class time learning actively, rather than passively, in a variety of ways. Be open to the fact that class may not always look like what you think a lecture has to look like!

You will have better results if you approach learning as a community activity. (After all, if that weren't important, you could take this class online!) The best resource you have is not the textbook, old exams, homework problems, or office hours, but your classmates. Use them! Work together in and outside of class, form study groups, ask for and give help. Conversely, remember that you are the best resource they have too, and communicating your ideas to others is a great learning technique and a crucially important skill. The second-best way to learn math is to teach it to someone else.

Finally, don't be afraid to make mistakes. Despite all our emphasis on grades, failure is a crucial part of the learning process, and you should not expect to get everything right the first time. It is also your responsibility to help create an environment in which your classmates can safely engage in the process of productive failure.

## Textbook and Materials

Textbook: Calculus of Several Variables, by Brian E. Blank. This book is available as a free pdf and will be posted on Canvas; you can print it out as desired. In return for this free textbook, please respect the author's requests (on page 4); in particular, posting the book online is not allowed.

Calculators: You may use any calculator or computer algebra system for your homework, but you must show your work. On take-home exams, you may also use a calculator or computer, but standards for showing your work will be higher. Calculator policies for the final exam will be determined later.

Canvas: Most, if not all, course materials will be posted on Canvas, and announcements will often be used to communicate important information. Please check Canvas regularly, and make sure you have it set to notify you when announcements are posted. Note that both sections 1 and 2 are combined into one Canvas course, so you may only see section 1 displayed (even if you're enrolled in section 2).

## Homework and Exams

Problem sets will be assigned approximately weekly; there will be approximately 11 of them. They will consist of a few graded problems, and several additional ungraded problems. You should consider the ungraded problems as equally important components of your homework! The graded problems alone will not provide enough practice to perform well on exams.

I strongly encourage you to collaborate on your problem sets, as long as you are able to solve each problem on your own after discussing it with your classmates. See the section "Teaching and Learning Philosophy" above for more thoughts on the role of collaboration in the learning process.

Midterm Exams: There will be two take-home midterm exams, which will be given in place of problem sets in weeks to be determined later. They will be somewhat longer and more involved, and grading will be stricter (in terms of clarity of writing and showing of work). Also, collaboration is not allowed on the midterm exams; you must solve and write up the problems yourself.

Final Exam: The final exam for both sections is from 8:30-10:30 PM on Friday, May 3rd. The final will not be given at any other time, and you must arrange your schedule so that you can take it. In particular, do not leave campus for the summer before the exam. The ONLY circumstances under which exceptions will be made are if you are seriously ill, a loved one is near death, or you have a conflict with another final exam.

Students requiring accommodations for a disability during exams or otherwise should register with Disability Resources as soon as possible. Send your VISA (which you will receive from Disability Resources) to Dr. Johnson at least two weeks in advance of the first exam so your accommodations can be arranged.

• 30% problem sets (with the lowest score dropped)
• 30% take-home midterms (15% each, no dropping)
• 40% final exam

• A+: At instructor's discretion
• A: 90 and up
• A-: 85-90
• B+: 80-85
• B: 75-80
• B-: 70-75
• C+: 65-70
• C: 60-65
• C-: 55-60
• D: 50-55 (D+ and D- ranges determined later)
• F: below 50
Note that scores will not be rounded; a total grade of 84.99 is still a B+, not an A-.

If you take the class on a credit/no credit (pass/fail) basis, you must earn at least a C- to pass.

## Useful Resources

Campus Resources

External Math Resources

• Wolfram Alpha is a great way to check your work. Do not use it, however, to do homework problems for you.
• Sage is a Python-based system intended as an open-source alternative to Wolfram Alpha, Mathematica, and similar systems.
• GNU Octave is an open-source alternative to Matlab.
• Khan Academy has been immensely popular with many of my students as a supplemental resource.

• Do as many problems as you can. Do every problem in the textbook, even, if that's consistent with your mental health and success in other courses.
• The textbook contains solutions for all of the exercises presented therein. Only resort to reading the solution when either you've already solved the problem yourself, or you have made an earnest effort and are well and truly stuck.
• Do homework sets as soon as we've covered the relevant material, not right before they're due.
• Read a section or two ahead before class, and attempt a few problems. This shouldn't make class boring; rather, it makes class an opportunity to clarify your thoughts and get a different perspective.

## Approximate Schedule

This schedule is very much an estimate, and we may deviate significantly from it. If you miss class, please confirm with another student so you can make sure you catch up on the correct material.

Week Dates Sections Assignments due (tentative) Notes
1 1/14-1/18 1.1-1.4