Math 268: Multidimensional Calculus

Spring 2018

Course Description

This course is a serious introduction to multidimensional calculus that makes use of matrices and linear transformations. Students will be expected to write proofs; however, some of the deeper results will be presented without proofs.

Tentative Syllabus

• Functions of several variables, regions and domains, limits and continuity.
• Partial derivatives, linearization, Jacobian.
• Chain rule, inverse and implicit functions and geometric applications.
• Higher derivatives, Taylor’s theorem, optimization, vector fields.
• Multiple integrals and change of variables, Leibnitz’s rule.
• Line integrals, Green’s theorem.
• Path independence and connectedness, conservative vector fields.
• Surfaces and orientability, surface integrals.
• Divergence theorem and Stokes’s theorem.

Textbook and References

There are plenty of references on Calculus and can be divided into isomorphism classes based on difficulty. (Translation: I’m not expanding my brief notes.)

• My brief lecture notes: for printing or for viewing online.
• Khan Academy. (Lots of examples, pictures, intuition; but not at the level of rigor that will be expected in this course.)
• Hermann, Strang Calculus Volume 3. (At a level a bit lower than this course; but available for free on OpenStax.)
• Lecture notes by Santiago Canez (also available on his website. (A bit deeper / more through than we will have time for in this course.)
• Advanced Calculus of Several Variables by C. H. Edwards, Jr (roughly chapters 2 through 5; again at a level slightly higher than we will have time for in this course.)
• Advanced Calculus (5th Edition) by Wilfred Kaplan. (roughly chapters 2 through 5). Note: When I initially recommended this book, it used to be a cheap paperback book. It is not so cheap anymore, and I do not recommend buying it.
• Multivariable Mathematics: Linear Algebra, Multivariable Calculus, and Manifolds by Shifrin.
• The more analytically inclined can also use any of the references used for 269 any of the references used for 269

Class Policies

Lectures

• If you must sleep, don’t snore!
• Be courteous when you use mobile devices.

Homework

• Homework must be turned in at the beginning of class on the due date.
• Late homework policy:
  • Late homework will NOT be accepted. In particular, homework turned in after class starts will NOT be accepted.
  • To account for unusual circumstances, the bottom 20% of your homework will not count towards your grade.
  • I will only consider making an exception to the above late homework policy if you have documented personal emergencies lasting at least 18 days.
• I recommend starting the homework early. Most students will not be able to do the homework in one evening.
• I view homework more as a learning exercise as opposed to a test. Feel free to collaborate, use books and whatever resources you can find. I recommend trying problems independently first, and then seeking help on problems you had trouble with.
• I also strongly urge you to fully understand solutions before turning them in. I will usually put a few homework problems on your exams with a devious twist. A through understanding of the solutions (even if you didn’t come up with it yourself) will invariably help you. But knowledge of the solution without understanding will almost never help you.
• Nearly perfect student solutions may be scanned and hosted here, with your identifying information removed. If you don’t want any part of your solutions used, please make a note of it in the margin of your assignment.

Exams

• All exams are closed book, in class.
• No calculators, computational aids, or internet enabled devices are allowed.
• The final time will be announced by the registrar here. Be aware of their schedule before making your travel plans.

Grading

• Homework will count as 20% of your grade. Moreover, between 25% and 50% of your exams will consist of (possibly modified) homework questions, so I advise you to really understand the homework.
• The remainder 80% of your grade will be determined by your exams weighted as the higher of:
  • 20% for each midterm, and 40% final,
  • or 30% for your better midterm and 50% for the final,
• If you miss a midterm for some reason, I will not give you a makeup. Instead, I will count your other midterm as 30% and final as 50% using the second option above.