Math 268: Multidimensional Calculus

Instructor Gautam Iyer.   WEH 6121.
Lectures MWF 9:30-10:20 in BH A53.
Office Hours (instructor) Mon 10:30--11:50, Fri 1:30--2:20
TA Thomas Swayze.  WEH 7205.
Office Hours (TA) Mon 3:30--4:30, Tue 11:00--12:00
Recitation Sec A: Tu 1:30-2:20 in PH A18B, and Sec B: 3:30-4:20 in WEH 6423
Homework due Wednesdays, at the beginning of class. Late homework will not be accepted
Midterm 1 Wed, Feb 14 (in class)
Midterm 2 Wed, Mar 28 (in class)
Final TBA
Mailing list math-268 (for course announcements and discussion. Please subscribe to this list.)

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


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


  • 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.


  • 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.


  • 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.