Advanced Calculus Of Several Variables By Edwards Homeworks

Math 268: Multidimensional Calculus

InstructorGautam 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
TAThomas Swayze.  WEH 7205.
Office Hours (TA) Mon 3:30--4:30, Tue 11:00--12:00
RecitationSec A: Tu 1:30-2:20 in PH A18B, and Sec B: 3:30-4:20 in WEH 6423
Homework dueWednesdays, at the beginning of class. Late homework will not be accepted
Midterm 1Wed, Feb 14 (in class)
Midterm 2Wed, Mar 28 (in class)
Mailing listmath-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.

This material is rather challenging. I am happy to report that in spite of the challenging nature of the material many of our undergraduates have been equal to the task of dealing well with it. You should not take this course unless you have had prior experience with proving things and with mathematical abstraction. We don't have higher level honors courses but if we did these courses would be among them. It is also absolutely necessary for more advanced work in mathematics and can come in handy in other branches of science and engineering.

In previous calculus courses you learned to differentiate and integrate functions of one or more real variable and to apply these techniques to solve problems of various sorts. You probably didn't pay may attention to mathematical rigor. In fact, when you studied things like integration formulae in several variables you probably didn't pay much attention to intellectual rigor either. I assume that part of the reason you are taking this course is to get all that stuff straight.

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