MA2712 – Multivariable Calculus

Problem Sets

Teaching the module multivariable calculus led me to compile a set of problems with fairly detailed solutions covering the basic topics of multivariable calculus: functions of several variables, partial derivatives, extreme value problems and double integrals.

The module covered chapters 15, 16 and 17 of

The sections of the problem set are in correspondence with the sections of the textbook. Other textbooks that served as source and inspiration are


Here are the problems and solutions in one file each.

  • Problems [pdf]
  • Solutions [pdf]
Solutions are also available section-by-section which might be more practical for use as handouts.
  1. Functions, graphs, and level surfaces [pdf]
  2. Introduction to partial derivatives [pdf]
  3. Linear approximations and tangent planes [pdf]
  4. The chain rule [pdf]
  5. Gradients and directional derivatives [pdf]
  6. Gradients and level surfaces [pdf]
  7. Maxima and minima [pdf]
  8. Constrained extrema and Lagrange multipliers [pdf]
  9. Review exercises [pdf]
  10. The double integral and iterated integrals [pdf]
  11. The double integral over general regions [pdf]
  12. Polar coordinates and the change of coordinates [pdf]
  13. Applications of the double integral [pdf]

LaTeX Source

The LaTeX source files are available on GitHub. Two external programs are needed to compile them. To create 3D figures I used Asymptote and 2D figures were created using TikZ with some external calls to gnuplot.


This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.

A copy of the licence can be found at