Teaching multivariable calculus and tensor calculus with computer algebra software

Abstract

To go from calculus of scalar functions of one variable to multivariate calculus of vector-valued functions is a steep learning curve for many students. It takes a lot of practice to get used to the new concepts such as the directional derivatives, the di.erentiability, the many types of first order di.erential operators, parameterization of surfaces and the fundamental theorems of integrals, e.g. the Divergence Theorem and the Stokes Theorem. Along the learning process of mastering the skills, the students often need to check whether the intermediate steps in the tedious calculations are correct. Unfortunately, this is beyond the capability of an ordinary calculator and the answers provided at the end of the books. This is where computer algebra software, such as Mathematica, can come to students' help. With the developed symbolic computation tools, the students can tweak a given problem, solve a new one by hand and then check the answer against the result obtained by using the computer algebra software.