# A proof based Multivariable Calculus and Linear Algebra

May I know how can I teach a proof-based Multivariable Calculus and linear algebra as a single course? While there are quite a few known books in the field such as:

1) Vector Calculus, Linear Algebra and Differential forms by Hubbard & Hubbard

and

2) Multivariable Mathematics by Theodore Shifrin, etc

which are proof based, I wonder if students can easily catch the proof part since most students have not taken Real Analysis. I believe I will need to spend sometime to explain about proof and in doing so, I wonder if I will be able to cover the whole general syllabus for Multivariable Calculus and Linear Algebra of the US education system? If I exclude the proof part and focus on the application and computation part, then there will be no issue at all.

Hence, has anyone tried doing a proof based course and can share experience on how to handle such course?

• What is the target audience? I suspect very low percentage of people taking the class are pure math types who need those proofs. Engineers don't need it. Scientists don't need it. Applied mathematicians don't need it. High school teachers don't need it. (And even for the pure math types, why not just wait until real analysis course!?) – guest Nov 29 '18 at 17:15
• FWIW, Caltech forces (very very very strong) students to incorporate lots of real analysis proofs and such into calc 1 and then gives students the choice to dump the proof emphasis in calc 2 and 3 (go into a non proof based track). Most of them do drop out of proof based. They are voting with their feet. Why push something people don't want? – guest Nov 29 '18 at 17:17
• In some sense, the older version of Thomas' Calculus or Strang's Calculus take such a path in a limited sense. They include an introduction to basic matrix math and give a better account of differentiability of real maps. I think treating linear algebra first without too much calculus is wise. On the other hand, I think having multivariate calculus with some matrix math is great. For linear algebra, I'd like to have multivariate calc. as a prereq. so I can use calc. III geometric insight as a backdrop for dimension theory and visualization. There are two separate courses here, at least two! – James S. Cook Nov 30 '18 at 0:28
• My guess is that proposing a proof-based Multivariable Calculus and linear algebra class to student without real analysis background is a mistake (not yours, obviously), but I am not sure the answer "you cannot" would be of any help. – Benoît Kloeckner Nov 30 '18 at 10:30
• There is (or was) a two semester course at Columbia University called Honors Mathematics I&II that rigorously covers Calculus and Linear Algebra with introductory material for students new to proofs. The texts were Calculus I&II by Apostol which are excellent, although the hardcover versions are wildly expensive. – sfmiller940 Dec 3 '18 at 19:46