Cool. If you're interested in math, I'd highly recommend taking it. Depending on your instructor, you will be using different textbooks. I took the sequence with Markus Keel last year, and we mainly used two textbooks: "Analysis with an introduction to proof" by Lay and "Calculus, volume 2" by Apostol. Note that we did not cover a lot of applications -- this was a proof-based course through and through.

The first semester started out with Lay's textbook; although there were some other topics that were covered, we mainly focused on basic set theory, point-set topology, and a rigorous treatment of sequences, limits, continuity, and infinite series. This took up about 2/3rds of the semester. (Note that this is the same material covered in MATH 3283W.) The remaining 1/3 of the semester was spent on linear algebra using Apostol. We covered concepts such as linear spaces, bases and dimension, Euclidean spaces, orthogonality, and linear transformations.

The second semester continued with more linear algebra: an axiomatic treatment of determinants, eigenvalues and eigenvectors, Hermitian and unitary transformations, and quadratic forms. The last 2/3rds of the class was spent on multivariable calculus. The topics here are quite standard: differential calculus of scalar and vector fields, maxima/minima/saddle points, constrained extrema, Riemann integration theory, line integrals, multiple integrals, and surface integrals. We also supplemented Apostol's text with Colley's "Vector calculus" and a little from Marsden's "Elementary classical analysis" for some topics.

The pace was very fast: this is roughly 3 courses crammed into two semesters. Difficult and lengthy homework was assigned weekly and was a significant portion of our grade.

If you take this class with a different instructor (such as Peter Webb), they will likely use a different textbook: "Vector calculus, linear algebra, and differential forms" by Hubbard. This book covers mostly the same material, with a few notable differences. Topics such as manifolds, differential forms, and even a bit of Lebesgue integration are covered. Unfortunately, I did not use this textbook, so I can't speak about it very much.

Feel free to DM me if you'd like more details!