Textbook for Multivariable and/or Vector Calculus I'm looking for a tetxbook that covers Multivariable Calculus and/or Vector Calculus theoretically.
I have done Analysis (single-variable) at the level of Introduction to Real Analysis by Bartle and Sherbet and Principles of Mathematical Analysis by Walter Rudin. Except for the theory of integration. I haven't done a lot of exercises from "Baby Rudin" but I have done a course on Topology so some of the exercises will be approachable now, though I may have to go and revise Topology to refresh my concepts/recall the theorems.
Anyway, I'm looking for a textbook that extends the material in the aforementioned books to Multivariable and/or Vector Calculus. Any suggestions?
 A: Check out Mathematical Analysis I (for single variable) and Mathematical Analysis II (for mulitivariable calculus) by Vladimir A. Zorich 
https://www.amazon.com/Mathematical-Analysis-Universitext-Vladimir-Zorich/dp/3540874518
Eastern European style. Typical reference in U.S. is Michael Spivak Calculus on Manifolds and more comprehensive lesser known Analysis on Manifolds by Munkres (the same Mukres who wrote the First Course in Topology).
Any serious understanding of Multivariable Calculus will have to involve concurrent course in Electromagnetic field theory (Classical Electrodynamics)
as well as understanding of Hodge star operator and at least a rudiments of De Rham-Hodge Theory. 
A: Honestly I would recommend that you use a University Calculus text, practice doing problems and read the proofs there. You will see that even the extremely non-rigorous proofs are hard to follow. The problem with tackling an intense book off hand is that you may go through the whole text without being able to do simple computations. 
A: Introduction to Analysis by Wade. In my opinion it contains very readable, rigorous, and user-friendly proofs of the vector calculus theorems (for $\mathbb R^3$). I think it is the most clearly written analysis text I have seen where the reader is required to do fewer "side calculations".
Spivak's Calculus on Manifolds is concise and a bit harder to read, but contains more general versions of the theorems for any dimension using differential forms.
A: I quite like our own Ted Shifrin's Multivariable Mathematics and Jerry Shurman's Calculus and Analysis in Euclidean Space.
