What book should I choose for self-studying Calculus II?. I'm looking for a book that has a good explanation of the content and also solved exercises (which is a very important thing that I'm usually missing).
I'm already studying Calculus I by A First Course in Calculus by Lang and Calculus and Analytical Geometry by Thomas and Finney. Do you think these are good books for Calculus II? I have a feeling Lang doesn't cover all the contents in my course (not sure about Thomas and Finney tho). What is in your opinion the best book for self-study. If there is a better book than the ones on this list please tell me. Thanks!!
NOTE:
Here are the contents of my course:
The algebraic and topological structure of $\mathbb{R}^n$.
Functions from $\mathbb{R}^n$ to $\mathbb{R}^m$: continuity and the notion of limit. Differential calculus. Partial derivatives. Chain rule. Taylor's theorem in $\mathbb{R}^n$ and applications to the study of extreme values. Inverse and implicit function theorems. Extreme values of functions with constrained variables.
Multiple integrals: Fubini's theorem, change of variables theorem, applications to the computation of physical quantities.
Line integrals: Integrals of scalar fields and vector fields. Fundamental theorem of calculus for line integrals, conservative fields, and scalar potentials. Green's theorem.
Surface integrals: surface integrals of a scalar field, flux of a vector field, divergence theorem and Stokes' theorem.
Best Answer
You are correct that Lang's A First Course in Calculus doesn't cover these topics. The sequel Calculus of Several Variables does cover the computational aspects of most of them, but without complete proofs. I don't know the book by Thomas and Finney.
Calculus, Vol. 2, by Apostol is a good book that covers most of these topics (Taylor's formula only up to the second order, I think) at a higher level of rigour, although some difficult proofs are omitted. It doesn't seem to include the implicit and inverse function theorems. However, you can complement your reading with sections of the same author's Mathematical Analysis.
A more demanding book that includes, but also goes far beyond, what you've listed is Mathematical Analysis, I, II, by Zorich.