There are lots of great CM books out there. Besides Taylor and Gregory some of my favorites are
- Scheck - Mechanics: From Newton's Laws to Deterministic Chaos
-Lanczos - The Variational Principles of Mechanics...
In my opinion the exercises are not that hard. I think there's a good balance between computational exercises and proofs... you souldn't have much trouble figuring out whether your answers are correct or not.
I'm really surprised no one has mentioned Gauge Fields, Knots and Gravity by John Baez...
I love that book, it's not very rigorous, but it provides a lot of insight, and for each topic gives a list of references, so that...
I'm confused haha.. you dislike hand-wavy books, but you are not looking for a pure math book...
You should also note that Hassani wrote two books, Math Methods For Students of Physics and Related Fields (maybe the one you looked at), and the one I mentioned earlier, which is a graduate...
Hassani gives a really nice treatment of Green Functions in his book
Mathematical Physics: A Modern Introduction to its Foundations
You could try
Ordinary Differential Equations by Arnold
or Differential Equations: A Dynamical Systems Approach by Hubbard & West
They're both great!
Besides Munkres I reccomend:
-Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach by Hubbard
- Advanced Calculus of Several Variables by Edwards...
Title: Analysis I,II,III
Author: Herbert Amann, Joachim Escher
Category: Mathematics > Analysis
Amazon link for the book: https://www.amazon.com/dp/3764371536/?tag=pfamazon01-20&tag=pfamazon01-20
Contents: Foundations, convergence, continuous functions, differentiation in one variable...
Well, I don't find proofs useless... I believe that the things you learn in physics make a lot more sense if you know proof based math.
For example, take Laplace's equation, I'm sure that the first method that comes to your mind is separation of variables, but why can you make the assumption...
I took my first proof based course as a freshman. At my university, if you study math, physics, computer science or actuarial science, with very few exceptions, all the math courses you'll take are going to be proof based.
Well, actually there are two "Hassani's"
There is "Mathematical Methods: For Students of Physics and Related Fields"
this one is not very rigorous (same as Boas), but it has some very nice explanations.
Edwards - Advanced Calculus of Several Variables
Hubbard - Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach
I would add to what micromass said:
Advanced Calculus of Several Variables - Edwards
Differential Equations and Their Applications - Braun
I also like ODE by Tenenbaum/Pollard, it's sometimes not very rigorous but contains all you need and is very clear!