No, it's not. Both are suitable for an undergrad course.. I think you may be referring to this one
https://www.amazon.com/dp/0387985794/?tag=pfamazon01-20&tag=pfamazon01-20
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
https://www.amazon.com/dp/3540219250/?tag=pfamazon01-20&tag=pfamazon01-20
-Lanczos - The Variational Principles of Mechanics...
Hassani is the one you're looking for! It's not as rigorous as a pure math book, but it is certainly not the typical hand-wavy math methods book. Szekeres is also a great book though...
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...
https://www.amazon.com/dp/9810220340/?tag=pfamazon01-20&tag=pfamazon01-20
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
https://www.amazon.com/dp/0387985794/?tag=pfamazon01-20&tag=pfamazon01-20
I'd also replace Born & Wolf (too advanced) and Hecht (too verbose) with Kenyon
https://www.amazon.com/dp/0199584605/?tag=pfamazon01-20&tag=pfamazon01-20
You could try
Ordinary Differential Equations by Arnold
https://www.amazon.com/dp/0262510189/?tag=pfamazon01-20&tag=pfamazon01-20
or Differential Equations: A Dynamical Systems Approach by Hubbard & West
https://www.amazon.com/dp/0387972862/?tag=pfamazon01-20&tag=pfamazon01-20
They're both great!
This is the one you are looking for!
Advanced Calculus of Several Variables by Edwards
https://www.amazon.com/dp/0486683362/?tag=pfamazon01-20&tag=pfamazon01-20