I am trying to properly understand multivariable calculus as a physicist and would like to know of any recommendations for textbooks to work through. Ultimately I want to be able to properly understand Maxwell's equations and the switching between integral and differential form.
When I mean 'properly understand', I mean to be able to both prove theorems (e.g. Stokes' Theorem) but understand properly why they are true. I have already been through the hand-wavy arguments about 'and this cancels out on the boundary therefore it's zero' and 'the divergence is like the spreading out of the field'. This isn't very satisfying. I have also been working my way through analysis textbooks - but the proofs there are often totally opaque and arbitrary, lending absolutely no insight into why multivariable calculus and Stokes' Theorem actually work.
To clarify, I can slog algebra and apply all the multivariable calculus theorems to 'derive' results but to be honest, I see the initial equation, work through the problem applying various rules and theorems having absolutely no idea what is happening, and end up at the final result none the wiser. What I'd like to do is properly understand all those stages in between.
I vaguely know that I could go in various directions - differential forms, tensors, differential geometry, calculus on manifolds... etc. What's the best path to take? Is there a textbook that will adequately address all these things?