How Can I Better Understand Advanced Calculus Proofs?

[NukeEng101]

I'm a sophomore at Rensselaer Polytechnic Institute and I'm taking MATH 4600 which is Advanced Calculus. I love the class and it is very interesting, we're taking what we learned in Multivariable Calculus, but just at a much higher level. However, my teacher does a lot of proofs behind why it's true and a lot of theory. I am having trouble understanding the proofs of the implicit theorem, the inversion theorem, and even just partial derivatives.

I know how to do the problems with actual examples, it's just the theory that's a little weak and I really want to improve on it to fully appreciate it a lot more. Thanks to anyone who could help give me pointers to understand the proofs a lot more!

 

[Robert1986]

Read more proofs.

 

[mathwonk]

give us more detail so we are not left with just the option of giving you the full proof of the inverse function theorem. or read spivak, calculus on manifolds.

 

[A. Bahat]

And make sure you try to see the "big picture" behind a proof. This makes all of the details usually easy. Ask yourself why each step is needed; what's the idea behind it? For example, the inverse function theorem is basically a corollary of the contraction mapping theorem. [You could take this line of thought even further. The contraction mapping theorem applies in any metric space, not just Euclidean space. By considering function spaces, we get the existence/uniqueness theorem for ODEs. Or by considering arbitrary Banach spaces we can generalize the inverse function theorem, necessary for infinite-dimensional settings like functional analysis. So it's really important that you get the fundamental idea behind the proof to be able to apply it in new situations like these!]