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Proofs In Advance Calculus

  1. Oct 4, 2012 #1
    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!
     
  2. jcsd
  3. Oct 5, 2012 #2
    Read more proofs.
     
  4. Oct 5, 2012 #3

    mathwonk

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    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.
     
  5. Oct 8, 2012 #4
    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!]
     
    Last edited: Oct 8, 2012
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