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Volume of parametrized manifold

  1. Apr 13, 2009 #1
    I don't think this is a difficult problem, but I am not sure about what is being asked in the question. I got it from Munkres' Analysis on Manifolds page 193 Q 2.


    1. The problem statement, all variables and given/known data

    Let A be open in R^k; let f : A-->R be of class C^r; let Y be the graph of f in R^(k+1), parametrized by the function @(x) = (x, f(x)). Express V(Y@) as an integral. (this is the volume of the graph of f)


    2. Relevant equations

    the volume of a parametrized manifold, or V(Y@) = integral over A of det((D@)^tr D@), where D@ is the derivative matrix of @ and D@^tr is the transpose of D@.

    3. The attempt at a solution

    I think I would just use the above equation to express V(Y@) as an integral. But then I would be done a little too quickly, so I thought that maybe I am supposed to simplify the matrix multiplication or something. Am I missing something here?

    Also, can someone tell me how to post using mathematical symbols?
     
  2. jcsd
  3. Apr 14, 2009 #2
    Can anyone confirm I am on the right track or am I way off? Or, at the very least, can someone tell me how to post using mathematical symbols. I am not very computer savvy. :)
     
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