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Multivariable Calculus: Chain Rule and Second Derivatives

  1. May 22, 2012 #1
    1. The problem statement, all variables and given/known data
    Here is the problem with the solution:
    http://dl.dropbox.com/u/64325990/MATH%20253/Capture.PNG [Broken]

    What I don't understand is circled in red, how did they combine dxdy with dydx? Is it with Clairaut's theorem? If it is can someone explain how it works in this case because the second derivative makes this very confusing. Additionally 4s^2 ≠ 4r^2 so how can you use Clairaut?
     
    Last edited by a moderator: May 6, 2017
  2. jcsd
  3. May 22, 2012 #2
    That's the "By continuity of partials". There's a theorem regarding continuity of partials and the equality of mixed partials. I'm sure it's in your text. You should check it out. It's kind of a big deal.

    Yes it's Clairaut.
     
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