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Symmetry of higher order partial derivatives

  1. Jul 20, 2010 #1
    As per Clariut's theorem, if the derivatives of a function up to the high order are continuous at (a,b), then we can apply mixed derivatives. I am looking at

    and I cannot understand in the example for non-symmetry, why the derivatives are not continuous and not equal. Can someone please explain that example?


  2. jcsd
  3. Jul 20, 2010 #2


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    Did you also check out the talk page of the Wikipedia article you linked?

    It is explained in more detail there.
  4. Jul 20, 2010 #3
    Yeah that counterexample is basically one huge ugly computation that you could verify yourself. Unfortunately there are a wealth of such ugly counterexamples in multivariable differentiation theory because there are no nice basic theorems. The only theorems I would consider nice are the inverse function theorem and the implicit function theorem. Unfortunately you have to expose yourself to this ugliness if you want to understand more beautiful areas of math such as complex analysis, harmonic analysis and PDE's, etc.
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