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Mixed Derivatives

  1. Nov 21, 2005 #1
    I am familiar with the standard rule of mixed partial derivatives in that the order in which you partially differentiate dosn't matter. I have just been considering whether the same rule applies if we take f(q,t) say where q=q(t) and we differentiate normally w.r.t t then partially w.r.t q. Is the order of these operations always immaterial in this case too? I can't find a counterexample but I havn't yet got a satisfactory insight into this problem.

    Anyone want to enlighten me?
     
  2. jcsd
  3. Nov 21, 2005 #2

    Galileo

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    As long as [tex]\frac{\partial^2 f}{\partial q \partial t}=\frac{\partial^2 f}{\partial t \partial q}[/tex] it doesn't matter. All derivatives must exist and be continuous ofcourse.

    You could apply the multivariate chain-rule to show it explicitly.
     
  4. Nov 21, 2005 #3
    Maybe I wasn't quite clear enough. I was wodering if:
    [tex]\frac{\partial^2 f}{\partial qdt \dt}=\frac{\partial^2 f}{\d dt \partial q}[/tex]
    always holds for f(q,t).
     
    Last edited: Nov 21, 2005
  5. Nov 21, 2005 #4

    TD

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    If the these mixed derivatives exist and are continuous, then they're the same.
     
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