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Conditions for change of order in derivative of a partial

  1. Feb 14, 2016 #1
    Sorry about the title, had a hard time trying to fit the question on the given space. The question is quite simple : If [itex] F = F(x_1,...,x_n,t) [/itex] , Under what conditions is [itex] \frac{d }{dt} \frac{\partial F }{\partial xi} = \frac{\partial }{\partial xi} \frac{dF }{dt} [/itex] true?
     
  2. jcsd
  3. Feb 14, 2016 #2

    andrewkirk

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    Did you mean to write total derivatives ##\frac{d}{dt}## instead of partials ##\frac{\partial}{\partial t}##?

    If you meant to write partials then there is only one set of conditions, which are set out in Schwarz's theorem here.

    If you meant to write ##\frac{d}{dt}## then there is an additional condition required, which is that ##\frac{\partial x_j}{\partial t}=0## for all ##j##.
     
  4. Feb 14, 2016 #3
    I meant [itex] \frac{d}{dt} [/itex]. I knew Schwarz's Theorem, but wasn't sure on when to safely interchange total and partial derivatives. You solved the problem, thanks!
     
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