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Commutativity of differentiation in a special case

  1. Aug 24, 2011 #1
    Problem

    I'd like to prove [itex] \frac{d}{dt}[\frac{\partial}{\partial{x}}f(x(t),y(t),t)]=\frac{\partial}{\partial{x}}[\frac{d}{dt}f(x(t),y(t),t)][/itex].

    Attempt
    [tex]\begin{equation*}\begin{split}
    \frac{d}{dt}[\frac{\partial}{\partial x}f(x(t),y(t),t)]=\frac{d}{dt}\lim_{\epsilon\to 0}\frac{f(x(t)+\epsilon,y(t),t)-f(x(t),y(t),t)}{\epsilon}\\
    =\lim_{\delta\to 0}\lim_{\epsilon\to 0} \frac{[f(x(t+\delta)+\epsilon,y(t+\delta),t+\delta)-f(x(t)+\epsilon,y(t),t)]-[f(x(t+\delta),y(t+\delta),t+\delta)-f(x(t),y(t),t)]}{\epsilon\delta}\end{split}\end{equation*}[/tex]

    So if my previous steps are correct, I need to show that the 2 limits are commutative, which I have no idea...
     
  2. jcsd
  3. Aug 24, 2011 #2

    hunt_mat

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    Homework Helper

    Did you know you can express the total derivative as a sum of partial derivatives?
     
  4. Aug 24, 2011 #3

    HallsofIvy

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    Are you required to go all the way back to the definition of derivative? If not, use the chain rule as hunt mat suggests.
     
  5. Aug 24, 2011 #4
    Thanks for the hints!

    [itex] \frac{d}{dt}\partial_{x}f(x(t), y(t), t)=(\sum\dot{x}_k \partial_{x_k}+\partial_t)\partial_{x}f=\sum\dot x_k\partial_{x_k}\partial_{x}f+\partial_t\partial_{x}f [/itex].

    At this point, I need to show the "equality of mixed partials". I found the proof of [itex] f_{xy}=f_{yx}[/itex] for [itex] f(x, y) [/itex] here: http://www.sju.edu/~pklingsb/clairaut.pdf

    My function is f(x(t), y(t), t) so a bit different.

    For proving [itex] f_{x(t)\hspace{1mm}t} = f_{t\hspace{1 mm}x(t)}[/itex] for f(x(t), t), I think the proof in the note works if I substitute c=x(a) and d=x(b).

    I also think the theorem can be extended to [itex] f(x_1(t), x_2(t), ..., x_n(t), t) [/itex] if I imagine slicing the n dimentional space by a plane parallel to [itex] x_i x_j [/itex] plane (or [itex] x_i\hspace{1 mm}t [/itex] plane).

    So I have [itex] f_{x_i x_j}=f_{x_j x_i}[/itex] and [itex] f_{x_i\hspace{1 mm}t}=f_{t\hspace{1 mm}x_i}[/itex] for [itex] f(x_1(t), ..., x_n(t), t) [/itex]. Is that okay?

    By the way, I don't understand why the author of the note wrote [itex] f_{xy}(x,y)-f_{yx}(x,y)\ge \frac{h}{2} [/itex]. Is that arbitrary? Can I use h/3 instead for example?
     
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