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  1. I understand that for $\frac{\partial f}{\partial x \partial y}$ we do as following $\frac{\partial f}{ \partial y}\cdot \frac{\partial f}{ \partial y}(f(x,y))$ how do I write it in Wolfram d^2/(dtdz) is correct?

  2. in general in which cases $\frac{\partial f}{\partial x \partial y}=\frac{\partial f}{\partial y \partial x}$ and when it do not?

gbox
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    If $f$ is an analytic function, then always. – Kaynex Feb 28 '16 at 21:16
  • I would use the $\LaTeX$ notation \frac{\partial^2 f}{\partial x \partial y} = \frac{\partial^2 f}{\partial y \partial x}. The topic you seem to be asking about is "equality of mixed partials". How to "write it in Wolfram" is probably off-topic here. – hardmath Feb 29 '16 at 04:33

1 Answers1

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  1. You can write "d^2/(dxdy) f(x,y)" for $\frac{\partial^2 f(x,y)}{\partial x \partial y}$ in wolfram alpha.
  2. This is :$\frac{\partial^2 f}{\partial x \partial y}=\frac{\partial }{ \partial x}\left( \frac{\partial f}{ \partial y}\right)$
  3. This equality always holds if the function is $\mathcal{C}^2$ (all the second derivatives exists and are continous). This is known as the Schwarz theroem. As far as I know this is not a necessary condition and we don't know exactly when the symetry occurs.
nicomezi
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