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Is the following statement $$\frac{\partial^2 f}{\partial x \, \partial y}=\frac{\partial^2 f}{\partial y \, \partial x}$$ always true? If not what are the conditions for this to be true?

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2 Answers

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Second order partial derivatives commute if $f$ is $\mathcal{C}^2$ (i.e. all the second partial derivatives exist and are continuous). This is sometimes called Schwarz's Theorem or Clairaut's Theorem; see here.

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This is true in general if $ f \in \mathcal{C}^2 $. This has a name: symmetry. More formally, it is known as Clariut's Theorem or Schwarz's theorem.

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