I have done an example where I needed to show that every conservative $C^2$ vector field is irrotational. However, there is something unclear in the solutions: Namely, I am uncertain what does the following sentence at the end of the solution mean:
"since second partial derivatives are independent of the order (for smooth functions)", and I was wondering how does that imply that the equality before that is 0?
1 Answer
$\begingroup$This is normally called symmetry of second derivatives, or Clairaut's theorem. It means that for functions that have continuous second partial derivatives, $$ \frac{\partial^2f}{\partial x \partial y} - \frac{\partial^2 f}{\partial y \partial x} = 0. $$ But each term in the line above is of precisely this form (Presumably the document's notation has $\phi_x = \partial_x \phi$ and so on).
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