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?
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).