Schwarz theorem

Schwarz's theorem, also known as Clairaut's theorem, states that if a function has continuous second derivatives, then the order in which you take partial derivatives does not matter. In practical terms, this means that for such functions, the mixed partial derivative with respect to x and y is the same as with respect to y and x—that is, \( \frac{\partial^2 f}{\partial x \partial y} = \frac{\partial^2 f}{\partial y \partial x} \). This symmetry simplifies calculations in calculus and ensures consistency in how derivatives are computed when dealing with multivariable functions, provided certain smoothness conditions are met.