Symmetry of Second Derivatives

Schwarz's Theorem: Symmetry of Second Derivatives

Let $f:D\subseteq {\mathbb{R}}^n\to \mathbb{R}$ be a real scalar field.

If $f$ is twice continuously partially differentiable on $D$, then for all $i,j\in \{1,\cdots ,n\}$

${\partial }_i{\partial }_{j}f={\partial }_j{\partial }_{i}f$

PROOF

TODO