Schwarz's Theorem

vector-calculus

Definition

Schwarz's Theorem

Let $D\subseteq {\mathbb{R}}^2$ be an open set and $f:D\to \mathbb{R}$ a function whose partial derivative $\frac{\partial }{\partial x}\frac{\partial }{\partial y}f$ and $\frac{\partial }{\partial y}\frac{\partial }{\partial x}f$ exist and are continuous in $D$. Then:

$$\frac{\partial }{\partial x}\frac{\partial }{\partial y}f=\frac{\partial }{\partial y}\frac{\partial }{\partial x}f$$

If $f$ is $m$-times partially differentiable function in $D$, then all partial derivative up to order $m$ are independent of the order of differentiation.

Note that the commutativity of the partial derivation operator follows from the theorem.