Clairaut’s theorem

Clairaut’s Theorem.

If $\mathrm{f}\mathrm{:}{\mathrm{R}}^n\mathrm{\to }{\mathrm{R}}^m$ is a function whose second partial derivatives exist and are continuous on a set $S\mathrm{\subseteq }{\mathrm{R}}^n$, then

$$\frac{{\partial }^{2}f}{\partial x_i\partial x_j}=\frac{{\partial }^{2}f}{\partial x_j\partial x_i}$$

on $S$, where $\mathrm{1}\mathrm{\le }i\mathrm{,}j\mathrm{\le }n$.

This theorem is commonly referred to as the equality of mixed partials.It is usually first presented in a vector calculus course,and is useful in this context for proving basic properties of the interrelations of gradient, divergence, and curl.For example, if $𝐅:{ℝ}^3\to {ℝ}^3$ is a function satisfying the hypothesis, then $\nabla \cdot (\nabla \times 𝐅)=0$.Or, if $f:{ℝ}^3\to ℝ$ is a function satisfying the hypothesis, then $\nabla \times \nabla f=\mathrm{𝟎}$.