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 $\mathbf{F}\colon\mathbb{R}^{3}\to\mathbb{R}^{3}$ is a function satisfying the hypothesis, then $\nabla\cdot(\nabla\times\mathbf{F})=0$. Or, if $f\colon\mathbb{R}^{3}\to\mathbb{R}$ is a function satisfying the hypothesis, then $\nabla\times\nabla f=\mathbf{0}$.