Higher Derivatives and Differentials

Abstract

Let $O \subset {\mathbb{R}}^{n}$ be open and f a real-valued function defined on O. If the function $\frac{\partial f} {\partial {x}_{i}}$ is defined on O, then we can consider the existence of its partial derivatives. If $\frac{\partial } {\partial {x}_{j}}( \frac{\partial f} {\partial {x}_{i}})(a)$ exists, then we write for this derivative $\frac{{\partial }^{2}f} {\partial {x}_{j}\partial {x}_{i}}(a)$ if i≠j, and $\frac{{\partial }^{2}f} {\partial {x}_{i}^{2}} (a)$ if i = j.