Definition:Differential of Mapping/Real-Valued Function

Notation
Substituting $\mathrm d x_i \left({x; h}\right) = h_i$ for $i = 1, 2, \ldots, n$, the following notation emerges:
 * $\displaystyle \mathrm d f := \sum_{i \mathop = 1}^n \frac{\partial f}{\partial x_i}\mathrm d x_i = \frac{\partial f}{\partial x_1}\mathrm d x_1 + \frac{\partial f}{\partial x_2}\mathrm d x_2 + \cdots + \frac{\partial f}{\partial x_n}\mathrm d x_n$