Definition:Differential of Mapping/Real-Valued Function

Definition
Let $f: \R^n \to \R$ be a real-valued function which is differentiable at a point $x = \left({x_1, x_2, \ldots, x_n}\right) \in \R^n$.

The differential of $f$ at $x$ is defined as:
 * $\displaystyle \mathrm d f \left({x; h}\right) := \sum_{i \mathop = 1}^n \frac{\partial f}{\partial x_i} h_i = \frac{\partial f}{\partial x_1} h_1 + \frac{\partial f}{\partial x_2} h_2 + \cdots + \frac{\partial f}{\partial x_n} h_n$

where:
 * $h = \left({h_1, h_2, \ldots, h_n}\right) \in \R^n$
 * $\dfrac {\partial f}{\partial x_i}$ is the partial derivative of $f$ with respect to $x_i$.

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