Definition:Differential of Mapping/Real-Valued Function/Point

Definition

Let $U \subset \R^n$ be an open set.

Let $f: U \to \R$ be a real-valued function.

Let $f$ be differentiable at a point $x \in U$.

$\ds \map {\d f} {x; h} := \sum_{i \mathop = 1}^n \frac {\map {\partial f} x} {\partial x_i} h_i = \frac {\map {\partial f} x} {\partial x_1} h_1 + \frac {\map {\partial f} x} {\partial x_2} h_2 + \cdots + \frac {\map {\partial f} x} {\partial x_n} h_n$

where:

$h = \tuple {h_1, h_2, \ldots, h_n} \in \R^n$

$\dfrac {\partial f} {\partial x_i}$ is the partial derivative of $f$ with respect to $x_i$.