Definition:Differential of Mapping/Real-Valued Function

Definition

At a point

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 $x_i$.

Notation

Substituting $\map {\d x_i} {x; h} = h_i$ for $i = 1, 2, \ldots, n$, the following notation emerges:

$\ds \d f := \sum_{i \mathop = 1}^n \frac {\partial f} {\partial x_i} \rd x_i = \frac {\partial f} {\partial x_1} \rd x_1 + \frac {\partial f} {\partial x_2} \rd x_2 + \cdots + \frac {\partial f} {\partial x_n} \rd x_n$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 13$: Partial Derivatives: $13.62$