Definition:Differential of Mapping/Real-Valued Function

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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 $x \in U$.

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

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:

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


Examples

Function of 2 Variables

Let $f: \R^2 \to \R$ be a real-valued function of $2$ variables.

Let $z = \map f {x, y}$ for all $\tuple {x, y} \in \R^2$.

Then the differential of $z$ is given by:

$\d z := \dfrac {\partial f} {\partial x} \d x + \dfrac {\partial f} {\partial y} \d y$


Also known as

The differential of a real-valued function $f$ as defined here is also known as the total differential of $f$.


Sources

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