Directional derivative

Builds on Euclidean space Partial derivative

Definition

Let $U \subseteq \mathbb{R}^n$ be an open set, let $f \colon U \to \mathbb{R}^m$ , let $a \in U$ , and let $v \in \mathbb{R}^n$ . The directional derivative of $f$ at $a$ in the direction $v$ is the limit $D_v f(a) = \lim_{t \to 0} \frac{f(a + t v) - f(a)}{t},$ when this limit exists in $\mathbb{R}^m$ . Other notations for it are $\partial_v f(a)$ and $\nabla_v f(a)$ .

Remarks

For $v = e_i$ , the directional derivative is the partial derivative $\partial f / \partial x_i$ at $a$ . Some books require that $v$ is a unit vector, so that $D_v f(a)$ is a rate of change per unit of distance. If $D_v f(a)$ exists and $c \in \mathbb{R}$ , then $D_{cv} f(a)$ exists and equals $c \, D_v f(a)$ . When $f$ is differentiable at $a$ , the directional derivatives are the values of the derivative: $D_v f(a) = Df(a)(v)$ , see this statement.

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Tag	`0Y7PVW4Q44`
URI	`m2://0Y7PVW4Q44`
Kind	definition
Writing	W1 stated
MSC	26B05
Defines	Directional derivative
Uses	Euclidean space, Partial derivative

Directional derivative

Definition

Remarks

Reading path

Where this leads

Further reading

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