Directional derivative Guide, Meaning , Facts, Information and Description

In mathematics, the directional derivative of a multivariate differentiable function along a given unit vector intuitively represents the rate of change of the function in the direction of that vector. It therefore generalizes the notion of a partial derivative in which the direction is always taken along one of the coordinate axes.

Table of contents

1 Definition 2 The Directional Derivative in Differential Geometry 3 See also

Definition

The directional derivative of a differentiable function along a unit vector is the function defined by the limit

It can be written in terms of the gradient of by

where denotes the dot product (Euclidean inner product). At any point , the directional derivative of intuitively represents the rate of change in in the direction of at the point .

The Directional Derivative in Differential Geometry

A vector field at a point naturally gives rise to linear functionals defined on by evaluating the directional derivative of a differentiable function along the unit vector where is the vector of the tangent space at assigned by the vector field. The value of the functional is then defined as the value of the corresponding directional derivative at in the direction of .

See also

• gradient
• partial derivative

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