directional derivative

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.

Definition

The directional derivative of a differentiable function

f(\vec{x}) = f(x_1, x_2, \ldots, x_n)

along a unit vector

\vec{v} = (v_1, \ldots, v_n)

is the function defined by the limit

D_{\vec{v}}{f} = \lim_{h \rightarrow 0}{\frac{f(\vec{x} + h\vec{v}) - f(\vec{x})}{h}}

It can be written in terms of the gradient

\nabla(f)

f

D_{\vec{v}}{f} = \nabla(f) \cdot \vec{v}

where

\cdot

denotes the dot product (Euclidean inner product). At any point

p

, the directional derivative of

f

intuitively represents the rate of change in

f

in the direction of

\vec{v}

at the point

p

The Directional Derivative in Differential Geometry

A vector field at a point

p

naturally gives rise to linear functionals defined on

p

by evaluating the directional derivative of a differentiable function

f

along the unit vector

\vec{v}/||\vec{v}||

where

\vec{v}

is the vector of the tangent space at

p

assigned by the vector field. The value of the functional is then defined as the value of the corresponding directional derivative at

p

in the direction of

\vec{v}

Directional Derivative

Definition

The Directional Derivative in Differential Geometry

See also