Directional derivative

In mathematics, the directional derivative of a dependent function of several variables is the instantaneous rate of change of this function in a predetermined direction by a vector.

A generalization of the directional derivative on infinite-dimensional spaces is the Gâteaux differential.

Definitions

Be an open set, and a unit vector ( ie ).

The ( two-sided ) directional derivative of a function at a point along ( or in the direction of ) is defined by the limit

If the limit exists.

An alternative formulation is as follows:

By means of a function is defined with. Its derivative at the point is just the directional derivative of at the point in direction.

When you remove the restriction, so there are two ways to interpret the term " derivative along or in the direction of":

Spellings

Instead are the spellings

Usual, to avoid confusion with, among others, the covariant derivatives of the differential geometry.

Is totally differentiable, then the directional derivative can be represented using the total derivative ( see the section properties). Spellings for this are

All forms and spelling without vector arrows are common and those in which points and vectors are distinguished by bold scalars.

One-sided directional derivative

The one-sided directional derivative of in the direction defined by

The double-sided directional derivative in direction exists if and only if

In which case

Properties

  • If you choose the direction vector of the coordinate unit vectors, we obtain the partial derivatives of the respective point:
  • The unilateral directional derivative is homogeneous as a function of a positive, that is, for all positive applies:
  • If is totally differentiable, then the directional derivative as a function of even linear and can be expressed by the gradient of:

Example

In the one-dimensional case, there are only two possible directions, namely to the left or to the right. Thus, the directional derivatives correspond to the usual one-sided derivatives. The derivatives in both directions may assume different values ​​, which clearly means that the function can have a kink. A simple example is the absolute value function. It is not differentiable at, but the one-sided directional derivative exists:

The absolute value is therefore equal to its one-sided directional derivative at 0 as a function of.

Linear map
682969
de