Directional derivative

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In mathematics, the directional derivative of a multivariate differentiable function along a given 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 parallel to one of the coordinate axes.

Contents

1 Definition
2 The directional derivative in differential geometry
3 Normal derivative
4 See also

Definition

The directional derivative of a scalar function [f(\vec) = f(x_1, x_2, \ldots, x_n)] along a vector [\vec = (v_1, \ldots, v_n)] is the function defined by the limit

[D_} = \lim_ + h\vec) - f(\vec)}}.]

If the function is differentiable, it can be written in terms of the gradient [\nabla(f)] of [f] by

[D_} = \nabla(f) \cdot \vec]

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] along [\vec] at the point [p]. Usually directions are taken to be normalized, so [\vec] is a unit vector, although the definition above works for arbitrary (even zero) vectors.

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 vector [\vec] where [\vec] 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].

Normal derivative

A normal derivative is a directional derivative taken in the direction normal (that is, orthogonal) to some surface in space, or more generally along a normal vector field orthogonal to some hypersurface. See for example Neumann boundary condition.

Directional derivative

Definition

The directional derivative in differential geometry

Normal derivative

See also