Directional derivative

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In mathematics, the directional derivative of a multivariate differentiable function along a given vector V at a given point P intuitively represents the instantaneous rate of change of the function, moving through P, in the direction of V. It therefore generalizes the notion of a partial derivative, in which the direction is always taken parallel to one of the coordinate axes.

The directional derivative is a special case of the Gâteaux derivative.

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[edit] Definition

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

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

Sometimes authors write Dv instead of \nabla_v. If the function f is differentiable at \vec{x}, then the directional derivative exists along any vector \vec{v}, and one has

\nabla_{\vec{v}}{f}(\vec{x}) = \nabla f(\vec{x}) \cdot \vec{v}

where the \nabla on the right denotes the gradient and \cdot is the Euclidean inner product. At any point \vec{x}, the directional derivative of f intuitively represents the rate of change in f along \vec{v} at the point \vec{x}. Usually directions are taken to be normalized, so \vec{v} is a unit vector, although the definition above works for arbitrary (even zero) vectors.[1]

[edit] Properties

Many of the familiar properties of the ordinary derivative hold for the directional derivative. These include, for any functions f and g defined in a neighborhood of, and differentiable at, p:

\nabla_v h\circ g (p) = h'(g(p)) \nabla_v g (p)

[edit] In differential geometry

Let M be a differentiable manifold and p a point of M. Suppose that f is function defined in a neighborhood of p, and differentiable at p. If v is a tangent vector to M at p, then the directional derivative of f along v, denoted variously as \nabla_v f(p) (see covariant derivative), Lvf(p) (see Lie derivative), or vp(f) (see Tangent space#Definition via derivations), can be defined as follows. Let γ : [-1,1] → M be a differentiable curve with γ(0) = p and γ′(0) = v. Then the directional derivative is defined by

\nabla_v f(p) = \left.\frac{d}{d\tau} f\circ\gamma(\tau)\right|_{\tau=0}

This definition can be proven independent of the choice of γ, provided γ is selected in the prescribed manner so that γ'(0) = v.

[edit] 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. If the normal direction is denoted by \vec{n}, then the directional derivative of a function ƒ is sometimes denoted as \frac{ \partial f}{\partial n}.

[edit] References

  1. ^ See Tom Apostol (1974). Mathematical Analysis. Addison-Wesley, 344-345. ISBN 0-201-00288-4. 

[edit] See also