Directional derivative
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Calculus 

Integral calculus

Specialized calculi

In mathematics, the directional derivative of a multivariate differentiable function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity specified by v. It therefore generalizes the notion of a partial derivative, in which the rate of change is taken along one of the coordinate curves, all other coordinates being constant.
The directional derivative is a special case of the Gâteaux derivative.
Contents
Definition
Generally applicable definition
The directional derivative of a scalar function
along a vector
is the function defined by the limit^{1}
If the function f is differentiable at x, then the directional derivative exists along any vector v, and one has
where the on the right denotes the gradient and is the dot product.^{2} At any point x, the directional derivative of f intuitively represents the rate of change of with respect to time when it is moving at a speed and direction given by v at the point x.
Variation using only direction of vector
Some authors define the directional derivative to be with respect to the vector v after normalization, thus ignoring its magnitude. In this case, one has
or in case f is differentiable at x,
This definition has some disadvantages: its applicability is limited to when the norm of a vector is defined and nonzero. It is incompatible with notation used in some other areas of mathematics, physics and engineering, but should be used when what is wanted is the rate of increase in f per unit distance.
Restriction to unit vector
Some authors restrict the definition of the directional derivative to being with respect to a unit vector. With this restriction, the two definitions above become the same.
Notation
Directional derivatives can be also denoted by:
where v is a parameterization of a curve to which v is tangent and which determines its magnitude.
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:
 The sum rule:
 The constant factor rule: For any constant c,
 The product rule (or Leibniz rule):
 The chain rule: If g is differentiable at p and h is differentiable at g(p), then
In differential geometry
Let M be a differentiable manifold and p a point of M. Suppose that f is a 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 (see covariant derivative), (see Lie derivative), or (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
This definition can be proven independent of the choice of γ, provided γ is selected in the prescribed manner so that γ′(0) = v.
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 , then the directional derivative of a function f is sometimes denoted as . In other notations
In the continuum mechanics of solids
Several important results in continuum mechanics require the derivatives of vectors with respect to vectors and of tensors with respect to vectors and tensors.^{3} The directional directive provides a systematic way of finding these derivatives.
The definitions of directional derivatives for various situations are given below. It is assumed that the functions are sufficiently smooth that derivatives can be taken.
Derivatives of scalar valued functions of vectors
Let be a real valued function of the vector . Then the derivative of with respect to (or at ) in the direction is defined as
for all vectors .
Properties:
 If then
 If then
 If then
Derivatives of vector valued functions of vectors
Let be a vector valued function of the vector . Then the derivative of with respect to (or at ) in the direction is the secondorder tensor defined as
for all vectors .
Properties:
 If then
 If then
 If then
Derivatives of scalar valued functions of secondorder tensors
Let be a real valued function of the second order tensor . Then the derivative of with respect to (or at ) in the direction is the second order tensor defined as
for all second order tensors .
Properties:
 If then
 If then
 If then
Derivatives of tensor valued functions of secondorder tensors
Let be a second order tensor valued function of the second order tensor . Then the derivative of with respect to (or at ) in the direction is the fourth order tensor defined as
for all second order tensors .
Properties:
 If then
 If then
 If then
 If then
See also
 Fréchet derivative
 Gâteaux derivative
 Derivative (generalizations)
 Lie derivative
 Differential form
 Structure tensor
 Tensor derivative (continuum mechanics)
 Del in cylindrical and spherical coordinates
Notes
 ^ R. Wrede, M.R. Spiegel (2010). Advanced Calculus (3rd edition ed.). Schaum's Outline Series. ISBN 9780071623667.
 ^ Technically, the gradient ∇f is a covector, and the "dot product" is the action of this covector on the vector v.
 ^ J. E. Marsden and T. J. R. Hughes, 2000, Mathematical Foundations of Elasticity, Dover.
References
 Hildebrand, F. B. (1976). Advanced Calculus for Applications. Prentice Hall. ISBN 0130111899.
 K.F. Riley, M.P. Hobson, S.J. Bence (2010). Mathematical methods for physics and engineering. Cambridge University Press. ISBN 9780521861533.
External links
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