Directional derivative

# Directional derivative

Discussion

Encyclopedia
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the directional derivative of a multivariate differentiable function
Differentiable function
In calculus , a differentiable function is a function whose derivative exists at each point in its domain. The graph of a differentiable function must have a non-vertical tangent line at each point in its domain...

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
Partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant...

, 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
Gâteaux derivative
In mathematics, the Gâteaux differential or Gâteaux derivative is a generalization of the concept of directional derivative in differential calculus. Named after René Gâteaux, a French mathematician who died young in World War I, it is defined for functions between locally convex topological vector...

.

## Definition

The directional derivative of a scalar function
along a unit vector
is the function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

defined by the limit
Limit (mathematics)
In mathematics, the concept of a "limit" is used to describe the value that a function or sequence "approaches" as the input or index approaches some value. The concept of limit allows mathematicians to define a new point from a Cauchy sequence of previously defined points within a complete metric...

(See other notations below.) If the function is differentiable at , then the directional derivative exists along any unit vector and one has

where the on the right denotes the gradient
In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....

and is the Euclidean inner product
Dot product
In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...

. At any point , the directional derivative of intuitively represents the rate of change
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

in along at the point .

One sometimes permits non-unit vectors, allowing the directional derivative to be taken in the direction of , where is any nonzero vector. In this case, one must modify the definitions to account for the fact that may not be normalized, so one has
or in case is differentiable at ,
Such notation for non-unit vectors (undefined for the zero vector), however, is incompatible with notation used elsewhere in mathematics, where the space of derivations in a derivation algebra is expected to be a vector space.

### Notation

Directional derivatives can be also denoted by:

### Properties

Many of the familiar properties of the ordinary derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

hold for the directional derivative. These include, for any functions f and g defined in a neighborhood of, and differentiable
Total derivative
In the mathematical field of differential calculus, the term total derivative has a number of closely related meanings.The total derivative of a function f, of several variables, e.g., t, x, y, etc., with respect to one of its input variables, e.g., t, is different from the partial derivative...

at, p:
• The sum rule
Sum rule in differentiation
In calculus, the sum rule in differentiation is a method of finding the derivative of a function that is the sum of two other functions for which derivatives exist. This is a part of the linearity of differentiation. The sum rule in integration follows from it...

:
• The constant factor rule
Constant factor rule in differentiation
In calculus, the constant factor rule in differentiation, also known as The Kutz Rule, allows you to take constants outside a derivative and concentrate on differentiating the function of x itself...

: For any constant c,
• The product rule
Product rule
In calculus, the product rule is a formula used to find the derivatives of products of two or more functions. It may be stated thus:'=f'\cdot g+f\cdot g' \,\! or in the Leibniz notation thus:...

(or Leibniz rule):
• The chain rule
Chain rule
In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions. That is, if f is a function and g is a function, then the chain rule expresses the derivative of the composite function in terms of the derivatives of f and g.In integration, the...

: If g is differentiable at p and h is differentiable at g(p), then

## In differential geometry

Let M be a differentiable manifold
Differentiable manifold
A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...

and p a point of M. Suppose that f is a function defined in a neighborhood of p, and differentiable
Total derivative
In the mathematical field of differential calculus, the term total derivative has a number of closely related meanings.The total derivative of a function f, of several variables, e.g., t, x, y, etc., with respect to one of its input variables, e.g., t, is different from the partial derivative...

at p. If v is a tangent vector
Tangent vector
A tangent vector is a vector that is tangent to a curve or surface at a given point.Tangent vectors are described in the differential geometry of curves in the context of curves in Rn. More generally, tangent vectors are elements of a tangent space of a differentiable manifold....

to M at p, then the directional derivative of f along v, denoted variously as (see covariant derivative
Covariant derivative
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given...

), (see Lie derivative
Lie derivative
In mathematics, the Lie derivative , named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a vector field or more generally a tensor field, along the flow of another vector field...

), 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
Hypersurface
In geometry, a hypersurface is a generalization of the concept of hyperplane. Suppose an enveloping manifold M has n dimensions; then any submanifold of M of n − 1 dimensions is a hypersurface...

. See for example Neumann boundary condition
Neumann boundary condition
In mathematics, the Neumann boundary condition is a type of boundary condition, named after Carl Neumann.When imposed on an ordinary or a partial differential equation, it specifies the values that the derivative of a solution is to take on the boundary of the domain.* For an ordinary...

. If the normal direction is denoted by , then the directional derivative of a function ƒ 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. 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 the vector defined as
for all vectors .

Properties:

1) If then

2) If then

3) 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 second order tensor defined as
for all vectors .

Properties:

1) If then

2) If then

3) If then

### Derivatives of scalar valued functions of second-order 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:

1) If then

2) If then

3) If then

### Derivatives of tensor valued functions of second-order 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:

1) If then

2) If then

3) If then

4) If then

• Fréchet derivative
Fréchet derivative
In mathematics, the Fréchet derivative is a derivative defined on Banach spaces. Named after Maurice Fréchet, it is commonly used to formalize the concept of the functional derivative used widely in the calculus of variations. Intuitively, it generalizes the idea of linear approximation from...

• Gâteaux derivative
Gâteaux derivative
In mathematics, the Gâteaux differential or Gâteaux derivative is a generalization of the concept of directional derivative in differential calculus. Named after René Gâteaux, a French mathematician who died young in World War I, it is defined for functions between locally convex topological vector...

• Derivative (generalizations)
Derivative (generalizations)
The derivative is a fundamental construction of differential calculus and admits many possible generalizations within the fields of mathematical analysis, combinatorics, algebra, and geometry.- Derivatives in analysis :...

• Lie derivative
Lie derivative
In mathematics, the Lie derivative , named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a vector field or more generally a tensor field, along the flow of another vector field...

• Differential form
Differential form
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...

• Structure tensor
Structure tensor
In mathematics, the structure tensor, also referred to as the second-moment matrix, is a matrix derived from the gradient of a function. It summarizes the predominant directions of the gradient in a specified neighborhood of a point, and the degree to which those directions are coherent...

• Tensor derivative (continuum mechanics)
Tensor derivative (continuum mechanics)
The derivatives of scalars, vectors, and second-order tensors with respect to second-order tensors are of considerable use in continuum mechanics. These derivatives are used in the theories of nonlinear elasticity and plasticity, particularly in the design of algorithms for numerical...

• Del in cylindrical and spherical coordinates
Del in cylindrical and spherical coordinates
This is a list of some vector calculus formulae of general use in working with various curvilinear coordinate systems.- Note :* This page uses standard physics notation. For spherical coordinates, \theta is the angle between the z axis and the radius vector connecting the origin to the point in...