In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the directional derivative of a multivariable
differentiable (scalar) function along a given
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
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.
The directional derivative of a
scalar function
In mathematics and physics, a scalar field is a function associating a single number to every point in a space – possibly physical space. The scalar may either be a pure mathematical number (dimensionless) or a scalar physical quantity (wi ...
''f'' with respect to a vector v at a point (e.g., position) x may be denoted by any of the following:
It therefore generalizes the notion of a
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 (as opposed to the total derivative, in which all variables are allowed to vary). Part ...
, in which the rate of change is taken along one of the
curvilinear
In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is invertible, l ...
coordinate curves
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is si ...
, all other coordinates being constant.
The directional derivative is a special case of the
Gateaux derivative
In mathematics, the Gateaux differential or Gateaux derivative is a generalization of the concept of directional derivative in differential calculus. Named after René Gateaux, a French mathematician who died young in World War I, it is defined ...
.
Definition
![Directional derivative contour plot](https://upload.wikimedia.org/wikipedia/commons/4/43/Directional_derivative_contour_plot.svg)
The ''directional derivative'' of a
scalar function
In mathematics and physics, a scalar field is a function associating a single number to every point in a space – possibly physical space. The scalar may either be a pure mathematical number (dimensionless) or a scalar physical quantity (wi ...
:
along a vector
:
is the
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
defined by the
limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
:
This definition is valid in a broad range of contexts, for example where the
norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...
of a vector (and hence a unit vector) is undefined.
For differentiable functions
If the function ''f'' is
differentiable
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
at x, then the directional derivative exists along any unit
vector v at x, and one has
:
where the
on the right denotes the ''
gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradi ...
'',
is the
dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
and v is a unit vector. This follows from defining a path
and using the definition of the derivative as a limit which can be calculated along this path to get:
:
Intuitively, the directional derivative of ''f'' at a point x represents the
rate of change of ''f'', in the direction of v with respect to time, when moving past x.
Using only direction of vector
The angle ''α'' between the tangent ''A'' and the horizontal will be maximum if the cutting plane contains the direction of the gradient ''A''.
In a
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
, some authors define the directional derivative to be with respect to an arbitrary nonzero vector v after
normalization
Normalization or normalisation refers to a process that makes something more normal or regular. Most commonly it refers to:
* Normalization (sociology) or social normalization, the process through which ideas and behaviors that may fall outside of ...
, thus being independent of its magnitude and depending only on its direction.
This definition gives the rate of increase of ''f'' per unit of distance moved in the direction given by v. In this case, one has
:
or in case ''f'' is differentiable at x,
:
Restriction to a unit vector
In the context of a function on a
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
, some texts restrict the vector v to being a
unit vector
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat").
The term ''direction vecto ...
. With this restriction, both the above definitions are equivalent.
Properties
Many of the familiar properties of the ordinary
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
hold for the directional derivative. These include, for any functions ''f'' and ''g'' defined in a
neighborhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
of, and
differentiable
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
at, p:
#
sum rule:
#
constant factor rule: For any constant ''c'',
#
product rule
In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v + ...
(or Leibniz's rule):
#
chain rule
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
: If ''g'' is differentiable at p and ''h'' is differentiable at ''g''(p), then
In differential geometry
Let be a
differentiable manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
and a point of . Suppose that is a function defined in a neighborhood of , and
differentiable
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
at . If is a
tangent vector
In mathematics, 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 R''n''. More generally, tangent vectors are eleme ...
to at , then the directional derivative of along , denoted variously as (see
Exterior derivative
On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The res ...
),
(see
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 different ...
),
(see
Lie derivative
In differential geometry, the Lie derivative ( ), named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by another vector fi ...
), or
(see ), can be defined as follows. Let be a differentiable curve with and . Then the directional derivative is defined by
:
This definition can be proven independent of the choice of , provided is selected in the prescribed manner so that .
The Lie derivative
The
Lie derivative
In differential geometry, the Lie derivative ( ), named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by another vector fi ...
of a vector field
along a vector field
is given by the difference of two directional derivatives (with vanishing torsion):
:
In particular, for a scalar field
, the Lie derivative reduces to the standard directional derivative:
:
The Riemann tensor
Directional derivatives are often used in introductory derivations of the
Riemann curvature tensor
In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds. ...
. Consider a curved rectangle with an infinitesimal vector
along one edge and
along the other. We translate a covector
along
then
and then subtract the translation along
and then
. Instead of building the directional derivative using partial derivatives, we use the
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 different ...
. The translation operator for
is thus
:
and for
,
:
The difference between the two paths is then
:
It can be argued that the noncommutativity of the covariant derivatives measures the curvature of the manifold:
:
where
is the Riemann curvature tensor and the sign depends on the
sign convention
In physics, a sign convention is a choice of the physical significance of signs (plus or minus) for a set of quantities, in a case where the choice of sign is arbitrary. "Arbitrary" here means that the same physical system can be correctly describ ...
of the author.
In group theory
Translations
In the
Poincaré algebra, we can define an infinitesimal translation operator P as
:
(the ''i'' ensures that P is a
self-adjoint operator
In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map ''A'' (from ''V'' to itse ...
) For a finite displacement λ, the
unitary
Unitary may refer to:
Mathematics
* Unitary divisor
* Unitary element
* Unitary group
* Unitary matrix
* Unitary morphism
* Unitary operator
* Unitary transformation
* Unitary representation
* Unitarity (physics)
* ''E''-unitary inverse semigrou ...
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
representation for translations is
:
By using the above definition of the infinitesimal translation operator, we see that the finite translation operator is an exponentiated directional derivative:
:
This is a translation operator in the sense that it acts on multivariable functions ''f''(x) as
:
Rotations
The
rotation operator also contains a directional derivative. The rotation operator for an angle ''θ'', i.e. by an amount ''θ'' = , ''θ'', about an axis parallel to
is
:
Here L is the vector operator that generates
SO(3)
In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space \R^3 under the operation of composition.
By definition, a rotation about the origin is a tr ...
:
:
It may be shown geometrically that an infinitesimal right-handed rotation changes the position vector x by
:
So we would expect under infinitesimal rotation:
:
It follows that
:
Following the same exponentiation procedure as above, we arrive at the rotation operator in the position basis, which is an exponentiated directional derivative:
:
Normal derivative
A normal derivative is a directional derivative taken in the direction normal (that is,
orthogonal
In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''.
By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
) to some surface in space, or more generally along a
normal vector
In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve at ...
field orthogonal to some
hypersurface
In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidean ...
. See for example
Neumann boundary condition
In mathematics, the Neumann (or second-type) boundary condition is a type of boundary condition, named after Carl Neumann.
When imposed on an ordinary or a partial differential equation, the condition specifies the values of the derivative appl ...
. If the normal direction is denoted by
, then the normal 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
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
with respect to vectors and tensors.
[J. E. Marsden and T. J. R. Hughes, 2000, ''Mathematical Foundations of Elasticity'', Dover.] The directional directive provides a systematic way of finding these derivatives.
See also
*
*
*
*
*
*
*
*
*
*
*
*
Notes
References
*
*
*
External links
Directional derivativesat
MathWorld
''MathWorld'' is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Dig ...
.
Directional derivativeat
PlanetMath
PlanetMath is a free, collaborative, mathematics online encyclopedia. The emphasis is on rigour, openness, pedagogy, real-time content, interlinked content, and also community of about 24,000 people with various maths interests. Intended to be c ...
.
{{Calculus topics
Differential calculus
Differential geometry
Generalizations of the derivative
Multivariable calculus
Scalars
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