TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, the directional derivative of a multivariate along a given
vector Vector may refer to: Biology *Vector (epidemiology) In epidemiology Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and determinants In mathematics, the determinant is a Scalar (mathematics ...
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 or scalar-valued function (mathematics), function associates a Scalar (mathematics), scalar value to every point (geometry), point in a space (mathematics), space – possibly physical space. The scal ...

''f'' with respect to a vector v at a point (e.g., position) x may be denoted by any of the following: : $\nabla_\left(\mathbf\right)=f\text{'}_\mathbf\left(\mathbf\right)=D_\mathbff\left(\mathbf\right)=Df\left(\mathbf\right)\left(\mathbf\right)=\partial_\mathbff\left(\mathbf\right)=\mathbf\cdot=\mathbf\cdot \frac$. It therefore generalizes the notion of a
partial derivative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

, 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, ...
coordinate curves, all other coordinates being constant. The directional derivative is a special case of the
Gateaux derivative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
.

Definition

The ''directional derivative'' of a
scalar function In mathematics and physics, a scalar field or scalar-valued function (mathematics), function associates a Scalar (mathematics), scalar value to every point (geometry), point in a space (mathematics), space – possibly physical space. The scal ...
:$f\left(\mathbf\right) = f\left(x_1, x_2, \ldots, x_n\right)$ along a vector :$\mathbf = \left(v_1, \ldots, v_n\right)$ 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 ...
$\nabla_$ defined by the
limit Limit or Limits may refer to: Arts and media * Limit (music), a way to characterize harmony * Limit (song), "Limit" (song), a 2016 single by Luna Sea * Limits (Paenda song), "Limits" (Paenda song), 2019 song that represented Austria in the Eurov ...
:$\nabla_\left(\mathbf\right) = \lim_.$ This definition is valid in a broad range of contexts, for example where the
norm Norm, the Norm or NORM may refer to: In academic disciplines * Norm (geology), an estimate of the idealised mineral content of a rock * Norm (philosophy), a standard in normative ethics that is prescriptive rather than a descriptive or explanat ...
of a vector (and hence a unit vector) is undefined.

For differentiable functions

If the function ''f'' is
differentiable In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
at x, then the directional derivative exists along any vector v, and one has :$\nabla_\left(\mathbf\right) = \nabla f\left(\mathbf\right) \cdot \mathbf$ where the $\nabla$ on the right denotes the ''
gradient In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Prod ...

'' and $\cdot$ is the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...
. This follows from defining a path $h\left(t\right)=x+tv$ and using the definition of the derivative as a limit which can be calculated along this path to get: :$\begin 0&=\lim_\frac t \\ &=\lim_\frac t - Df\left(x\right)\left(v\right)=\nabla_v f\left(x\right)-Df\left(x\right)\left(v\right) \\ \rightarrow & \nabla f\left(\mathbf\right) \cdot \mathbf=Df\left(x\right)\left(v\right)=\nabla_\left(\mathbf\right) \end$ Intuitively, the directional derivative of ''f'' at a point x represents the 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 classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
, 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 :$\nabla_\left(\mathbf\right) = \lim_,$ or in case ''f'' is differentiable at x, :$\nabla_\left(\mathbf\right) = \nabla f\left(\mathbf\right) \cdot \frac .$

Restriction to a unit vector

In the context of a function on a
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
, some texts restrict the vector v to being a
unit vector In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
. With this restriction, both the above definitions are equivalent.

Properties

Many of the familiar properties of the ordinary
derivative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

hold for the directional derivative. These include, for any functions ''f'' and ''g'' defined in a
neighborhood A neighbourhood (British English British English (BrE) is the standard dialect A standard language (also standard variety, standard dialect, and standard) is a language variety that has undergone substantial codification of grammar ...
of, and
differentiable In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
at, p: # sum rule: #:$\nabla_ \left(f + g\right) = \nabla_ f + \nabla_ g.$ # constant factor rule: For any constant ''c'', #:$\nabla_ \left(cf\right) = c\nabla_ f.$ #
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 (mathematics), functions. For two functions, it may be stated in Notation for differentiatio ...
(or Leibniz's rule): #:$\nabla_ \left(fg\right) = g\nabla_ f + f\nabla_ g.$ #
chain rule In calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of ...
: If ''g'' is differentiable at p and ''h'' is differentiable at ''g''(p), then #:$\nabla_\left(h\circ g\right)\left(\mathbf\right) = h\text{'}\left(g\left(\mathbf\right)\right) \nabla_ g \left(\mathbf\right).$

In differential geometry

Let be a
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's s ...
and a point of . Suppose that is a function defined in a neighborhood of , and
differentiable In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
at . If is a
tangent vector In mathematics, a tangent vector is a Vector (geometry), vector that is tangent to a curve or Surface (mathematics), surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in R''n''. ...
to at , then the directional derivative of along , denoted variously as (see
Exterior derivative On a differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-interse ...
), $\nabla_ f\left(\mathbf\right)$ (see
Covariant derivative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
), $L_ f\left(\mathbf\right)$ (see
Lie derivative In differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integral ...
), or $_\left(f\right)$ (see ), can be defined as follows. Let be a differentiable curve with and . Then the directional derivative is defined by :$\nabla_ f\left(\mathbf\right) = \left.\frac f\circ\gamma\left(\tau\right)\_.$ 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 Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integral ...
of a vector field $W^\mu\left(x\right)$ along a vector field $V^\mu\left(x\right)$ is given by the difference of two directional derivatives (with vanishing torsion): :$\mathcal_V W^\mu=\left(V\cdot\nabla\right) W^\mu-\left(W\cdot\nabla\right) V^\mu.$ In particular, for a scalar field $\phi\left(x\right)$, the Lie derivative reduces to the standard directional derivative: :$\mathcal_V \phi=\left(V\cdot\nabla\right) \phi.$

The Riemann tensor

Directional derivatives are often used in introductory derivations of the
Riemann curvature tensor In the mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...
. Consider a curved rectangle with an infinitesimal vector ''δ'' along one edge and ''δ''′ along the other. We translate a covector ''S'' along ''δ'' then ''δ''′ and then subtract the translation along ''δ''′ and then ''δ''. Instead of building the directional derivative using partial derivatives, we use the
covariant derivative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
. The translation operator for ''δ'' is thus :$1+\sum_\nu \delta^\nu D_\nu=1+\delta\cdot D,$ and for ''δ''′, :$1+\sum_\mu \delta\text{'}^\mu D_\mu=1+\delta\text{'}\cdot D.$ 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 ''R'' is the Riemann curvature tensor and the sign depends on the
sign convention In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succes ...
of the author.

In group theory

Translations

In the Poincaré algebra, we can define an infinitesimal translation operator P as :$\mathbf=i\nabla.$ (the ''i'' ensures that P is a
self-adjoint operator In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
) For a finite displacement λ, the
unitary Unitary may refer to: Mathematics * E-unitary inverse semigroup, ''E''-unitary inverse semigroups * Unitary element * Unitary matrix * Unitary operator * Unitary transformation * Unitary group * Unitary divisor * Unitary representation * Unitarity ...
Hilbert space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
representation Representation may refer to: Law and politics *Representation (politics) Political representation is the activity of making citizens "present" in public policy making processes when political actors act in the best interest of citizens. This def ...
for translations is :$U\left(\boldsymbol\right)=\exp\left\left(-i\boldsymbol\cdot\mathbf\right\right).$ By using the above definition of the infinitesimal translation operator, we see that the finite translation operator is an exponentiated directional derivative: :$U\left(\boldsymbol\right)=\exp\left\left(\boldsymbol\cdot\nabla\right\right).$ This is a translation operator in the sense that it acts on multivariable functions ''f''(x) as :$U\left(\boldsymbol\right) f\left(\mathbf\right)=\exp\left\left(\boldsymbol\cdot\nabla\right\right) f\left(\mathbf\right) = f\left(\mathbf+\boldsymbol\right).$

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 $\hat = \boldsymbol/\theta$ is :$U\left(R\left(\mathbf\right)\right)=\exp\left(-i\mathbf\cdot\mathbf\right).$ Here L is the vector operator that generates
SO(3) In mechanics Mechanics (Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximatel ...
: :$\mathbf=\begin 0& 0 & 0\\ 0& 0 & 1\\ 0& -1 & 0 \end\mathbf+\begin 0 &0 & -1\\ 0& 0 &0 \\ 1 & 0 & 0 \end\mathbf+\begin 0&1 &0 \\ -1&0 &0 \\ 0 & 0 & 0 \end\mathbf.$ It may be shown geometrically that an infinitesimal right-handed rotation changes the position vector x by :$\mathbf\rightarrow \mathbf-\delta\boldsymbol\times\mathbf.$ So we would expect under infinitesimal rotation: :$U\left(R\left(\delta\boldsymbol\right)\right) f\left(\mathbf\right) = f\left(\mathbf-\delta\boldsymbol\times\mathbf\right)=f\left(\mathbf\right)-\left(\delta\boldsymbol\times\mathbf\right)\cdot\nabla f.$ It follows that :$U\left(R\left(\delta\mathbf\right)\right)=1-\left(\delta\mathbf\times\mathbf\right)\cdot\nabla.$ Following the same exponentiation procedure as above, we arrive at the rotation operator in the position basis, which is an exponentiated directional derivative: :$U\left(R\left(\mathbf\right)\right)=\exp\left(-\left(\mathbf\times\mathbf\right)\cdot\nabla\right).$

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 notion of perpendicularity to the linear algebra of bilinear forms. Two elements ''u'' and ''v'' of a vector space with bilinear form ''B'' are orthogonal when . Depending on the bili ...
) to some surface in space, or more generally along a
normal vector In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

field orthogonal to some
hypersurface In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of f ...
. 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 differential equation, ordinary or a partial differential equation, the condition specifies the v ...
. If the normal direction is denoted by $\mathbf$, then the normal derivative of a function ''f'' is sometimes denoted as $\frac$. 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
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.

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Derivative (generalizations) In mathematics, 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 In real, c ...
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Fréchet derivative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
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Gateaux derivative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
Lie derivative In differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integral ...
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Material derivative In continuum mechanics Continuum mechanics is a branch of mechanics Mechanics (Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country ...
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Differential form In the mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population ...
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Structure tensor In mathematics, the structure tensor, also referred to as the second-moment matrix, is a matrix (mathematics), matrix derived from the gradient of a function (mathematics), function. It describes the distribution of the gradient in a specified ...
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Tensor derivative (continuum mechanics) The derivatives Derivative may refer to: In mathematics and economics *Brzozowski derivative in the theory of formal languages *Derivative in calculus, a quantity indicating how a function changes when the values of its inputs change. *Formal de ...
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Del in cylindrical and spherical coordinates This is a list of some vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar ...

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