TheInfoList

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 * Product differentiation, in ...
, the gradient of a scalar-valued
differentiable function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

of several variables is the
vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. For instance, a vector field in the plane can be visualised as a collection of arrows with a given magnitude and direction, each att ...

(or
vector-valued function A vector-valued function, also referred to as a vector function, is a function (mathematics), mathematical function of one or more variables whose range of a function, range is a set of multidimensional Euclidean vector, vectors or infinite-dimensi ...

) $\nabla f$ whose value at a point $p$ is the
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 risk factor, determinants of health and disease conditions in defined pop ...
whose components are the
partial derivative In , a partial derivative of a is its with respect to one of those variables, with the others held constant (as opposed to the , in which all variables are allowed to vary). Partial derivatives are used in and . The partial derivative of a fu ...

s of $f$ at $p$. That is, for $f \colon \R^n \to \R$, its gradient $\nabla f \colon \R^n \to \R^n$ is defined at the point $p = \left(x_1,\ldots,x_n\right)$ in ''n-''dimensional space as the vector: :$\nabla f\left(p\right) = \begin \frac\left(p\right) \\ \vdots \\ \frac\left(p\right) \end.$ The
nabla symbol ∇ The nabla symbol The nabla is a triangular symbol resembling an inverted Greek delta:Indeed, it is called ''anadelta'' ( ανάδελτα) in Modern Greek Modern Greek (, , or , ''Kiní Neoellinikí Glóssa''), generally referred to by ...
$\nabla$, written as an upside-down triangle and pronounced "del", denotes the . The gradient vector can be interpreted as the "direction and rate of fastest increase". If the gradient of a function is non-zero at a point , the direction of the gradient is the direction in which the function increases most quickly from , and the
magnitude Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (mathematics), the relative size of an object *Norm (mathematics), a term for the size or length of a vector *Order of ...
of the gradient is the rate of increase in that direction, the greatest
absoluteAbsolute may refer to: Companies * Absolute Entertainment, a video game publisher * Absolute Radio, (formerly Virgin Radio), independent national radio station in the UK * Absolute Software Corporation, specializes in security and data risk managem ...

directional derivative. Further, the gradient is the zero vector at a point if and only if it is a
stationary point In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
(where the derivative vanishes). The gradient thus plays a fundamental role in
optimization theory Nelder-Mead minimum search of Simionescu's function. Simplex vertices are ordered by their values, with 1 having the lowest ( best) value., alt= Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming i ...
, where it is used to maximize a function by
gradient ascent Gradient descent (also often called steepest descent) is a :First order methods, first-order Iterative algorithm, iterative Mathematical optimization, optimization algorithm for finding a local minimum of a differentiable function. The idea is to ...
. The gradient is dual to the
total derivative In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
$df$: the value of the gradient at a point is a
tangent vector :''For a more general — but much more technical — treatment of tangent vectors, see tangent space.'' In mathematics, a tangent vector is a Vector (geometry), vector that is tangent to a curve or Surface (mathematics), surface at a given point. T ...
– a vector at each point; while the value of the derivative at a point is a ''co''tangent vector – a linear function on vectors. They are related in that 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 ...
of the gradient of at a point with another tangent vector equals the
directional derivative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
of at of the function along ; that is, $\nabla f(p) \cdot \mathbf v = \frac(p) = df_(\mathbf)$. The gradient admits multiple generalizations to more general functions on
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 surface. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of su ...

s; see .

# Motivation

Consider a room where the temperature is given by a
scalar field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

, , so at each point the temperature is , independent of time. At each point in the room, the gradient of at that point will show the direction in which the temperature rises most quickly, moving away from . The magnitude of the gradient will determine how fast the temperature rises in that direction. Consider a surface whose height above sea level at point is . The gradient of at a point is a plane vector pointing in the direction of the steepest slope or
grade Grade or grading may refer to: Arts and entertainment * Grade (band) Grade is a melodic hardcore band from Canada, often credited as pioneers in blending metallic hardcore with the hon and melody of emo, and - most notably - the alternating scr ...
at that point. The steepness of the slope at that point is given by the magnitude of the gradient vector. The gradient can also be used to measure how a scalar field changes in other directions, rather than just the direction of greatest change, by taking a
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 ...
. Suppose that the steepest slope on a hill is 40%. A road going directly uphill has slope 40%, but a road going around the hill at an angle will have a shallower slope. For example, if the road is at a 60° angle from the uphill direction (when both directions are projected onto the horizontal plane), then the slope along the road will be the dot product between the gradient vector and a
unit vector In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
along the road, namely 40% times the
cosine In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in al ...

of 60°, or 20%. More generally, if the hill height function is
differentiable In calculus (a branch of mathematics), a differentiable function of one Real number, real variable is a function whose derivative exists at each point in its Domain of a function, domain. In other words, the Graph of a function, graph of a differen ...

, then the gradient of dotted with a
unit vector In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
gives the slope of the hill in the direction of the vector, the
directional derivative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
of along the unit vector.

# Notation

The gradient of a function $f$ at point $a$ is usually written as $\nabla f \left(a\right)$. It may also be denoted by any of the following: * $\vec f \left(a\right)$ : to emphasize the vector nature of the result. * * $\left. \frac\_$ * $\partial_i f$ and $f_$ :
Einstein notation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
.

# Definition

The gradient (or gradient vector field) of a scalar function is denoted or where ( nabla) denotes the vector
differential operator 300px, A harmonic function defined on an annulus. Harmonic functions are exactly those functions which lie in the kernel of the Laplace operator, an important differential operator. In mathematics, a differential operator is an Operator (mathe ...
,
del Del, or nabla, is an operator used in mathematics (particularly in vector calculus Vector calculus, or vector analysis, is concerned with derivative, differentiation and integral, integration of vector fields, primarily in 3-dimensional ...

. The notation is also commonly used to represent the gradient. The gradient of is defined as the unique vector field whose dot product with any
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 risk factor, determinants of health and disease conditions in defined pop ...
at each point is the directional derivative of along . That is, :$\big\left(\nabla f\left(x\right)\big\right)\cdot \mathbf = D_f\left(x\right).$ Formally, the gradient is ''dual'' to the derivative; see relationship with derivative. When a function also depends on a parameter such as time, the gradient often refers simply to the vector of its spatial derivatives only (see Spatial gradient). The magnitude and direction of the gradient vector are
independent Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s * Independent ...
of the particular .

## Cartesian coordinates

In the three-dimensional
Cartesian coordinate system A Cartesian coordinate system (, ) in a plane Plane or planes may refer to: * Airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft A fixed-wing aircraft is a heavier-than-air flying machine Early fly ...
with a
Euclidean metric In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore occas ...
, the gradient, if it exists, is given by: :$\nabla f = \frac \mathbf + \frac \mathbf + \frac \mathbf,$ where , , are the
standard Standard may refer to: Flags * Colours, standards and guidons * Standard (flag), a type of flag used for personal identification Norm, convention or requirement * Standard (metrology), an object that bears a defined relationship to a unit of ...
unit vectors in the directions of the , and coordinates, respectively. For example, the gradient of the function :$f\left(x,y,z\right)= 2x+3y^2-\sin\left(z\right)$ is :$\nabla f = 2\mathbf+ 6y\mathbf -\cos\left(z\right)\mathbf.$ In some applications it is customary to represent the gradient as a
row vector In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and th ...
or
column vector In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and ...
of its components in a rectangular coordinate system; this article follows the convention of the gradient being a column vector, while the derivative is a row vector.

## Cylindrical and spherical coordinates

In
cylindrical coordinates 240px, A cylindrical coordinate system with origin , polar axis , and longitudinal axis . The dot is the point with radial distance , angular coordinate , and height . A cylindrical coordinate system is a three-dimensional coordinate system that s ...

with a Euclidean metric, the gradient is given by:. :$\nabla f\left(\rho, \varphi, z\right) = \frac\mathbf_\rho + \frac\frac\mathbf_\varphi + \frac\mathbf_z,$ where is the axial distance, is the azimuthal or azimuth angle, is the axial coordinate, and , and are unit vectors pointing along the coordinate directions. In
spherical coordinates File:3D Spherical 2.svg, 240px, Spherical coordinates as often used in ''mathematics'': radial distance , azimuthal angle , and polar angle . The meanings of and have been swapped compared to the physics convention. As in physics, (rho) is of ...

, the gradient is given by: :$\nabla f\left(r, \theta, \varphi\right) = \frac\mathbf_r + \frac\frac\mathbf_\theta + \frac\frac\mathbf_\varphi,$ where is the radial distance, is the azimuthal angle and is the polar angle, and , and are again local unit vectors pointing in the coordinate directions (that is, the normalized covariant basis). For the gradient in other
orthogonal coordinate system In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no genera ...
s, see Orthogonal coordinates (Differential operators in three dimensions).

## General coordinates

We consider general coordinates, which we write as , where is the number of dimensions of the domain. Here, the upper index refers to the position in the list of the coordinate or component, so refers to the second component—not the quantity squared. The index variable refers to an arbitrary element . Using
Einstein notation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, the gradient can then be written as: :$\nabla f = \fracg^ \mathbf_j$ ( Note that its
dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality ** . . . see more cases in :Duality theories * Dual ...
is $\mathrmf= \frac\mathbf^i$ ), where $\mathbf_i = \partial \mathbf/\partial x^i$ and $\mathbf^i = \mathrmx^i$ refer to the unnormalized local covariant and contravariant bases respectively, $g^$ is the inverse metric tensor, and the Einstein summation convention implies summation over ''i'' and ''j''. If the coordinates are orthogonal we can easily express the gradient (and the differential) in terms of the normalized bases, which we refer to as $\hat_i$ and $\hat^i$, using the scale factors (also known as Lamé coefficients) $h_i= \lVert \mathbf_i \rVert = 1\, / \lVert \mathbf^i \rVert$ : :$\nabla f = \sum_^n \, \frac\frac\mathbf_i$ ( and $\mathrmf = \sum_^n \, \frac\frac\mathbf^i$ ), where we cannot use Einstein notation, since it is impossible to avoid the repetition of more than two indices. Despite the use of upper and lower indices, $\mathbf_i$, $\mathbf^i$, and $h_i$ are neither contravariant nor covariant. The latter expression evaluates to the expressions given above for cylindrical and spherical coordinates.

# Relationship with derivative

## Relationship with total derivative

The gradient is closely related to the
total derivative In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
(
total differential In calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. ...
) $df$: they are
transpose In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces an ...

(
dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality ** . . . see more cases in :Duality theories * Dual ...
) to each other. Using the convention that vectors in $\R^n$ are represented by
column vector In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and ...
s, and that covectors (linear maps $\R^n \to \R$) are represented by
row vector In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and th ...
s, the gradient $\nabla f$ and the derivative $df$ are expressed as a column and row vector, respectively, with the same components, but transpose of each other: :$\nabla f\left(p\right) = \begin\frac\left(p\right) \\ \vdots \\ \frac\left(p\right) \end ;$ :$df_p = \begin\frac\left(p\right) & \cdots & \frac\left(p\right) \end .$ While these both have the same components, they differ in what kind of mathematical object they represent: at each point, the derivative is a
cotangent vector In differential geometry Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The Differential ...
, a
linear form 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 ...

(
covector In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
) which expresses how much the (scalar) output changes for a given infinitesimal change in (vector) input, while at each point, the gradient is a
tangent vector :''For a more general — but much more technical — treatment of tangent vectors, see tangent space.'' In mathematics, a tangent vector is a Vector (geometry), vector that is tangent to a curve or Surface (mathematics), surface at a given point. T ...
, which represents an infinitesimal change in (vector) input. In symbols, the gradient is an element of the tangent space at a point, $\nabla f\left(p\right) \in T_p \R^n$, while the derivative is a map from the tangent space to the real numbers, $df_p \colon T_p \R^n \to \R$. The tangent spaces at each point of $\R^n$ can be "naturally" identified with the vector space $\R^n$ itself, and similarly the cotangent space at each point can be naturally identified with the
dual vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
$\left(\R^n\right)^*$ of covectors; thus the value of the gradient at a point can be thought of a vector in the original $\R^n$, not just as a tangent vector. Computationally, given a tangent vector, the vector can be ''multiplied'' by the derivative (as matrices), which is equal to taking 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 ...
with the gradient: :$\left(df_p\right)\left(v\right) = \begin\frac\left(p\right) & \cdots & \frac\left(p\right) \end \beginv_1 \\ \vdots \\ v_n\end = \sum_^n \frac\left(p\right) v_i = \begin\frac\left(p\right) \\ \vdots \\ \frac\left(p\right) \end \cdot \beginv_1 \\ \vdots \\ v_n\end = \nabla f\left(p\right) \cdot v$

### Differential or (exterior) derivative

The best linear approximation to a differentiable function :$f \colon \R^n \to \R$ at a point in is a linear map from to which is often denoted by or and called the differential or
total derivative In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
of at . The function , which maps to , is called the
total differential In calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. ...
or
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 ...
of and is an example of a
differential 1-form In the mathematics, 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 unified approach to define integran ...
. Much as the derivative of a function of a single variable represents the
slope In mathematics, the slope or gradient of a line Line, lines, The Line, or LINE may refer to: Arts, entertainment, and media Films * ''Lines'' (film), a 2016 Greek film * ''The Line'' (2017 film) * ''The Line'' (2009 film) * ''The Line'', ...

of the
tangent 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 ...

to the
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discret ...

of the function, the directional derivative of a function in several variables represents the slope of the tangent
hyperplane 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 ...
in the direction of the vector. The gradient is related to the differential by the formula :$\left(\nabla f\right)_x\cdot v = df_x\left(v\right)$ for any , where $\cdot$ 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 ...
: taking the dot product of a vector with the gradient is the same as taking the directional derivative along the vector. If is viewed as the space of (dimension ) column vectors (of real numbers), then one can regard as the row vector with components :$\left\left( \frac, \dots, \frac\right\right),$ so that is given by
matrix multiplication 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 ...

. Assuming the standard Euclidean metric on , the gradient is then the corresponding column vector, that is, :$\left(\nabla f\right)_i = df^\mathsf_i.$

### Linear approximation to a function

The best
linear approximation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
to a function can be expressed in terms of the gradient, rather than the derivative. The gradient of a
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
from the Euclidean space to at any particular point in characterizes the best
linear approximation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
to at . The approximation is as follows: :$f\left(x\right) \approx f\left(x_0\right) + \left(\nabla f\right)_\cdot\left(x-x_0\right)$ for close to , where is the gradient of computed at , and the dot denotes the dot product on . This equation is equivalent to the first two terms in the multivariable Taylor series expansion of at .

## Relationship with Fréchet derivative

Let be an
open set 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 . If the function is differentiable, then the differential of is the
Fréchet derivative In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of . Thus is a function from to the space such that :$\lim_ \frac = 0,$ where · is the dot product. As a consequence, the usual properties of the derivative hold for the gradient, though the gradient is not a derivative itself, but rather dual to the derivative: ;
Linearity Linearity is the property of a mathematical relationship (''function (mathematics), function'') that can be graph of a function, graphically represented as a straight Line (geometry), line. Linearity is closely related to ''Proportionality (math ...

The gradient is linear in the sense that if and are two real-valued functions differentiable at the point , and and are two constants, then is differentiable at , and moreover :$\nabla\left\left(\alpha f+\beta g\right\right)\left(a\right) = \alpha \nabla f\left(a\right) + \beta\nabla g \left(a\right).$ ;
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 ...
If and are real-valued functions differentiable at a point , then the product rule asserts that the product is differentiable at , and :$\nabla \left(fg\right)\left(a\right) = f\left(a\right)\nabla g\left(a\right) + g\left(a\right)\nabla f\left(a\right).$ ;
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 ...
Suppose that is a real-valued function defined on a subset of , and that is differentiable at a point . There are two forms of the chain rule applying to the gradient. First, suppose that the function is a
parametric curve In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
; that is, a function maps a subset into . If is differentiable at a point such that , then :$\left(f\circ g\right)\text{'}\left(c\right) = \nabla f\left(a\right)\cdot g\text{'}\left(c\right),$ where ∘ is the
composition operator In mathematics, the composition operator C_\phi with symbol \phi is a linear operator defined by the rule :C_\phi (f) = f \circ\phi where f \circ\phi denotes function composition. The study of composition operators is covered bAMS category 47B33 ...
: . More generally, if instead , then the following holds: :$\nabla \left(f\circ g\right)\left(c\right) = \big\left(Dg\left(c\right)\big\right)^\mathsf \big\left(\nabla f\left(a\right)\big\right),$ where T denotes the transpose
Jacobian matrix 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 * Produ ...
. For the second form of the chain rule, suppose that is a real valued function on a subset of , and that is differentiable at the point . Then :$\nabla \left(h\circ f\right)\left(a\right) = h\text{'}\big\left(f\left(a\right)\big\right)\nabla f\left(a\right).$

# Further properties and applications

## Level sets

A level surface, or
isosurface An isosurface is a three-dimensional analog of an isoline. It is a surface that represents points of a constant value (e.g. pressure, temperature, velocity, density) within a volume Volume is the quantity of three-dimensional space enclosed ...
, is the set of all points where some function has a given value. If is differentiable, then the dot product of the gradient at a point with a vector gives the directional derivative of at in the direction . It follows that in this case the gradient of 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 the
level set 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 ...
s of . For example, a level surface in three-dimensional space is defined by an equation of the form . The gradient of is then normal to the surface. More generally, any embedded
hypersurface In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that ...
in a Riemannian manifold can be cut out by an equation of the form such that is nowhere zero. The gradient of is then normal to the hypersurface. Similarly, an affine algebraic hypersurface may be defined by an equation , where is a polynomial. The gradient of is zero at a singular point of the hypersurface (this is the definition of a singular point). At a non-singular point, it is a nonzero normal vector.

## Conservative vector fields and the gradient theorem

The gradient of a function is called a gradient field. A (continuous) gradient field is always a
conservative vector field In vector calculus, a conservative vector field is a vector field that is the gradient of some function (mathematics), function. Conservative vector fields have the property that the line integral is path independent; the choice of any path betwee ...
: its
line integral In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
along any path depends only on the endpoints of the path, and can be evaluated by the gradient theorem (the fundamental theorem of calculus for line integrals). Conversely, a (continuous) conservative vector field is always the gradient of a function.

# Generalizations

## Jacobian

The
Jacobian matrix 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 * Produ ...
is the generalization of the gradient for vector-valued functions of several variables and
differentiable map In mathematics, a differentiable function of one Real number, real variable is a Function (mathematics), function whose derivative exists at each point in its Domain of a function, domain. In other words, the Graph of a function, graph of a differ ...
s between
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 ...
s or, more generally,
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 surface. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of su ...

s. A further generalization for a function between
Banach 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 th ...
s is the
Fréchet derivative In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. Suppose is a function such that each of its first-order partial derivatives exist on . Then the Jacobian matrix of is defined to be an matrix, denoted by $\mathbf_\mathbb\left(\mathbb\right)$ or simply $\mathbf$. The th entry is $\mathbf J_ = \frac$. Explicitly : $\mathbf J = \begin \dfrac & \cdots & \dfrac \end = \begin \nabla^\mathsf f_1 \\ \vdots \\ \nabla^\mathsf f_m \end = \begin \dfrac & \cdots & \dfrac\\ \vdots & \ddots & \vdots\\ \dfrac & \cdots & \dfrac \end.$

## Gradient of a vector field

Since the total derivative of a vector field is a
linear mapping In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
from vectors to vectors, it is a
tensor 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 ...

quantity. In rectangular coordinates, the gradient of a vector field is defined by: :$\nabla \mathbf=g^\frac \mathbf_i \otimes \mathbf_k,$ (where the
Einstein summation notation In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention that implies summation over a set of indexed terms in a formula, thus achieving notational br ...
is used and the
tensor product 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 ...

of the vectors and is a dyadic tensor of type (2,0)). Overall, this expression equals the transpose of the Jacobian matrix: :$\frac = \frac.$ In curvilinear coordinates, or more generally on a curved
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 surface. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of su ...
Christoffel symbols In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surface (topology), surfaces or other manifolds endowed with a metric ...
: :$\nabla \mathbf=g^\left\left(\frac+_f^l\right\right) \mathbf_i \otimes \mathbf_k,$ where are the components of the inverse
metric tensor 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 ...
and the are the coordinate basis vectors. Expressed more invariantly, the gradient of a vector field can be defined by the
Levi-Civita connection In Riemannian manifold, Riemannian or pseudo-Riemannian manifold, pseudo Riemannian geometry (in particular the Lorentzian manifold, Lorentzian geometry of General Relativity, general relativity), the Levi-Civita connection is the unique affine co ...
and metric tensor:. :$\nabla^a f^b = g^ \nabla_c f^b ,$ where is the connection.

## Riemannian manifolds

For any
smooth function In mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mat ...

on a Riemannian manifold , the gradient of is the vector field such that for any vector field , :$g\left(\nabla f, X\right) = \partial_X f,$ that is, :$g_x\big\left(\left(\nabla f\right)_x, X_x \big\right) = \left(\partial_X f\right) \left(x\right),$ where denotes the
inner product In mathematics, an inner product space or a Hausdorff space, Hausdorff pre-Hilbert space is a vector space with a binary operation called an inner product. This operation associates each pair of vectors in the space with a Scalar (mathematics), ...
of tangent vectors at defined by the metric and is the function that takes any point to the directional derivative of in the direction , evaluated at . In other words, in a
coordinate chartIn topology, a branch of mathematics, a topological manifold is a topological space which locally resembles real numbers, real ''n''-dimension (mathematics), dimensional Euclidean space. Topological manifolds are an important class of topological spa ...
from an open subset of to an open subset of , is given by: :$\sum_^n X^ \big\left(\varphi\left(x\right)\big\right) \frac\left(f \circ \varphi^\right) \Bigg, _,$ where denotes the th component of in this coordinate chart. So, the local form of the gradient takes the form: :$\nabla f = g^ \frac _i .$ Generalizing the case , the gradient of a function is related to its exterior derivative, since :$\left(\partial_X f\right) \left(x\right) = \left(df\right)_x\left(X_x\right) .$ More precisely, the gradient is the vector field associated to the differential 1-form using the
musical isomorphism 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 ...
:$\sharp=\sharp^g\colon T^*M\to TM$ (called "sharp") defined by the metric . The relation between the exterior derivative and the gradient of a function on is a special case of this in which the metric is the flat metric given by the dot product.

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Curl Curl or CURL may refer to: Science and technology * Curl (mathematics) In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the p ...
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Divergence In vector calculus Vector calculus, or vector analysis, is concerned with derivative, differentiation and integral, integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes ...

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Four-gradientIn differential geometry Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The Differential g ...
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Hessian matrix In mathematic Mathematics (from Greek: ) includes the study of such topics as quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in te ...

# References

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