Gradient

In the three-dimensional Cartesian coordinate system with a Euclidean metric, the gradient, if it exists, is given by:

${\displaystyle \nabla f={\frac {\partial f}{\partial x}}\mathbf {i} +{\frac {\partial f}{\partial y}}\mathbf {j} +{\frac {\partial f}{\partial z}}\mathbf {k} ,}$

where i, j, k are the standard unit vectors in the directions of the x, y and z coordinates, respectively. For example, the gradient of the function

${\displaystyle f(x,y,z)=2x+3y^{2}-\sin(z)}$

is

${\displaystyle \nabla f=2\mathbf {i} +6y\mathbf {j} -\cos(z)\mathbf {k} .}$

In some applications it is customary to represent the gradient as a row vector or column vector 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.

In cylindrical coordinates with a Euclidean metric, the gradient is given by:[5]

${\displaystyle \nabla f(\rho ,\varphi ,z)={\frac {\partial f}{\partial \rho }}\mathbf {e} _{\rho }+{\frac {1}{\rho }}{\frac {\partial f}{\partial \varphi }}\mathbf {e} _{\varphi }+{\frac {\partial f}{\partial z}}\mathbf {e} _{z},}$

where ρ is the axial distance, φ is the azimuthal or azimuth angle, z is the axial coordinate, and e_ρ, e_φ and e_z are unit vectors pointing along the coordinate directions.

In spherical coordinates, the gradient is given by:[5]

${\displaystyle \nabla f(r,\theta ,\varphi )={\frac {\partial f}{\partial r}}\mathbf {e} _{r}+{\frac {1}{r}}{\frac {\partial f}{\partial \theta }}\mathbf {e} _{\theta }+{\frac {1}{r\sin \theta }}{\frac {\partial f}{\partial \varphi }}\mathbf {e} _{\varphi },}$

where r is the radial distance, φ is the azimuthal angle and θ is the polar angle, and e_r, e_θ and e_φ are again local unit vectors pointing in the coordinate directions (that is, the normalized covariant basis).

For the gradient in other orthogonal coordinate systems, see Orthogonal coordinates (Differential operators in three dimensions).

We consider general coordinates, which we write as x¹, …, xⁱ, …, xⁿ, where n 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 x² refers to the second component—not the quantity x squared. The index variable i refers to an arbitrary element xⁱ. Using Einstein notation, the gradient can then be written as:

$${\displaystyle \nabla f={\frac {\partial f}{\partial x^{i}}}g^{ij}\mathbf {e} _{j}}$$

(Note that its dual is ${\textstyle \mathrm {d} f={\frac {\partial f}{\partial x^{i}}}\mathbf {e} ^{i}}$),

where ${\displaystyle \mathbf {e} _{i}=\partial \mathbf {x} /\partial x^{i}}$ and ${\displaystyle \mathbf {e} ^{i}=\mathrm {d} x^{i}}$ refer to the unnormalized local covariant and contravariant bases respectively, ${\displaystyle g^{ij}}$ 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 ${\displaystyle {\hat {\mathbf {e} }}_{i}}$ and ${\displaystyle {\hat {\mathbf {e} }}^{i}}$, using the scale factors (also known as Lamé coefficients) ${\displaystyle h_{i}=\lVert \mathbf {e} _{i}\rVert ={\sqrt {g_{ii}}}=1\,/\lVert \mathbf {e} ^{i}\rVert }$ :

$${\displaystyle \nabla f={\frac {\partial f}{\partial x^{i}}}g^{ij}{\hat {\mathbf {e} }}_{j}{\sqrt {g_{jj}}}=\sum _{i=1}^{n}\,{\frac {\partial f}{\partial x^{i}}}{\frac {1}{h_{i}}}\mathbf {\hat {e}} _{i}}$$

(and ${\textstyle \mathrm {d} f=\sum _{i=1}^{n}\,{\frac {\partial f}{\partial x^{i}}}{\frac {1}{h_{i}}}\mathbf {\hat {e}} ^{i}}$),

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, ${\displaystyle \mathbf {\hat {e}} _{i}}$, ${\displaystyle \mathbf {\hat {e}} ^{i}}$, and ${\displaystyle h_{i}}$ are neither contravariant nor covariant.

The latter expression evaluates to the expressions given above for cylindrical and spherical coordinates.

The gradient is closely related to the total derivative (total differential) ${\displaystyle df}$: they are transpose (dual) to each other. Using the convention that vectors in ${\displaystyle \mathbb {R} ^{n}}$ are represented by column vectors, and that covectors (linear maps ${\displaystyle \mathbb {R} ^{n}\to \mathbb {R} }$) are represented by row vectors,[lower-alpha 1] the gradient ${\displaystyle \nabla f}$ and the derivative ${\displaystyle df}$ are expressed as a column and row vector, respectively, with the same components, but transpose of each other:

${\displaystyle \nabla f(p)={\begin{bmatrix}{\frac {\partial f}{\partial x_{1}}}(p)\\\vdots \\{\frac {\partial f}{\partial x_{n}}}(p)\end{bmatrix}};}$

${\displaystyle df_{p}={\begin{bmatrix}{\frac {\partial f}{\partial x_{1}}}(p)&\cdots &{\frac {\partial f}{\partial x_{n}}}(p)\end{bmatrix}}.}$

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, a linear form (covector) 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, which represents an infinitesimal change in (vector) input. In symbols, the gradient is an element of the tangent space at a point, ${\displaystyle \nabla f(p)\in T_{p}\mathbb {R} ^{n}}$, while the derivative is a map from the tangent space to the real numbers, ${\displaystyle df_{p}\colon T_{p}\mathbb {R} ^{n}\to \mathbb {R} }$. The tangent spaces at each point of ${\displaystyle \mathbb {R} ^{n}}$ can be "naturally" identified[lower-alpha 4] with the vector space ${\displaystyle \mathbb {R} ^{n}}$ itself, and similarly the cotangent space at each point can be naturally identified with the dual vector space ${\displaystyle (\mathbb {R} ^{n})^{*}}$ of covectors; thus the value of the gradient at a point can be thought of a vector in the original ${\displaystyle \mathbb {R} ^{n}}$, 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 with the gradient:

${\displaystyle (df_{p})(v)={\begin{bmatrix}{\frac {\partial f}{\partial x_{1}}}(p)&\cdots &{\frac {\partial f}{\partial x_{n}}}(p)\end{bmatrix}}{\begin{bmatrix}v_{1}\\\vdots \\v_{n}\end{bmatrix}}=\sum _{i=1}^{n}{\frac {\partial f}{\partial x_{i}}}(p)v_{i}={\begin{bmatrix}{\frac {\partial f}{\partial x_{1}}}(p)\\\vdots \\{\frac {\partial f}{\partial x_{n}}}(p)\end{bmatrix}}\cdot {\begin{bmatrix}v_{1}\\\vdots \\v_{n}\end{bmatrix}}=\nabla f(p)\cdot v}$

The best linear approximation to a differentiable function

${\displaystyle f\colon \mathbb {R} ^{n}\to \mathbb {R} }$

at a point x in Rⁿ is a linear map from Rⁿ to R which is often denoted by df_x or Df(x) and called the differential or total derivative of f at x. The function df, which maps x to df_x, is called the total differential or exterior derivative of f and is an example of a differential 1-form.

Much as the derivative of a function of a single variable represents the slope of the tangent to the graph of the function,[6] the directional derivative of a function in several variables represents the slope of the tangent hyperplane in the direction of the vector.

The gradient is related to the differential by the formula

${\displaystyle (\nabla f)_{x}\cdot v=df_{x}(v)}$

for any v ∈ Rⁿ, where ${\displaystyle \cdot }$ is the dot product: taking the dot product of a vector with the gradient is the same as taking the directional derivative along the vector.

If Rⁿ is viewed as the space of (dimension n) column vectors (of real numbers), then one can regard df as the row vector with components

${\displaystyle \left({\frac {\partial f}{\partial x_{1}}},\dots ,{\frac {\partial f}{\partial x_{n}}}\right),}$

so that df_x(v) is given by matrix multiplication. Assuming the standard Euclidean metric on Rⁿ, the gradient is then the corresponding column vector, that is,

${\displaystyle (\nabla f)_{i}=df_{i}^{\mathsf {T}}.}$

The best linear approximation to a function can be expressed in terms of the gradient, rather than the derivative. The gradient of a function f from the Euclidean space Rⁿ to R at any particular point x₀ in Rⁿ characterizes the best linear approximation to f at x₀. The approximation is as follows:

${\displaystyle f(x)\approx f(x_{0})+(\nabla f)_{x_{0}}\cdot (x-x_{0})}$

for x close to x₀, where (∇f )_x0 is the gradient of f computed at x₀, and the dot denotes the dot product on Rⁿ. This equation is equivalent to the first two terms in the multivariable Taylor series expansion of f at x₀.

Let U be an open set in Rⁿ. If the function f : U → R is differentiable, then the differential of f is the Fréchet derivative of f. Thus ∇f is a function from U to the space Rⁿ such that

$${\displaystyle \lim _{h\to 0}{\frac {|f(x+h)-f(x)-\nabla f(x)\cdot h|}{\|h\|}}=0,}$$

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

The gradient is linear in the sense that if f and g are two real-valued functions differentiable at the point a ∈ Rⁿ, and α and β are two constants, then αf + βg is differentiable at a, and moreover

$${\displaystyle \nabla \left(\alpha f+\beta g\right)(a)=\alpha \nabla f(a)+\beta \nabla g(a).}$$

Product rule

If f and g are real-valued functions differentiable at a point a ∈ Rⁿ, then the product rule asserts that the product fg is differentiable at a, and

$${\displaystyle \nabla (fg)(a)=f(a)\nabla g(a)+g(a)\nabla f(a).}$$

Chain rule

Suppose that f : A → R is a real-valued function defined on a subset A of Rⁿ, and that f is differentiable at a point a. There are two forms of the chain rule applying to the gradient. First, suppose that the function g is a parametric curve; that is, a function g : I → Rⁿ maps a subset I ⊂ R into Rⁿ. If g is differentiable at a point c ∈ I such that g(c) = a, then

$${\displaystyle (f\circ g)'(c)=\nabla f(a)\cdot g'(c),}$$

where ∘ is the composition operator: (f ∘ g)(x) = f(g(x)).

More generally, if instead I ⊂ R^k, then the following holds:

$${\displaystyle \nabla (f\circ g)(c)={\big (}Dg(c){\big )}^{\mathsf {T}}{\big (}\nabla f(a){\big )},}$$

where (Dg)^T denotes the transpose Jacobian matrix.

For the second form of the chain rule, suppose that h : I → R is a real valued function on a subset I of R, and that h is differentiable at the point f(a) ∈ I. Then

$${\displaystyle \nabla (h\circ f)(a)=h'{\big (}f(a){\big )}\nabla f(a).}$$

A level surface, or isosurface, is the set of all points where some function has a given value.

If f is differentiable, then the dot product (∇f )_x ⋅ v of the gradient at a point x with a vector v gives the directional derivative of f at x in the direction v. It follows that in this case the gradient of f is orthogonal to the level sets of f. For example, a level surface in three-dimensional space is defined by an equation of the form F(x, y, z) = c. The gradient of F is then normal to the surface.

More generally, any embedded hypersurface in a Riemannian manifold can be cut out by an equation of the form F(P) = 0 such that dF is nowhere zero. The gradient of F is then normal to the hypersurface.

Similarly, an affine algebraic hypersurface may be defined by an equation F(x₁, ..., x_n) = 0, where F is a polynomial. The gradient of F 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.

The gradient of a function is called a gradient field. A (continuous) gradient field is always a conservative vector field: its line integral 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.

The Jacobian matrix is the generalization of the gradient for vector-valued functions of several variables and differentiable maps between Euclidean spaces or, more generally, manifolds.[7][8] A further generalization for a function between Banach spaces is the Fréchet derivative.

Suppose f : Rⁿ → R^m is a function such that each of its first-order partial derivatives exist on ℝⁿ. Then the Jacobian matrix of f is defined to be an m×n matrix, denoted by ${\displaystyle \mathbf {J} _{\mathbb {f} }(\mathbb {x} )}$ or simply ${\displaystyle \mathbf {J} }$. The (i,j)th entry is ${\displaystyle \mathbf {J} _{ij}={\frac {\partial f_{i}}{\partial x_{j}}}}$. Explicitly

$${\displaystyle \mathbf {J} ={\begin{bmatrix}{\dfrac {\partial \mathbf {f} }{\partial x_{1}}}&\cdots &{\dfrac {\partial \mathbf {f} }{\partial x_{n}}}\end{bmatrix}}={\begin{bmatrix}\nabla ^{\mathsf {T}}f_{1}\\\vdots \\\nabla ^{\mathsf {T}}f_{m}\end{bmatrix}}={\begin{bmatrix}{\dfrac {\partial f_{1}}{\partial x_{1}}}&\cdots &{\dfrac {\partial f_{1}}{\partial x_{n}}}\\\vdots &\ddots &\vdots \\{\dfrac {\partial f_{m}}{\partial x_{1}}}&\cdots &{\dfrac {\partial f_{m}}{\partial x_{n}}}\end{bmatrix}}.}$$

Since the total derivative of a vector field is a linear mapping from vectors to vectors, it is a tensor quantity.

In rectangular coordinates, the gradient of a vector field f = ( f¹, f², f³) is defined by:

${\displaystyle \nabla \mathbf {f} =g^{jk}{\frac {\partial f^{i}}{\partial x^{j}}}\mathbf {e} _{i}\otimes \mathbf {e} _{k},}$

(where the Einstein summation notation is used and the tensor product of the vectors e_i and e_k is a dyadic tensor of type (2,0)). Overall, this expression equals the transpose of the Jacobian matrix:

${\displaystyle {\frac {\partial f^{i}}{\partial x^{j}}}={\frac {\partial (f^{1},f^{2},f^{3})}{\partial (x^{1},x^{2},x^{3})}}.}$

In curvilinear coordinates, or more generally on a curved manifold, the gradient involves Christoffel symbols:

${\displaystyle \nabla \mathbf {f} =g^{jk}\left({\frac {\partial f^{i}}{\partial x^{j}}}+{\Gamma ^{i}}_{jl}f^{l}\right)\mathbf {e} _{i}\otimes \mathbf {e} _{k},}$

where g^jk are the components of the inverse metric tensor and the e_i are the coordinate basis vectors.

Expressed more invariantly, the gradient of a vector field f can be defined by the Levi-Civita connection and metric tensor:[9]

${\displaystyle \nabla ^{a}f^{b}=g^{ac}\nabla _{c}f^{b},}$

where ∇_c is the connection.

For any smooth function f on a Riemannian manifold (M, g), the gradient of f is the vector field ∇f such that for any vector field X,

${\displaystyle g(\nabla f,X)=\partial _{X}f,}$

that is,

${\displaystyle g_{x}{\big (}(\nabla f)_{x},X_{x}{\big )}=(\partial _{X}f)(x),}$

where g_x( , ) denotes the inner product of tangent vectors at x defined by the metric g and ∂_X f is the function that takes any point x ∈ M to the directional derivative of f in the direction X, evaluated at x. In other words, in a coordinate chart φ from an open subset of M to an open subset of Rⁿ, (∂_X f )(x) is given by:

${\displaystyle \sum _{j=1}^{n}X^{j}{\big (}\varphi (x){\big )}{\frac {\partial }{\partial x_{j}}}(f\circ \varphi ^{-1}){\Bigg |}_{\varphi (x)},}$

where X^j denotes the jth component of X in this coordinate chart.

So, the local form of the gradient takes the form:

${\displaystyle \nabla f=g^{ik}{\frac {\partial f}{\partial x^{k}}}{\textbf {e}}_{i}.}$

Generalizing the case M = Rⁿ, the gradient of a function is related to its exterior derivative, since

${\displaystyle (\partial _{X}f)(x)=(df)_{x}(X_{x}).}$

More precisely, the gradient ∇f is the vector field associated to the differential 1-form df using the musical isomorphism

${\displaystyle \sharp =\sharp ^{g}\colon T^{*}M\to TM}$

(called "sharp") defined by the metric g. The relation between the exterior derivative and the gradient of a function on Rⁿ is a special case of this in which the metric is the flat metric given by the dot product.