This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations. Please improve this article by introducing more precise citations. (December 2011)

In the above two images, the scalar field is in black and white, black representing higher values, and its corresponding gradient is represented by blue arrows.

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

A generalization of the gradient for functions on a Euclidean space that have values in another Euclidean space is the Jacobian. A further generalization for a function from one Banach space to another is the Fréchet derivative.

Interpretations[link]

Gradient of the 2-d function Failed to parse (Missing texvc executable; please see math/README to configure.): f(x,y)= xe^{-x^2 - y^2} is plotted as blue arrows over the pseudocolor plot of the function

Consider a room in which the temperature is given by a scalar field, Failed to parse (Missing texvc executable; please see math/README to configure.): T , so at each point Failed to parse (Missing texvc executable; please see math/README to configure.): (x,y,z)

the temperature is Failed to parse (Missing texvc executable; please see math/README to configure.): T(x,y,z) . (We will assume that the temperature does not change over time.) At each point in the room, the gradient of Failed to parse (Missing texvc executable; please see math/README to configure.): T

at that point will show the direction the temperature rises most quickly. The magnitude of the gradient will determine how fast the temperature rises in that direction.

Consider a surface whose height above sea level at a point Failed to parse (Missing texvc executable; please see math/README to configure.): (x, y)

is Failed to parse (Missing texvc executable; please see math/README to configure.): H(x, y)

. The gradient of Failed to parse (Missing texvc executable; please see math/README to configure.): H

at a point is a vector pointing in the direction of the steepest slope or grade 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. Suppose that the steepest slope on a hill is 40%. If a road goes directly up the hill, then the steepest slope on the road will also be 40%. If, instead, the road goes around the hill at an angle, then it will have a shallower slope. For example, if the angle between the road and the uphill direction, projected onto the horizontal plane, is 60°, then the steepest slope along the road will be 20%, which is 40% times the cosine of 60°.

This observation can be mathematically stated as follows. If the hill height function Failed to parse (Missing texvc executable; please see math/README to configure.): H

is differentiable, then the gradient of Failed to parse (Missing texvc executable; please see math/README to configure.): H
dotted with a unit vector gives the slope of the hill in the direction of the vector. More precisely, when Failed to parse (Missing texvc executable; please see math/README to configure.): H
is differentiable, the dot product of the gradient of Failed to parse (Missing texvc executable; please see math/README to configure.): H
with a given unit vector is equal to the directional derivative of Failed to parse (Missing texvc executable; please see math/README to configure.): H
in the direction of that unit vector.

Definition[link]

The gradient of the function f(x,y) = −(cos²x + cos²y)² depicted as a projected vector field on the bottom plane

The gradient (or gradient vector field) of a scalar function Failed to parse (Missing texvc executable; please see math/README to configure.): f(x_1, x_2, x_3, \dots, x_n)

is denoted Failed to parse (Missing texvc executable; please see math/README to configure.): \nabla f
or Failed to parse (Missing texvc executable; please see math/README to configure.): \vec{\nabla} f
where Failed to parse (Missing texvc executable; please see math/README to configure.): \nabla
(the nabla symbol) denotes the vector differential operator, del. The notation Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{grad}(f)
is also commonly used for the gradient. The gradient of f is defined as the unique vector field whose dot product with any unit vector v at each point x is the directional derivative of f along v.  That is,

Failed to parse (Missing texvc executable; please see math/README to configure.): (\nabla f(x))\cdot \mathbf{v} = D_{\mathbf v}f(x).

In a rectangular coordinate system, the gradient is the vector field whose components are the partial derivatives of f:

Failed to parse (Missing texvc executable; please see math/README to configure.): \nabla f = \frac{\partial f}{\partial x_1 }\mathbf{e}_1 + \cdots + \frac{\partial f}{\partial x_n }\mathbf{e}_n

where the e_i are the orthogonal unit vectors pointing in the coordinate directions. When a function also depends on a parameter such as time, the gradient often refers simply to the vector of its spatial derivatives only.

In the three-dimensional Cartesian coordinate system, this is given by

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\partial f}{\partial x} \mathbf{i} + \frac{\partial f}{\partial y} \mathbf{j} + \frac{\partial f}{\partial z} \mathbf{k}

where Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf i,\mathbf j, \mathbf k

are the standard unit vectors. For example, the gradient of the function

Failed to parse (Missing texvc executable; please see math/README to configure.): f(x,y,z)= \ 2x+3y^2-\sin(z)

is:

Failed to parse (Missing texvc executable; please see math/README to configure.): \nabla f= \frac{\partial f}{\partial x} \mathbf{i} + \frac{\partial f}{\partial y} \mathbf{j} + \frac{\partial f}{\partial z} \mathbf{k} = 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.

Gradient and the derivative or differential[link]

Linear approximation to a function[link]

The gradient of a function Failed to parse (Missing texvc executable; please see math/README to configure.): f

from the Euclidean space Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{R}^n
to Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{R}
at any particular point x0 in Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{R}^n
characterizes the best linear approximation to f at x0. The approximation is as follows:

Failed to parse (Missing texvc executable; please see math/README to configure.): f(x) \approx f(x_0) + (\nabla f)_{x_0}\cdot(x-x_0)

for Failed to parse (Missing texvc executable; please see math/README to configure.): x

close to Failed to parse (Missing texvc executable; please see math/README to configure.): x_0

, where Failed to parse (Missing texvc executable; please see math/README to configure.): (\nabla f)_{x_0}

is the gradient of f computed at Failed to parse (Missing texvc executable; please see math/README to configure.): x_0

, and the dot denotes the dot product on Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{R}^n . This equation is equivalent to the first two terms in the multi-variable Taylor Series expansion of f at x₀.

Differential or (exterior) derivative[link]

The best linear approximation to a function Failed to parse (Missing texvc executable; please see math/README to configure.): f: \mathbb{R}^n \to \mathbb{R}

at a point Failed to parse (Missing texvc executable; please see math/README to configure.): x
in Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{R}^n
is a linear map from Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{R}^n
to Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{R}
which is often denoted by Failed to parse (Missing texvc executable; please see math/README to configure.): \mathrm{d}f_x
or Failed to parse (Missing texvc executable; please see math/README to configure.): Df(x)
and called the differential or (total) derivative of Failed to parse (Missing texvc executable; please see math/README to configure.): f
at Failed to parse (Missing texvc executable; please see math/README to configure.): x

. The gradient is therefore related to the differential by the formula Failed to parse (Missing texvc executable; please see math/README to configure.): (\nabla f)_x\cdot v = \mathrm d f_x(v)

for any Failed to parse (Missing texvc executable; please see math/README to configure.): v \in \mathbb{R}^n . The function Failed to parse (Missing texvc executable; please see math/README to configure.): \mathrm{d}f , which maps Failed to parse (Missing texvc executable; please see math/README to configure.): x

to Failed to parse (Missing texvc executable; please see math/README to configure.): \mathrm{d}f_x

, is called the differential or exterior derivative of Failed to parse (Missing texvc executable; please see math/README to configure.): f

and is an example of a differential 1-form. 

If Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{R}^n

is viewed as the space of (length Failed to parse (Missing texvc executable; please see math/README to configure.): n

) column vectors (of real numbers), then one can regard Failed to parse (Missing texvc executable; please see math/README to configure.): \mathrm{d}f

as the row vector

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathrm{d}f = \left( \frac{\partial f}{\partial x_1}, \dots, \frac{\partial f}{\partial x_n}\right)

so that Failed to parse (Missing texvc executable; please see math/README to configure.): \mathrm{d}f_x(v)

is given by matrix multiplication. The gradient is then the corresponding column vector, i.e., Failed to parse (Missing texvc executable; please see math/README to configure.): \nabla f = \mathrm{d} f^T

.

Gradient as a derivative[link]

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 Failed to parse (Missing texvc executable; please see math/README to configure.): \nabla f

is a function from U to the space R such that

Failed to parse (Missing texvc executable; please see math/README to configure.): \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:

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

Failed to parse (Missing texvc executable; please see math/README to configure.): \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)(x) = f(x)g(x) of the functions f and g is differentiable at a, and

Failed to parse (Missing texvc executable; please see math/README to configure.): \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

Failed to parse (Missing texvc executable; please see math/README to configure.): (f\circ g)'(c) = \nabla f(a)\cdot g'(c),

where Failed to parse (Missing texvc executable; please see math/README to configure.): \circ

is the composition operator.

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

Failed to parse (Missing texvc executable; please see math/README to configure.): \nabla (f\circ g)(c) = (\nabla f(a)) (Dg(c))^T

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 c = f(a) ∈ I. Then

Failed to parse (Missing texvc executable; please see math/README to configure.): \nabla (h\circ f)(a) = h'(c)\nabla f(a).

Further properties and applications[link]

Level sets[link]

If the partial derivatives of f are continuous, then the dot product Failed to parse (Missing texvc executable; please see math/README to configure.): (\nabla f)_x\cdot 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.

Let us consider a function f at a point P. If we draw a surface through this point P and the function has the same value at all points on this surface,then this surface is called a 'level surface'.

Conservative vector fields and the gradient theorem[link]

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.

Riemannian manifolds[link]

For any smooth function f on a Riemannian manifold (M,g), the gradient of f is the vector field Failed to parse (Missing texvc executable; please see math/README to configure.): \nabla f

such that for any vector field Failed to parse (Missing texvc executable; please see math/README to configure.): X

,

Failed to parse (Missing texvc executable; please see math/README to configure.): g(\nabla f, X) = \partial_X f, \qquad \text{i.e.,}\quad g_x((\nabla f)_x, X_x ) = (\partial_X f) (x)

where Failed to parse (Missing texvc executable; please see math/README to configure.): g_x( \cdot, \cdot )

denotes the inner product of tangent vectors at x defined by the metric g and

Failed to parse (Missing texvc executable; please see math/README to configure.): \partial_X f

(sometimes denoted 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 Failed to parse (Missing texvc executable; please see math/README to configure.): \varphi
from an open subset of M to an open subset of Rn, Failed to parse (Missing texvc executable; please see math/README to configure.): (\partial_X f)(x)
is given by:

Failed to parse (Missing texvc executable; please see math/README to configure.): \sum_{j=1}^n X^{j} (\varphi(x)) \frac{\partial}{\partial x_{j}}(f \circ \varphi^{-1}) \Big|_{\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:

Failed to parse (Missing texvc executable; please see math/README to configure.): \nabla f= g^{ik}\frac{\partial f}{\partial x^{k}}\frac{\partial}{\partial x^{i}}.

Generalizing the case M=Rⁿ, the gradient of a function is related to its exterior derivative, since Failed to parse (Missing texvc executable; please see math/README to configure.): (\partial_X f) (x) = df_x(X_x) . More precisely, the gradient Failed to parse (Missing texvc executable; please see math/README to configure.): \nabla f

is the vector field associated to the differential 1-form df using the musical isomorphism Failed to parse (Missing texvc executable; please see math/README to configure.): \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.

Cylindrical and spherical coordinates[link]

In cylindrical coordinates, the gradient is given by (Schey 1992, pp. 139–142):

Failed to parse (Missing texvc executable; please see math/README to configure.): \nabla f(\rho, \phi, z) = \frac{\partial f}{\partial \rho}\mathbf{e}_\rho+ \frac{1}{\rho}\frac{\partial f}{\partial \phi}\mathbf{e}_\phi+ \frac{\partial f}{\partial z}\mathbf{e}_z

where Failed to parse (Missing texvc executable; please see math/README to configure.): \phi

is the azimuthal angle, Failed to parse (Missing texvc executable; please see math/README to configure.): z
is the axial coordinate, and eρ, eφ and ez are unit vectors pointing along the coordinate directions.

In spherical coordinates (Schey 1992, pp. 139–142):

Failed to parse (Missing texvc executable; please see math/README to configure.): \nabla f(r, \theta, \phi) = \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 \phi}\mathbf{e}_\phi

where Failed to parse (Missing texvc executable; please see math/README to configure.): \phi

is the azimuth angle and Failed to parse (Missing texvc executable; please see math/README to configure.): \theta
is the zenith angle.

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

Gradient of a vector[link]

In rectangular coordinates, the gradient of a vector Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{f}=({{f}_{1}},{{f}_{2}},{{f}_{3}})

is defined by

Failed to parse (Missing texvc executable; please see math/README to configure.): \nabla \mathbf{f}=\frac{\partial {{f}_{i}}}{\partial {{x}_{j}}}{{\mathbf{e}}_{i}}{{\mathbf{e}}_{j}}

or the Jacobian matrix Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\partial ({{f}_{1}},{{f}_{2}},{{f}_{3}})}{\partial ({{x}_{1}},{{x}_{2}},{{x}_{3}})} .

In curvilinear coordinates, the gradient involves Christoffel symbols.

See also[link]

• Gradient theorem
• Del
• Divergence
• Curl
• Skew gradient

References[link]

• Korn, Theresa M.; Korn, Granino Arthur (2000), Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review, New York: Dover Publications, pp. 157–160, ISBN 0-486-41147-8, OCLC 43864234 .

• Schey, H.M. (1992), Div, Grad, Curl, and All That (2nd ed.), W.W. Norton, ISBN 0-393-96251-2, OCLC 25048561 .

• Dubrovin, B.A.; A.T. Fomenko, S.P. Novikov (1991), Modern Geometry--Methods and Applications: Part I: The Geometry of Surfaces, Transformation Groups, and Fields (Graduate Texts in Mathematics) (2nd ed.), Springer, pp. 14–17, ISBN 978-0-387-97663-1 

External links[link]

Look up gradient in Wiktionary, the free dictionary.

• Khan Academy Gradient lesson 1
• Kuptsov, L.P. (2001), "Gradient", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4, http://www.encyclopediaofmath.org/index.php?title=G/g044680 
• Weisstein, Eric W., "Gradient" from MathWorld.