The gradientMathworldPlanetmath is a first-order differential operatorMathworldPlanetmath that maps scalar functions to vector fields. It is a generalizationPlanetmathPlanetmath of the ordinary derivativePlanetmathPlanetmath, and as such conveys information about the rate of change of a function relative to small variations in the independent variables. The gradient of a function f is customarily denoted by f or by gradf.

1 Definition: Euclidean space

Let f:n be continuously differentiable. The gradient of f, denoted by f, is defined by the property:

D𝐯f=f𝐯for all vectors 𝐯n. (1)

The middle dot is the dot productMathworldPlanetmath, and D𝐯 is the directional derivativeMathworldPlanetmath with respect to 𝐯.

If x1,,xn are EuclideanPlanetmathPlanetmath coordinatesMathworldPlanetmathPlanetmath, corresponding to the orthonormal basis 𝐞1,,𝐞n, then

f=i=1nfxi𝐞i. (2)

The formulaMathworldPlanetmathPlanetmath (2) is sometimes given as the definition of f. We prefer to define f by the coordinate-free formula (1) instead, because then the geometric interpretationsMathworldPlanetmathPlanetmath (see below) become obvious, and (1) also indicates how we would go about calculating the gradient in other curvilinear coordinate systems. Formula (1) also makes it clear that the gradient is a physical vector, depending only on the inner product structure of n, and not on the specific coordinate systemMathworldPlanetmath used to calculate it.

There is the issue of whether the f as defined by (1) exists; but this is proved easily enough, by substituting the concrete expression (2) and seeing that it satisfies (1).

The gradient can be considered to be a vector-valued differential operator, written as


or, in the context of Euclidean 3-space, as


where 𝐢,𝐣,𝐤 are the unit vectorsMathworldPlanetmath lying along the positive direction of the x,y,z axes, respectively.

2 Geometric and physical interpretations

  1. (a)

    The direction of the vector f is the direction of the greatest positive change, or increase, in f. The magnitude of f is the magnitude of this increase. This follows immediately from (1):


    where θ is the angle between f and 𝐯. So among all unit directions 𝐯 of change, if 𝐯 is perpendicularMathworldPlanetmathPlanetmathPlanetmathPlanetmath to f then the change D𝐯f is zero; if 𝐯 is parallelMathworldPlanetmathPlanetmath to f then the change is maximized.

    Similarly, -f is the direction of the greatest negative change, or decrease, in f.

  2. (b)

    If M is the hypersurface in n defined by


    then f(p) is the normal to the hypersurface M at the point p. For kerDf(p) is the tangent spacePlanetmathPlanetmath TpM to M at p, that is, D𝐯f(p)=0 for all 𝐯TpM, and by definition (1), f(p) must be perpendicular to all 𝐯TpM.

    Note that Df0 is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to f0. Consequently, f also gives an orientation to the hypersurface M.

    For example, if f(𝐱)=𝐱-1 for 𝐱n, M is the (n-1)-dimensional sphere of unit radius, embedded in n. Its normal, f(𝐱)=𝐱/𝐱, as one would expect, points outward radially.

  3. (c)

    As a simple case of (b), consider the surface z=f(x,y) in 3, with Cartesian coordinatesMathworldPlanetmath (x,y,z). Think of this surface as describing a hill, with height z. Then the direction of the gradient vector f is the direction of steepest ascent of the hill, while its magnitude


    is the slope or steepness in that direction.

    If a ball is placed on the hill at a point (x,y,z), theoretically it should roll down the hill in the direction of the gradient vector -f(x,y). This may be easily derived by considering the mechanical forces on the ball. The direction of -f(x,y) is, in fact, the projection to the xy-plane of an outward normal vector to the hill at (x,y,z); the normal vector is involved because the movement of the ball arises from the normal force from the hill.

  4. (d)

    Suppose the surface z=f(x,y) in (c) describes a bowl instead of a hill, and we place a marble at any point (x,y,z) on this bowl. We would expect the marble to roll down to a local minimumMathworldPlanetmath point of f(x,y). Since the marble should roll down in the direction of -f, we might hope that we can find local minima of a given function f by following the path mapped out by the gradients -f. Formally, this method of finding local extrema (with some modifications) is called gradient descent.

  5. (e)

    If U is the potential function corresponding to a conservative physical force, then 𝐅=-U is the corresponding force field.

    Consequently, the gradient theoremMathworldPlanetmath,


    simply gives the formula for the change in the potential energy U when an object “does work” along a path γ in a conservative force field 𝐅.

3 Definition: Riemannian geometry

It is obvious how (1) can be generalized to the setting of Riemannian manifoldsMathworldPlanetmath: the dot product of n must be replaced by the Riemannian metric, and the analogue of D𝐯f is the directional derivative 𝐯[f], for tangent vectors 𝐯 on the Riemannian manifold. Thus for a smooth scalar-valued function f on a Riemannian manifold,

𝐗=gradfdfp(𝐯)=𝐯[f]=𝐗,𝐯p. (3)

We can calculate 𝐗 explicitly as follows. If xi are local coordinates on the manifold (not necessarily orthonormal), set 𝐗=Xixi (the Einstein summation convention is being used). Let gij and gij be the covariant and contravariant metric tensors, respectively. Then from (3),


and taking inversesMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath,

Xi=gijfxj. (4)

4 Duality with differential one-forms

Definitions (1) and (3) exhibit f as the vector field dual to the differential formMathworldPlanetmath df. The isomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath is given by applying the inner product or Riemannian metric. This isomorphism is, of course, linear; in particular it leads to the identity

f=fxixi, (5)

which is the dual to the standard formula of differentialMathworldPlanetmath one-forms:


Using (3) and (4), we have

xi=gijxj,xi=gijxj. (6)

So the isomorphism between vector fields and one-forms is expressed by changing the ’s in (6) to d’s, and vice versa. That is,

dxigijxj,xigijdxj. (7)

It is commonly said that this isomorphism is expressed by “raising and lowering the indices of a tensor field, using contractionsPlanetmathPlanetmath with gij and gij”.

Notice that when xi are orthonormal coordinates on n, equation (5) reduces to equation (2), because gij=𝐞i𝐞j=δij (Kronecker deltaMathworldPlanetmath).

The formulae presented in this sectionMathworldPlanetmathPlanetmath are useful in the Euclidean setting as well, for deriving the formulae for the gradient in various curvilinear coordinate systems (

5 Differential identities

Several properties of the one-dimensional derivative generalize to a multi-dimensional setting

(af+bg) =af+bg Linearity
(fg) =fg+gf Product ruleMathworldPlanetmath
(fϕ)(p) =Dϕ(p)*f(ϕ(p)) Chain ruleMathworldPlanetmath
(hf)(p) =h(f(p))f(p) Another Chain rule

The function h is h:. The notation (Dϕ)* denotes the transposeMathworldPlanetmath of the Jacobian matrix, in Euclidean coordinates, of ϕ:mn. In the abstract setting, (Dϕ)* is the adjointPlanetmathPlanetmath to the tangent map Dϕ between the tangent bundles of two Riemannian manifolds.

These identities can be proved directly from the definition, but the first three are really just the duals of the following well-known identities for differential forms:

d(af+bg) =adf+bdg
d(fg) =fdg+gdf
d(ϕ*f) =ϕ*df

and so may be derived by changing the d’s here to ’s! (Though the third identity may take a bit of thought.)

The following identity


is a special case of the differential forms identity d2=0. Conversely, if curlg=0 on a simply connected domain, then there is f such that g=gradf. See laminar field for details.

6 The symbolism

(This discussion does not really belong here, but should be moved to the nabla entry.)

Using the formalism, the divergenceMathworldPlanetmath operator can be expressed as , the curl operator as ×, and the Laplacian operator as 2. To wit, for a given vector field


and a given function f we have

𝐀 =Axx+Ayy+Azz
×𝐀 =(Azy-Ayz)𝐢+(Axz-Azx)𝐣+(Ayx-Axy)𝐤
2f =2fx2+2fy2+2fz2.


Title gradient
Canonical name Gradient
Date of creation 2013-03-22 11:59:20
Last modified on 2013-03-22 11:59:20
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 20
Author CWoo (3771)
Entry type Definition
Classification msc 58A10
Classification msc 26B12
Classification msc 26B10
Related topic Derivative2
Related topic RiemannianMetric
Related topic GradientInCurvilinearCoordinates
Related topic NablaNabla
Related topic DifferentialForms
Related topic FirstOrderOperatorsInRiemannianGeometry
Related topic VectorField
Related topic HessianMatrix
Related topic DerivativeNotation
Related topic JacobianMatrix
Related topic PartialDerivative
Related topic TiltCurve
Defines gradient