gradient (original) (raw)

Summary.

The gradient is a first-order differential operator that maps scalar functions to vector fields. It is a generalization of the ordinaryderivative, 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 grad⁡f.

Contents:

• 1 Definition: Euclidean space
• 2 Geometric and physical interpretations
• 3 Definition: Riemannian geometry
• 4 Duality with differential one-forms
• 5 Differential identities
• 6 The ∇ symbolism

1 Definition: Euclidean space

If x1,…,xn are Euclidean coordinates, corresponding to the orthonormal basis 𝐞1,…,𝐞n, then

The formula (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 interpretations (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 ℝn, and not on the specific coordinate system 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

where𝐢,𝐣,𝐤 are the unit vectors lying along the positivedirection of the x,y,z axes, respectively.

2 Geometric and physical interpretations

1. (a)
   Similarly,-∇⁡f is the direction of the greatest negative change, or decrease, in f.
2. (b)
   If M is the hypersurface in ℝn defined by

   M={p∈ℝn:f⁢(p)=0,D⁡f⁢(p)≠0},
   then ∇⁡f⁢(p) is the normal to the hypersurface M at the point p. For ker⁡D⁡f⁢(p) is the tangent space Tp⁢Mto M at p, that is, D𝐯⁡f⁢(p)=0 for all 𝐯∈Tp⁢M, and by definition (1), ∇⁡f⁢(p) must be perpendicular to all 𝐯∈Tp⁢M.
   Note that D⁡f≠0 is equivalent to ∇⁡f≠0. Consequently, ∇⁡f also gives an orientation to the hypersurface M.
   For example, if f⁢(𝐱)=∥𝐱∥-1 for 𝐱∈ℝn,M is the (n-1)-dimensional sphere of unit radius, embedded in ℝn. Its normal, ∇⁡f⁢(𝐱)=𝐱/∥𝐱∥, as one would expect, points outward radially.

3. (c)
   As a simple case of (b), consider thesurface z=f⁢(x,y) in ℝ3, with Cartesian coordinates (x,y,z). Think of this surface as describing a hill, with height z. Then the direction of the gradient vector ∇⁡fis the direction of steepest ascent of the hill, while its magnitude

   ∥∇⁡f∥=(∂⁡f∂⁡x)2+(∂⁡f∂⁡y)2
   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 projectionto 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 minimum 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 aconservative physical force, then 𝐅=-∇⁡Uis the corresponding force field.
   Consequently, the gradient theorem,

   ∫γ𝐅⋅d𝐬=-∫γ∇U⋅d𝐬=-U(γ(b))+U(γ(a)),γ:[a,b]→ℝ3
   simply gives the formula for the change in the potential energy Uwhen 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 manifolds: the dot product of ℝn must be replaced by the Riemannian metric, and the analogue of D𝐯⁡f 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,

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

and taking inverses,

4 Duality with differential one-forms

Notice that when xi are orthonormal coordinates on ℝn, equation (5) reduces to equation (2), becausegi⁢j=𝐞i⋅𝐞j=δi⁢j (Kronecker delta).

The formulae presented in this section are useful in the Euclidean setting as well, for deriving the formulae for the gradient in various curvilinear coordinate systems (http://planetmath.org/GradientInCurvilinearCoordinates).

5 Differential identities

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:

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 curl⁡g=0 on a simply connected domain, then there is f such that g=grad⁡f. 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 divergence 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 fwe have

References