gradient
Summary.
The gradient is a first-order differential operator
that mapsscalar functions to vector fields. It is a generalization
of the ordinaryderivative
, and as such conveys information about the rate of changeof a function relative to small variations in the independentvariables. The gradient of a function is customarily denoted by or by .
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
Let be continuously differentiable.The gradient of , denoted by ,is definedby the property:
(1) |
The middle dot is the dot product,and is the directional derivative
with respect to .
If are Euclidean coordinates
,corresponding to the orthonormal basis ,then
(2) |
The formula (2) is sometimes given as the definition of .We prefer to define by the coordinate-free formula (1) instead,because then the geometric interpretations
(see below) become obvious,and (1) also indicates how we would go aboutcalculating the gradient in other curvilinear coordinate systems.Formula (1) also makes it clear that the gradientis a physical vector, depending only on the inner product structure of ,and not on the specific coordinate system
used to calculate it.
There is the issue of whether the 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 avector-valued differential operator, written as
or, in the context of Euclidean 3-space, as
where are the unit vectors lying along the positivedirection of the axes, respectively.
2 Geometric and physical interpretations
- (a)
The direction of the vector is the direction of the greatest positive change, or increase, in .The magnitude of is the magnitude of this increase.This follows immediately from (1):
where is the angle between and .So among all unit directions of change, if is perpendicular
to then the change is zero; if is parallel
to then the change is maximized.
Similarly, is the direction of the greatest negative change, or decrease, in .
- (b)
If is the hypersurface in defined by
then is the normal to the hypersurface at the point . For is the tangent space
to at , that is, for all ,and by definition (1), must be perpendicular to all .
Note that is equivalent
to . Consequently, also givesan orientation to the hypersurface .
For example, if for , is the -dimensional sphere of unit radius, embedded in .Its normal, , as one would expect, points outward radially.
- (c)
As a simple case of (b), consider thesurface in , with Cartesian coordinates
.Think of this surface as describing a hill, with height .Then the direction of the gradient vector is the direction of steepest ascent of the hill, while itsmagnitude
is the slope or steepness in that direction.
If a ball is placed on the hill at a point ,theoretically it should roll down the hill in the direction of the gradientvector . This may be easily derived by considering the mechanical forceson the ball. The direction of is, in fact, the projectionto the xy-plane of an outward normal vector to the hill at ;the normal vector is involved because the movement of the ballarises from the normal force from the hill.
- (d)
Suppose the surface in (c) describes a bowl instead of a hill,and we place a marble at any point on this bowl.We would expect the marble to roll down to a local minimum
point of .Since the marble should roll down in the direction of ,we might hope that we can find local minimaof a given function by following the path mapped outby the gradients . Formally, this method offinding local extrema (with some modifications) is called gradient descent.
- (e)
If is the potential function corresponding to aconservative physical force, then is the corresponding force field.
Consequently, the gradient theorem
,
simply gives the formula for the change in the potential energy when an object “does work” along a path in a conservative force field .
3 Definition: Riemannian geometry
It is obvious how (1)can be generalized to the setting of Riemannian manifolds:the dot product of must be replacedby the Riemannian metric, and the analogueof is the directional derivative , for tangent vectors on the Riemannian manifold.Thus for a smooth scalar-valued function on a Riemannian manifold,
(3) |
We can calculate explicitly as follows.If are local coordinates on the manifold (not necessarily orthonormal),set (the Einstein summation convention is being used).Let and be the covariant and contravariant metric tensors, respectively.Then from (3),
and taking inverses,
(4) |
4 Duality with differential one-forms
Definitions (1) and (3)exhibit as the vector fielddual to the differential form .The isomorphism
is given by applying the inner product or Riemannian metric.This isomorphism is, of course, linear;in particular it leads to the identity
(5) |
which is the dual to the standard formula of differential one-forms:
Using (3) and (4),we have
(6) |
So the isomorphism between vector fields and one-formsis expressed by changingthe ’s in (6) to ’s, and vice versa. That is,
(7) |
It is commonly said that this isomorphism is expressed by“raising and lowering the indices of a tensor field,using contractions with and ”.
Notice that when are orthonormal coordinateson , equation (5)reduces to equation (2), because (Kronecker delta).
The formulae presented in this section are useful in the Euclidean setting aswell, for deriving the formulae for the gradient in various curvilinear coordinate systems (http://planetmath.org/GradientInCurvilinearCoordinates).
5 Differential identities
Several properties of the one-dimensional derivative generalize to amulti-dimensional setting
Linearity | ||||
Product rule![]() | ||||
Chain rule![]() | ||||
Another Chain rule |
The function is .The notation denotes the transpose of the Jacobian matrix,in Euclidean coordinates, of .In the abstract setting, is the adjoint
to the tangent map 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 dualsof the following well-known identities for differential forms:
and so may be derived by changing the ’s hereto ’s! (Though the third identity may take a bit of thought.)
The following identity
is a special case of the differential forms identity .Conversely, if on a simply connected domain, then thereis such that . See laminar field for details.
6 The symbolism
(This discussion does not really belong here, but should be movedto the nabla entry.)
Using the formalism,the divergence operator can be expressed as, the curl operator as , and theLaplacian operator as . To wit, for a given vector field
and a given function we have
References
- 1 Michael Spivak. A Comprehensive Introduction to Differential Geometry
,Volume I. Publish or Perish, 1979.