The gradient is a first-order differential operator that maps scalar functions to vector fields. It is a generalization of the ordinary derivative, 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 is customarily denoted by or by .
1 Definition: Euclidean space
Let be continuously differentiable. The gradient of , denoted by , is defined by the property:
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 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 , and not on the specific coordinate system used to calculate it.
2 Geometric and physical interpretations
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):
Similarly, is the direction of the greatest negative change, or decrease, in .
If is the hypersurface in defined by
For example, if for , is the -dimensional sphere of unit radius, embedded in . Its normal, , as one would expect, points outward radially.
As a simple case of (b), consider the surface 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 its magnitude
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 gradient vector . This may be easily derived by considering the mechanical forces on the ball. The direction of is, in fact, the projection to the xy-plane of an outward normal vector to the hill at ; the normal vector is involved because the movement of the ball arises from the normal force from the hill.
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 minima of a given function by following the path mapped out by the gradients . Formally, this method of finding local extrema (with some modifications) is called gradient descent.
If is the potential function corresponding to a conservative 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 replaced by the Riemannian metric, and the analogue of is the directional derivative , for tangent vectors on the Riemannian manifold. Thus for a smooth scalar-valued function on a Riemannian manifold,
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 Duality with differential one-forms
Definitions (1) and (3) exhibit as the vector field dual 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
So the isomorphism between vector fields and one-forms is expressed by changing the ’s in (6) to ’s, and vice versa. That is,
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
Several properties of the one-dimensional derivative generalize to a multi-dimensional setting
|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 duals of the following well-known identities for differential forms:
and so may be derived by changing the ’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 . Conversely, if on a simply connected domain, then there is such that . See laminar field for details.
6 The symbolism
(This discussion does not really belong here, but should be moved to the nabla entry.)
|Date of creation||2013-03-22 11:59:20|
|Last modified on||2013-03-22 11:59:20|
|Last modified by||CWoo (3771)|