Gradient

The gradient is a mathematical operator, specifically a differential operator, which can be applied to a scalar field and in this case called a gradient vector field provides. The gradient is perpendicular to the equipotential surface ( level set ) of the scalar field at a point P, and the magnitude of the gradient is the largest rate of change of the scalar field at the point P at.

Interpreted, for example, the relief map of a scene as a function of the any location associating the height at this point, then the gradient of h at the point is a vector in the xy plane, pointing in the direction of the steepest ascent of h at this location and its length is a measure of the slope ( slope). In order to distinguish between the operator and the result of his application is referred to such gradients of scalar field sizes in some sources as Gradientvektoren.

The gradient is along with other differential operators such as divergence and curl of vector analysis, a branch of multivariate analysis, examined. They are made with the same vector operator, with the so-called nabla operator (to indicate that the nabla operator can be alternatively understood as a vector, or sometimes ).

Definition

Uniquely determined vector The operator is the total differential and the Cartan derivative.

Coordinate representation

Said gradient in different coordinate systems, different representations.

Cartesian coordinates

In the Euclidean standard scalar is the column vector

The entries are the partial derivatives of in - direction. Often one writes in Cartesian coordinates as well (pronounced " nabla f" ) rather than with the nabla operator. In three dimensions, said gradient is thus the representation

Cylindrical and spherical coordinates

• Representation in three-dimensional cylindrical coordinates:

• Representation in three-dimensional spherical coordinates:

These are special cases of the gradient on Riemannian manifolds. For this generalization, see: exterior derivative.

Orthogonal coordinates

In general orthogonal coordinates, the gradient has the representation

Being indicative of the magnitude and the direction of the vector.

Coordinate -free representation as volume derivative

By means of the integral set of Gaussian, the gradient, similar to the divergence ( source density) and the rotation ( vortex density) are presented as volume derivation. This representation has the advantage that it is coordinate- independent. For this reason, the gradient in the field of engineering sciences is often defined directly so.

If a region of space with piecewise smooth boundary and the volume then by the gradient of the scalar field in the point by means of the volume derivative

Be calculated. This means the outer surface vector element of said outwardly facing surface normal vector and the scalar element.

In order to limit the formation region of space is contracted to the point, so that its contents to zero. Is replaced by a pressure gradient appears as a force density. The cross plots of the previous section result from the volume derivative, if one chooses the particular volume element, such as sphere or cylinder as a region of space.

Geometric interpretation

Geometrically, the gradient of a scalar field at a point is a vector that points in the direction of the steepest ascent of the scalar. This corresponds to the magnitude of the vector of the strength of the increase. One is located at a local minimum or maximum ( extremum ) or a saddle point, the gradient at this point especially the zero vector, provided that the end point is located in the interior of the observed region.

With the aid of the gradient can also be an increase in any direction, the direction of discharge called determine which - in contrast to the gradient - again is a scalar.

Properties

For all constants and scalar fields is considered:

Linearity

Product rule

Related to the directional derivative

Under the directional derivative is defined as the derivative, ie, the rise of a scalar field in the direction of a normalized vector in more detail:

Is in an environment of differentiable, then one can calculate the directional derivative as the scalar product of the gradient of:

Integrability

An important relationship for gradient fields in n dimensions is the statement that these always are " integrable ", in the following sense: It is valid for all i and k (i, k = 1, ..., n):

This directly verifiable relationship - identical in three dimensions with the rotational freedom of the field - is necessary for the existence of a " potential function " (more precisely: the function). Gi and Gk, the components of the vector field. The integrability condition also implies that for all closed paths W vanishes in the line integral, which in mechanics and electrodynamics has great significance.

Examples

The following gradients often occur in physics. It is used the position vector.

Note that the last of the gradient, not only affects. It is therefore written as.

Applications

Vector gradient

Definition

In mathematics, the gradient is defined only for functions with image area. In physics and engineering, however, a so-called vector gradient is introduced for functions. This defined hereinafter vector gradient is the same as the Jacobian matrix.

For ( with standard scalar product )

A function whose components describe the change of the components of in the direction of. If you write instead of now and suppresses the brackets, we obtain the so-called vector gradient of the defining property

This therefore represents the matrix multiplication of the Jacobian matrix of a matrix with the column vector of a matrix.

This definition may not be a mathematically useful generalization of the gradient is, for example, the fact that the definition of the Jacobian matrix is independent of the scalar product, as this significantly enter into the definition of the gradient.

The vector gradient can be interpreted as transposed dyadic product of the nabla operator and a vector:

In index notation:

The vector gradient is used, inter alia, in continuum mechanics ( eg, in the Navier -Stokes equations).

In the literature there are other definitions of the vector gradient, namely a transposed Jacobian matrix or as the dot product of a vector with the Nablaoperator

Total differential

Consider a vector field is an infinitesimal displacement:

The complete or total differential of a vector field is:

The total differential of a scalar field and a vector field thus have the same shape.

Properties

The calculation rules are those of the Jacobian matrix. denotes the vector gradient here.

For all constants and vector fields apply:

Linearity

Product rule

Especially for vector fields can be above relationship still transform:

Examples

The two formulas are used for example in the Cartesian multipole expansion.