Divergence

In mathematics, the divergence is a differential operator that maps a vector field is a scalar field. While a vector field, a vector is assigned to each point in a scalar field each point is a scalar, ie a number assigned. If one interprets the vector field as flow field, it indicates the divergence for each point in space, how much more of a neighborhood of this point flows out than flows into it. Using the divergence can find out if and where the vector field has sources ( divergence greater than zero) or sinks ( divergence less than zero ) So. Is the divergence everywhere equal to zero, refers to the field as a source free.

The divergence is used together with other differential operators such as gradient and rotation in the vector analysis, a branch of multivariate analysis, examined.

In physics, the divergence is used for example in the formulation of the Maxwell equations and the equation of continuity.

• 5.1 in n-dimensional space
• 5.2 In the three-dimensional space
• 5.3 Examples

• 6.1 statement
• 6.2 Point-like source

• 9.1 Definition
• 9.2 transport rate and geometric interpretation

For example in physics

Consider, for example, a calm water surface, on which a thin beam strikes oil. The movement of the oil on the surface can be described by a two-dimensional ( time-based ) vector field, that is, at each point at each time point is expressed as a vector having magnitude and direction of the flow rate of the oil film. The location at which the beam strikes the surface, an " oil well ", since there flows away from oil without supply on the surface. At this point the divergence is positive. In contrast, refers to a point at which the oil, for example, on the edge flows out of the basin, as a sink. At this point the divergence is negative.

Definition

The divergence of a differentiable vector field is a scalar field. Or it is written as. It signifies the formal nabla operator and the operator symbol of divergence. In the case of a three-dimensional vector field divergence is defined in Cartesian coordinates as

In general, for an n-dimensional vector field:

The divergence can be formally as the scalar product between and interpret, ie as the sum of component-wise products.

The divergence as the "source density "

If one interprets a vector field as flow field, its total differential describes an acceleration field. If at one point the acceleration matrix is diagonalizable, then each eigenvalue describes the acceleration in the direction of the associated eigenvector. Each positive eigenvalue thus describes the intensity of a directional source and each negative eigenvalue the directional intensity of a sink. Adding these eigenvalues ​​, we obtain the resulting intensity of a source or sink. Since the sum of the eigenvalues ​​is just the trace of the acceleration matrix, the source intensity is

Measured.

The divergence can be interpreted in this sense as " source density " (as opposed to " vortex density ", see rotation).

Coordinate -free representation

For the interpretation of divergence as the "source density " is the following coordinate- free definition in the form of a volume discharge important ( here for the case n = 3)

It is an arbitrary volume, for example a ball or a parallelepiped; is its content. It is integrated on the edge of this volume element, the outward normal, and the associated surface element.

For n > 3, this statement can be easily generalized by considering n-dimensional volumes and their (n-1 )-dimensional boundary surfaces. With specialization in infinitesimal cube or cuboid one obtains the well-known representation in Cartesian coordinates

In orthogonal curvilinear coordinates, for example, spherical coordinates or elliptic coordinates ( that is, for that ), in which, where not so, but which have the physical dimension of " length", is considered, however, some general

The points at the end contain more terms that follow by continuous cyclic permutations generated according to the scheme, etc., is written in the.

Derivation of the Cartesian representation

For the calculation of the Cartesian representation of the divergence from the coordinate -free representation, consider an infinitesimal cube.

Now we apply the mean value theorem of integral calculus, where the primed quantities are in the interval.

Thus, only the sum of the difference quotient remains

The partial to the border crossing derivatives are:

Covariant behavior under rotations and displacements

The divergence operator commutes with spatial rotations and translations of a vector field, ie the order of these operations makes no difference.

Rationale: If the vector field is rotated in space or (parallel) is moved, you need only the surface and volume elements to rotate in the same way, to get back to the same scalar expression in the above given coordinate independent representation. The scalar field rotates and thus shifts in the same way as the vector field.

A " separation theorem "

For n = 3 - dimensional vector fields which are at least twice continuously differentiable in all space and sufficiently rapidly at infinity go to zero, is that they fall into a vortex- free part and a source-free part. For the irrotational part is that it can be represented by its source density as follows:

For the source-free part, the same applies if the scalar potential is replaced by a so-called vector potential and substituted at the same time the expressions or ( = source density ) by the operations or ( = density of vertebral ).

This process is part of the Helmholtz theorem.

Properties

In the n- dimensional space

Be a constant, an open subset, a scalar field and two vector fields. Then, the following rules apply:

• The divergence is linear, which means that it is

• For the divergence of the product rule

• The divergence of the vector field corresponding to the trace of the Jacobian matrix of, which means that it applies This representation is koordinateninvariant because the trace of a linear map is invariant under a change of basis.

In three-dimensional space

If so there is also a product rule for the cross product, this is

Which is meant by the rotation. Differentiable paths for all it follows

For any differentiable.

Examples

In Cartesian coordinates, one finds immediately

For the Coulomb field can be found, if it is set in the first product rule, and

The formula for the divergence in spherical coordinates this result is of course even easier to obtain.

After Corollary fields of the following type are free of sources:

Gaussian integral theorem

Statement

Plays an important role in the divergence of the statement 's theorem. It states that the flow through a closed surface equals the integral of the divergence of the vector field within said volume, and thus allows the conversion of a volume integral in a surface integral:

Wherein the normal vector of the surface. Clearly for the case of a flow so that it describes the relationship between the flow through this space and the flow of the sources and sinks within the associated volume.

Point-like source

If, in the Gaussian integral theorem coulombartige the field and is chosen as the integration surface is a spherical surface of radius around the origin, then the integrand is constant and equal. Because the surface of the sphere, it follows

Thus, the integral set provides information about that also includes the point in contrast to the derivative expressions ( product rule or spherical coordinates ) is the volume integral of. This can be summed up with the result of the deduction invoice for a distribution equation:

Cylindrical and spherical coordinates

In cylindrical coordinates valid for the divergence of a vector field:

In spherical coordinates valid for the divergence of a vector field:

Derivation of spherical coordinates, that bypasses the differentiation of basis vectors: one introduces the test function and writes a volume integral even in Cartesian and spherical coordinates once. Using known expressions for gradient and volume element that results after multiplying the basis vectors

The derivatives of are partially integrated, with boundary terms vanish. On the right side the volume element must mitdifferenziert and are then restored in two terms ( expand ). The results

From the equality of the integrals for all test functions follows, that the expressions for the divergence are equal.

Inverse

Under certain conditions, there is a right or left inverse of the divergence. So there is for an open and bounded domain with boundary lipschitzstetigem an operator, such that for each with

Holds, where the corresponding Sobolev space and inscribed. means Bogowskii operator.

Divergence on Riemannian manifolds

In the Properties section has already been said that the divergence using the trace of the Jacobian matrix can be expressed and that this representation is koordinateninvariant. For this reason, we used this property to define the divergence on Riemannian manifolds. Using this definition, one can for example define the Laplace operator on Riemannian manifolds coordinate- free. This is then called Laplace -Beltrami operator.

Definition

Let be a Riemannian manifold and a vector field. Then the divergence is due to

Defined. This is a vector field and the operator is the Levi- Civita connection, which generalizes the nabla operator. Evaluates to you, it is and can be for all the form from linear algebra known track.

Transport rate and geometric interpretation

For the flow of a vector field of the transport rate applies

This is the Riemann - Lebesgue volume measure on the manifold, a relatively compact measurable subset and a smooth function. Is interpreted as the density of a conserved quantity, then it follows the continuity equation. For one obtains

The divergence is therefore the density of volumetric change rate relative to the flow. The divergence at a point indicates how fast the content of an infinitesimal volume element changes at this point, it moves with the flow. As a corollary it follows that a vector field is divergence-free if and only if the flux generated is volume -preserving.