Divergence theorem

The Gaussian integral theorem, and Gauss - Ostrogradski or divergence theorem, is a result from the vector analysis. It provides a connection between the divergence of a vector field and the field given by the flow here by a closed surface.

Named after Gauss integral theorem follows as a special case of the theorem of Stokes, which also generalizes the main theorem of differential and integral calculus.

Wording of the sentence

It is a compact set with smooth boundary sections, the edge is oriented by an external unit normal field. Furthermore, the vector field is continuously differentiable with on a open set. Then we have

Example

If the closed unit ball in, then applies as well.

For with the vector field.

It follows

As well as

In the calculation was used that applies to all, and that the ball unit, the volume and the surface.

Conclusions

Other identities can be derived from the Gaussian integral theorem. For simplicity, the following notation is used: and.

• Applying the Gaussian integral theorem on the product of a scalar with a vector field, then one obtains

• Applying the Gaussian integral theorem to the cross product of two vector fields and, then one obtains

• Applying the Gaussian integral theorem to a real function on the interval derivation, then we obtain the main theorem of differential and integral calculus. The evaluation of the integral at the interval ends in the main clause corresponds to the evaluation of the boundary integral of the divergence theorem.

Applications

Liquids, gases, electrodynamics

The set is used for the description of the conservation of mass, momentum and energy on any volume: the integral of the source distribution multiplied ( sum of the divergence of a vector field ) on the volume in the interior of a case with a constant results in the whole flow ( the shell integral) of total flow through the shell of this volume.

Gravity

The surface integral is the mass - 4πG times the inside as long as the composition is distributed in a radially symmetrical (constant density at a given distance from the center ), and irrespective of any ( also radially distributed symmetrically ) compositions outside: the gravitational field is obtained. In particular: The whole sphere outside a sphere has no (additional) influence, if its mass is distributed radially symmetrical. Only the sum of sources and sinks in the inner area work (→ see Newtonian Schalentheorem ).

Partial integration in multidimensional

The Gaussian integral theorem leads to a formula for partial integration in multidimensional

Importance

The Gaussian integral theorem is used in many areas of physics application, especially in electrodynamics and fluid dynamics.

In the latter case, the meaning of the sentence is particularly illustrative. Suppose that the vector field describes running water in a certain region of space. Then the divergence of just describes the strength of all sources and sinks in individual points. If you now wish to know how much water flows out of a total of a particular area, it is intuitively clear that you have the following two options:

• It examines and measures how much water enters through the surface of off and on. This corresponds to the flow rate of the vertical components of the surface as a surface integral.
• It accounted for ( measured ) inside the volume defined by how much water disappears altogether within in depressions (holes) and how much from sources ( water inflow ) is added. Thus adding the effects of sources and sinks. Alternatively, and this is then realized by the equivalent volume integral of the divergence.

The Gaussian integral theorem states that actually always run both ways absolutely equivalent to the target. He has thus the character of a conservation law of energy.

History

The set was probably first formulated by Joseph Louis Lagrange in 1762 and independently later by Carl Friedrich Gauss ( 1813), George Green (1825 ) and Mikhail Ostrogradski (1831 ) rediscovered. Ostrogradski also provided the first formal proof.