Helmholtz decomposition

The Helmholtz theorem, also Helmholtz decomposition, Stokes - Helmholtz decomposition or fundamental theorem of vector analysis, (after Hermann von Helmholtz ) states that for certain areas of the space can be written as a direct sum of divergence free functions and gradient fields.

Definitions

For an area of the space of divergence- free functions is known, wherein the space of the test functions, and refers to the standard. The decomposition

With Helmholtz decomposition is called, so far there is the decomposition. In this case there is a projection with which a so-called Helmholtz - projection.

If the half-space, a bounded domain with boundary or an outdoor space with boundary, then there exists the decomposition. For the disassembly for any areas with boundary exists.

Has a rim, applies, wherein the outer is normal.

Mathematical Application

Lösbarkeitstheorie in the Navier -Stokes equations, the Helmholtz projection plays an important role. The Helmholtz - projection is applied to the linearized Navier -Stokes equations, we obtain the Stokes equation

For. There was previously two unknowns, namely and there is now only one unknown. Both equations, the Stokes and the linearized equations, but they are equivalent.

The operator is called the Stokes operator.

Physical viewing

The Helmholtz theorem states that it is possible to represent ( almost) arbitrary vector field as a superposition of a rotation- free ( irrotational ) field and a divergence- free ( source-free ) field. An irrotational field, however, can in turn be represented by a scalar potential, a divergence- free field by a vector potential.

And

Then follows

And

It is also possible the vector field by superposition (addition ) of two different potentials and express ( the Helmholtz theorem).

The two complementary potentials can be obtained by the following integrals from the field:

Where the volume containing the fields.

The mathematical condition for the application of the Helmholtz 's theorem is next to the differentiability of the vector field that it is for than against it, that is. Otherwise, the above integrals diverge, so can no longer be calculated.

This theorem is particularly in electrodynamics of interest as they write with his help, the Maxwell equations in the potential image and can be solved easily. For all physically relevant problems doing the mathematical prerequisites are met.

Redundancy

While the original vector field can be described at each point of by components required for the scalar and vector potential components together. This redundancy can be used for eliminating by the source -free portion of the vector field of the toroidal- poloidal decomposition is subjected to, which ultimately suffice three Skalarpotentiale for description.