Helmholtz theorem

The Helmholtz theorem , also Helmholtz decomposition , Stokes-Helmholtz decomposition or the fundamental theorem of vector analysis , (after Hermann von Helmholtz ) states that for certain areas the space can be written as a direct sum of divergence-free functions and gradient fields . ${\displaystyle \Omega \subset \mathbb {R} ^{n}}$ ${\displaystyle L^{p}}$

Definitions

For an area is the space of divergence-free functions called, where the space of test functions and the norm referred. The decomposition ${\displaystyle \Omega \subset \mathbb {R} ^{n}}$ ${\displaystyle L_{\sigma }^{p}(\Omega )={\overline {\{u\in C_{c}^{\infty }(\Omega ):\nabla \cdot u=0\}}}^{\|\cdot \|_{p}}}$ ${\displaystyle C_{c}^{\infty }(\Omega )}$ ${\displaystyle \|\cdot \|_{p}}$ ${\displaystyle p}$

${\displaystyle L^{p}(\Omega )=L_{\sigma }^{p}(\Omega )\oplus G_{p}}$

with is called the Helmholtz decomposition, insofar as the decomposition exists. In this case, there is a projection of the so-called. Helmholtz projection. ${\displaystyle G_{p}=\{u=\nabla \phi :\phi \in L_{\text{loc}}^{1}(\Omega )\ {\text{und}}\ \nabla \phi \in L^{p}(\Omega )\}}$ ${\displaystyle P}$ ${\displaystyle PL^{p}(\Omega )=L_{\sigma }^{p}(\Omega )}$

If the half-space , a bounded area with -rand or an outer space with -rand, then the decomposition exists. For there is a decomposition for any regions with a margin. ${\displaystyle \Omega }$ ${\displaystyle C^{2}}$ ${\displaystyle C^{2}}$ ${\displaystyle p=2}$ ${\displaystyle C^{2}}$

Has an -rand, where is the outer normal . ${\displaystyle \Omega }$ ${\displaystyle C^{1}}$ ${\displaystyle L_{\sigma }^{p}(\Omega )=\{u\in L^{p}(\Omega ):\operatorname {div} u=0\ {\text{und}}\ u\cdot \nu =0\ {\text{auf}}\ \partial \Omega \}}$ ${\displaystyle \nu }$

Mathematical application

The Helmholtz projection plays an important role in the solvability theory of the Navier-Stokes equations . If the Helmholtz projection is applied to the linearized incompressible Navier-Stokes equation, the Stokes equation is obtained

${\displaystyle u_{t}-P\Delta u=f}$

for . Before there were two unknowns, namely and , there is now only one unknown. However, both equations, the Stokes equation and the linearized equation, are equivalent. ${\displaystyle u,f\in L_{\sigma }^{p}(\Omega )}$ ${\displaystyle u}$ ${\displaystyle p}$

The operator is called the Stokes operator . ${\displaystyle P\Delta }$

Physical consideration

The Helmholtz theorem states that it is possible to represent an (almost) arbitrary vector field as a superposition of a rotation-free ( eddy-free ) field and a divergence-free ( source-free ) field . A rotation-free field can, however, in turn be represented by a scalar potential , a divergence-free field by a vector potential . ${\displaystyle \mathbf {f} (\mathbf {r} )}$ ${\displaystyle \mathbf {a} (\mathbf {r} )}$ ${\displaystyle \mathbf {b} (\mathbf {r} )}$ ${\displaystyle \phi (\mathbf {r} )}$ ${\displaystyle \mathbf {A} (\mathbf {r} )}$

${\displaystyle \mathbf {a} (\mathbf {r} )=-\operatorname {grad} (\phi (\mathbf {r} ))}$

and

${\displaystyle \mathbf {b} (\mathbf {r} )=\operatorname {rot} (\mathbf {A} (\mathbf {r} ))}$

then follows

${\displaystyle \operatorname {rot} (\mathbf {a} (\mathbf {r} ))=-\operatorname {rot} (\operatorname {grad} (\phi (\mathbf {r} )))\equiv 0}$

and

${\displaystyle \operatorname {div} (\mathbf {b} (\mathbf {r} ))=\operatorname {div} (\operatorname {rot} (\mathbf {A} (\mathbf {r} )))\equiv 0}$

It is therefore possible to express the vector field by superposition (addition) of two different potentials and (the Helmholtz theorem). ${\displaystyle \mathbf {f} (\mathbf {r} )}$ ${\displaystyle \phi (\mathbf {r} )}$ ${\displaystyle \mathbf {A} (\mathbf {r} )}$

${\displaystyle \mathbf {f} (\mathbf {r} )=\mathbf {a} (\mathbf {r} )+\mathbf {b} (\mathbf {r} )=-\operatorname {grad} (\phi (\mathbf {r} ))+\operatorname {rot} (\mathbf {A} (\mathbf {r} ))}$

The two complementary potentials can be obtained from the field using the following integrals : ${\displaystyle \mathbf {f} (\mathbf {r} )}$

${\displaystyle \phi (\mathbf {r} )={\frac {1}{4\pi }}\int _{V}{\frac {\operatorname {div} (\mathbf {f} (\mathbf {r} '))}{|\mathbf {r} -\mathbf {r} '|}}\mathrm {d} ^{3}r'}$

${\displaystyle \mathbf {A} (\mathbf {r} )={\frac {1}{4\pi }}\int _{V}{\frac {\operatorname {rot} (\mathbf {f} (\mathbf {r} '))}{|\mathbf {r} -\mathbf {r} '|}}\mathrm {d} ^{3}r'}$

Where is the volume containing the fields. ${\displaystyle V}$

The mathematical condition for the application of Helmholtz's theorem is next to the differentiation of the vector field that it faster than against it, that is . Otherwise the above integrals diverge and can no longer be calculated. ${\displaystyle \mathbf {f} (\mathbf {r} ),}$ ${\displaystyle r\to \infty }$ ${\displaystyle {\frac {1}{r}}}$ ${\displaystyle 0}$ ${\displaystyle \lim _{r\to \infty }\mathbf {f} (\mathbf {r} )r=0}$

This theorem is of particular interest in electrodynamics , since with its help the Maxwell equations can be written in the potential image and solved more easily. The mathematical prerequisites are met for all physically relevant problems.

redundancy

While the original vector field is to be described by components at each point , components are necessary for the scalar and the vector potential together . This redundancy can be eliminated by subjecting the source-free portion of the vector field to the toroidal-poloidal decomposition , whereby a total of three scalar potentials are sufficient for the description. ${\displaystyle \Omega }$ ${\displaystyle n}$ ${\displaystyle n+1}$ ${\displaystyle n=3}$

Individual evidence

1. ↑ Tribikram Kundu: Ultrasonic and Electromagnetic NDE for Structure and Material Characterization . CRC Press, 2012, ISBN 1-4398-3663-9 , pp. 37 ( limited preview in Google Book search).
2. ^ GP Galdi, An introduction to the mathematical theory of the Navier-Stokes equations. Vol. I, Springer Tracts in Natural Philosophy, vol. 38, Springer-Verlag, New York, 1994, ISBN 0-387-94172-X