The Helmholtz decomposition of decreasing and weakly increasing vector fields

Abstract

Helmholtz' decomposition theorem for vector fields is presented usually with too strong restrictions on the fields. Based on the work of Blumenthal of 1905 it is shown that the decomposition of vector fields is not only possible for asymptotically weakly decreasing vector fields, but even for vector fields, which asymptotically increase sublinearly. Use is made of a regularization of the Green's function and the mathematics of the proof is formulated as simply as possible. We also show a few examples for the decomposition of vector fields including the electric dipole radiation.

Helmholtz decomposition theorem and Blumenthal's extension by regularization

“…The Helmholtz vector potential fulfills ∇ • A B H = 0 and ∇ × A B H = B , the same conditions as for the vector potential A C in the Coulomb gauge. We indeed obtain for this example A B H ( x) = A C ( x) [16,21].…”

“…We do not present explicit steps of the proof (for this see [16]), but in order to show that the different integrals, which arise in (3.2) and (3.3), exist and are finite, we separate the volume of integration into an inner volume of a sphere with radius R ≫ r and the outer domain r ′ R. Now, the large r behavior of the corresponding G i is taken into account to prove the convergence. We note that the singularities at x and at zero do not lead to a diverging contribution to the integral, as long as i 2.…”

“…An exception should be made in the case when i = 2 is chosen. Then, the difference in the vector fields could be a linear harmonic function resulting in a uniqueness up to a linear term (see again for more details in [16]).…”

Complete general solutions for equilibrium equations of isotropic strain gradient elasticity

“…For the further analysis it will be enough to assume that in the infinite domains representation (15) can be introduced under assumptions of weak decay conditions for the displacement field u u u that are used in classical elasticity [6], though the validity of Helmholtz theorem for the fields with sub-linear growth have been also established [52,53].…”

The Helmholtz decomposition revisited

“…Its vector potential is (for r 0 > ) proportional to the magnetic field B,  and its scalar potential is the quasistatic (acausal) potential in Coulomb gauge. The calculation for the decomposition is found in Petrascheck and Folk [3].…”