Helmholtz Decomposition - Fields With Prescribed Divergence and Curl

Fields With Prescribed Divergence and Curl

The term "Helmholtz Theorem" can also refer to the following. Let C be a solenoidal vector field and d a scalar field on R3 which are sufficiently smooth and which vanish faster than 1/r2 at infinity. Then there exists a vector field F such that

and

if additionally the vector field F vanishes as r → ∞, then F is unique.

In other words, a vector field can be constructed with both a specified divergence and a specified curl, and if it also vanishes at infinity, it is uniquely specified by its divergence and curl. This theorem is of great importance in electrostatics, since Maxwell's equations for the electric and magnetic fields in the static case are of exactly this type. The proof is by a construction generalizing the one given above: we set

where represents the Newtonian potential operator. (When acting on a vector field, such as ∇ × F, it is defined to act on each component.)