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.)
Read more about this topic: Helmholtz Decomposition
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