Editing Helmholtz decomposition (section)

=== Fields with prescribed divergence and curl ===
The term "Helmholtz theorem" can also refer to the following. Let {{math|'''C'''}} be a [[solenoidal vector field]] and ''d'' a scalar field on {{math|'''R'''<sup>3</sup>}} which are sufficiently smooth and which vanish faster than {{math|1/''r''<sup>2</sup>}} at infinity. Then there exists a vector field {{math|'''F'''}} such that

<math display="block">\nabla \cdot \mathbf{F} = d \quad \text{ and } \quad \nabla \times \mathbf{F} = \mathbf{C};</math>

if additionally the vector field {{math|'''F'''}} vanishes as {{math|''r'' → ∞}}, then {{math|'''F'''}} is unique.<ref name="griffiths1999"/>

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.<ref name="griffiths1999"/> The proof is by a construction generalizing the one given above: we set

<math display="block">\mathbf{F} = \nabla(\mathcal{G} (d)) - \nabla \times (\mathcal{G}(\mathbf{C})),</math>

where <math>\mathcal{G}</math> represents the [[Newtonian potential]] operator. (When acting on a vector field, such as {{math|∇ × '''F'''}}, it is defined to act on each component.)