Helmholtz Theorem

A vector field is uniquely determined by its divergence and curl, given the boundary conditions.

A vector field without curl is called irrotational and a vector field without divergence is called solenoidal.

If \(E\) is irrotational, \(E = - \nabla V\) for some \(V\).

If \(B\) is an solenoidal field then \(B = \nabla \times A\) for some \(A\).

In general any vector field is sum of an irrotational and solenoidal field.

Generalization of Helmholtz Theorem is the wiki:Polar factorization theorem (twitter)

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