Helmholtz decomposition
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The Helmholtz decomposition of a vector field which is twice continuously differentiable, with rapid enough decay at infinity, splits the vector field into a sum of gradient and curl as follows:
- <math>\mathbf{F} = - \nabla\,\mathcal{G} (\nabla \cdot \mathbf{F}) + \nabla \times \mathcal{G}(\nabla \times \mathbf{F})<math>
where <math>\mathcal{G}<math> represents the Newtonian potential.
If ∇·F=0, we say F is solenoidal or divergence-free and thus the Helmholtz decomposition of F collapses to
- <math>\mathbf{F} = \nabla \times \mathcal{G}(\nabla \times \mathbf{F}) = \nabla \times \mathbf{P}<math>
In this case, P is known as the vector potential for F.
Likewise, if ∇×F=0 then F is said to be curl-free or irrotational and thus the Helmholtz decomposition of F collapses then to
- <math>\mathbf{F} = - \nabla\,\mathcal{G} (\nabla \cdot \mathbf{F}) = - \nabla \phi<math>
In this case, φ is known as the scalar potential for F.