Chemistry - Helmholtz decomposition

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:

$\mathbf{F} = - \nabla\,\mathcal{G} (\nabla \cdot \mathbf{F}) + \nabla \times \mathcal{G}(\nabla \times \mathbf{F})$

where $\mathcal{G}$ represents the Newtonian potential.

If ∇·F=0, we say F is solenoidal or divergence-free and thus the Helmholtz decomposition of F collapses to

$\mathbf{F} = \nabla \times \mathcal{G}(\nabla \times \mathbf{F}) = \nabla \times \mathbf{P}$

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

$\mathbf{F} = - \nabla\,\mathcal{G} (\nabla \cdot \mathbf{F}) = - \nabla \phi$

In this case, φ is known as the scalar potential for F.

Categories: Multivariate calculus