Helmholtz decomposition

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In mathematics, in the area of vector calculus, Helmholtz's theorem, also known as the fundamental theorem of vector calculus, states that any sufficiently smooth, rapidly decaying vector field can be resolved into irrotational (curl-free) and solenoidal (divergence-free) component vector fields.

This implies that any vector field <math>\mathbf{F}</math> can be considered to be generated by a pair of potentials: a scalar potential <math>\phi</math> and a vector potential <math>\mathbf{A}</math>.

The resulting Helmholtz decomposition of a vector field, which is twice continuously differentiable and 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 <math>\nabla\cdot\mathbf{F}=0</math>, we say <math>\mathbf{F}</math> is solenoidal or divergence-free and thus the Helmholtz decomposition of <math>\mathbf{F}</math> collapses to

<math>\mathbf{F} = \nabla \times \mathcal{G}(\nabla \times \mathbf{F}) = \nabla \times \mathbf{A}</math>

In this case, <math>\mathbf{A}</math> is known as the vector potential for <math>\mathbf{F}</math>.

Likewise, if <math>\nabla\times\mathbf{F}=\mathbf{0}</math> then <math>\mathbf{F}</math> is said to be curl-free or irrotational and thus the Helmholtz decomposition of <math>\mathbf{F}</math> collapses then to

<math>\mathbf{F} = - \nabla\,\mathcal{G} (\nabla \cdot \mathbf{F}) = - \nabla \phi.</math>

In this case, <math>\phi</math> is known as the scalar potential for <math>\mathbf{F}</math>.

In general the negative gradient of the scalar potential is equated with the irrotational component, and the curl of the vector potential is equated with the solenoidal component:

<math> \mathbf{F} = -\nabla \varphi + \nabla \times \mathbf{A} </math>.

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