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the Helmholtz decomposition.

Generalization to arbitrary dimensions

Informally speaking, in , the Helmholtz decomposition can be expressed by where is any scalar function that solves the Poisson equation , where is the divergence of the vector field in , and is divergence free: . Thus the existence of Helmholtz decomposition a consequence of the existence of the solution of the Poisson equation .

Matrix approach

The generalization to dimensions cannot be done with a vector potential, since the rotation

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