On the Helmholtz decomposition for polyadics

A polyadic field of rank n is the tensor product of n vector fields. Helmholtz showed that a vector field, which is a polyadic field of rank 1, is nonuniquely decomposable into the gradient of a scalar function plus the rotation of a vector function. We show here that a polyadic field of rank n is, again nonuniquely, decomposable into a term consisting of n successive applications of the gradient to a scalar function, plus a term that consists of (n — 1) successive applications of the gradient to the rotation of a vector function, plus a term that consists of (n — 2) successive applications of the gradient to the rotation of the rotation of a dyadic function and so on, until the last (n + l)th term, which consists of n successive applications of the rotation operator to a polyadic function of rank n. Obviously, the n = 1 case recovers the Helmholtz decomposition theorem. For dyadic fields a more symmetric representation is provided and formulae that provide the potential representation functions are given. The special cases of symmetric and antisymmetric dyadics are discussed in detail. Finally, the multidivergence type relations, which reduce the number of independent scalar representation functions to n2, are presented.

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