Maxwell's multipole representation of traceless symmetric tensors and its application to functions of high-order tensors

It is known from the theory of group representations that a general tensor can be expressed as a sum of traceless symmetric tensors. In this paper, based on Sylvester's theorem, it is shown that a general traceless symmetric tensor of any finite orderm in three dimensions can be expressed as the traceless symmetric part of tensor product ofm unit vectors (called the multipoles) multiplied by a positive scalar. The above two basic structures of tensors allow us to easily give complete and irreducible representations for tensor functions with high–order tensor variables, since those for tensor functions of vectors are well established in the literature. Examples are given for scalar–valued functions of a single fourth–order tensor of the elastic type, and of a number of vectors and second–order tensors. In 1970 Backus gave an alternative proof of Sylvester's theorem, which shows how to compute the multipoles. Since Backus's result is not so ‘well known’ to the community of researchers working on continuum mechanics, in the present paper a direct (without using Sylvester's theorem) and constructive establishment of Maxwell's multipole representation is provided, which is closer in spirit to a more modern approach to this topic.