Decomposition of symmetric tensor fields in $\mathbb{R}^3$

In the article, we introduce generalizations of the curl operator acting on three-dimensional symmetric $m$-tensor fields and establish properties of them. For the spaces of three-dimensional tensor fields, new detailed decompositions are obtained. Each term in the decompositions is constructed using of one function. Decompositions of this kind play a special role, in particular, in the study of tomographic integral operators acting on symmetric $m$-tensor fields, $m\geqslant1$, and in the construction of algorithms for solving the emerging inverse problems.