Decomposition of Symmetric Tensor Fields in $\mathbb {R}^3$

We introduce generalizations of the curl operator acting on three-dimensional symmetric \( m \)-tensor fields and establish their properties. For the spaces of three-dimensional tensor fields, we obtain new detailed decompositions in which each term is constructed using one function. Decompositions of this kind play a special role, in particular, when studying tomographic integral operators acting on symmetric \( m \)-tensor fields, \( m\geqslant 1 \), and constructing algorithms for solving the arising inverse problems.