📢Independent Components and Tensor Symmetry

The number of independent components for a type (3,0) tensor is not a fixed value but is severely constrained by its symmetry properties. For a totally anti-symmetric tensor, all components with repeated indices are zero. The non-zero components are determined by the unique set of three distinct indices, so the number of independent components is found by counting combinations without repetition. In contrast, for a totally symmetric tensor, the order of indices doesn't matter, and repetitions are allowed. The number of independent components is found by counting combinations with repetition (a multiset). This fundamental difference in counting is the key distinction between the two tensor types and highlights how symmetry directly dictates a tensor's degrees of freedom.

Independent Components and Tensor Symmetry | Audiosviadean on Notion