Counting Independent Tensor Components Based on Symmetry

The number of independent components for a type (3,0) tensor is not a fixed value of $N^3$ 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, given by the formula $\binom{N}{3}$. 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), which leads to the formula $\binom{N+2}{3}$. 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.

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🏗️Structural clarification of Proof and Derivation

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