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title: Counting Independent Tensor Components Based on Symmetry

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# Counting Independent Tensor Components Based on Symmetry

Symmetry and Independent Components: A tensor's symmetry properties directly determine its number of independent components. While a general type $(3,0)$ tensor in $N$ dimensions has $N^3$ components, symmetry relations significantly reduce this number.

Anti-symmetric Tensor: For an anti-symmetric tensor $T^{a b c}$, the components are only nonzero if all three indices are distinct. The value of a component is then determined by the unique set of three indices. The number of independent components is therefore the number of ways to choose 3 unique indices from $N$, given by the combination formula $\binom{N}{3}$.

Symmetric Tensor: For a symmetric tensor $S^{a b c}$, the order of indices is irrelevant, and repeated indices are allowed. The number of independent components is the number of ways to choose 3 indices from $N$ with replacement, which is a stars and bars problem represented by the formula $\binom{N+2}{3}$.

A Clear Mathematical Distinction: The contrast between the two counting methodscombinations without repetition for anti-symmetric tensors and combinations with repetition for symmetric tensors-is the fundamental reason for the vastly different number of independent components for each tensor type.

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