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title: Symmetric and Antisymmetric Tensor Contractions

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# Symmetric and Antisymmetric Tensor Contractions
- The contraction of a symmetric tensor with an antisymmetric tensor is always zero. This fundamental property simplifies many tensor calculations.

- Any tensor can be uniquely decomposed into a symmetric part and an antisymmetric part. This decomposition, $T_{a b}=T_{\{a b\}}+T_{[a b]}$, is a powerful tool for analyzing tensor properties.
- When an arbitrary tensor $T_{a b}$ is contracted with a symmetric tensor formed by the outer product of a vector with itself, $v^a v^b$, the result depends solely on the symmetric part of $T_{a b}$. The antisymmetric part of the tensor does not contribute to the final value.

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