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:<math>T_{i_1i_2\dots i_r} = T_{i_{\sigma 1}i_{\sigma 2}\dots i_{\sigma r}}.</math>
The space of symmetric tensors of [[Tensor (intrinsic definition)#Tensor rank|rank]] ''r'' on a finite-dimensional [[vector space]] is [[natural isomorphism|naturally isomorphic]] to the dual of the space of [[homogeneous polynomial]]s of degree ''r'' on ''V''. Over [[field (mathematics)|fields]] of [[characteristic zero]], the [[graded vector space]] of all symmetric tensors can be naturally identified with the [[symmetric algebra]] on ''V''. A related concept is that of the [[antisymmetric tensor]] or [[alternating form]]. Symmetric tensors occur widely in [[engineering]], [[physics]] and [[mathematics]].
==Definition==
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