PHYMAT.PRO - Diego E. Gallardo

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Tensor criteria

Quotient Theorem:

Sometimes it is necessary to establish whether or not an entity T is a tensor. By definition T is a tensor only if T obeys the transformation rules for tensors. However, verifying this can be a long process. An alternative method consists in the Quotient Theorem: if the internal product TV is a tensor for any given contravariant vector V, then T is a tensor.

There are a few useful derivations from this theorem:

If is an invariant for every contravariant vector Vⁱ, then T_i is a covariant tensor (order 1).
If are components of a covariant vector for every contravariant vector Vⁱ, then T_ij is a covariant tensor (order 2).
If is an invariant for every two contravariant vectors Uⁱ and Vⁱ, then T_ij is a covariant tensor (order 2).
If T_ij is symetrical and is an invariant for every contravariant vector Vⁱ, then T_ij is a covariant tensor (order 2).