Remark:—
It is convenient to treat the scalar Invariant either as a contravariant or a co-variant tensor of zero rank.
Mixed tensor. We can also define a tensor of the second rank of the type
(12) Aμν = AμBν
which is co-variant with reference to μ and contravariant with reference to ν. Its transformation law is
"(13)."
Naturally there are mixed tensors with any number of co-variant indices, and with any number of contra-variant indices. The co-variant and contra-variant tensors can be looked upon as special cases of mixed tensors.
Symmetrical tensors:—
A contravariant or a co-variant tensor of the second or higher rank is called symmetrical when any two components obtained by the mutual interchange of two indices are equal. The tensor Aμν or Aμν is symmetrical, when we have for any combination of indices