Mixed tensor. We can also define a tensor of the second rank of the type
(12) A_{μ}^{ν} = A_{μ}B^ν
which is co-variant with reference to μ and contravariant with reference to ν. Its transformation law is
(13) A_{σ}^{τ´} = ([part]x_{τ´}/[part]x_{β}) · ([part]x_{α}/[part]x_{σ´}) A_{alpha}^{beta}.
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 A^{μν} or A_{μν} is symmetrical, when
we have for any combination of indices
(14) A^{μν} = A^{νμ}
or
(14a) A_{μν} = A_{νμ}.
It must be proved that a symmetry so defined is a property independent of the system of reference. It follows in fact from (9) remembering (14)
A^{στ´} = ([part]x_{σ´}/[part]x_{μ}) ([part]x´_{τ}?]/[part]x_{ν}) A^{μν} = ([part]x_{σ´}/[part]x_{μ}) ([part]x_{τ´}/[part]x_{ν}) A^{νμ} = A^{τσ´}
