of tensor according to the generalisation of (12) in combination with (6) or out of the generalisation of (13).
Inner and mixed multiplication of Tensors.
This consists in the combination of outer multiplication
with reduction. Examples:—From the co-variant tensor of
the second rank A_{μν} and the contravariant tensor of
the first rank B^{σ} we get by outer multiplication the
mixed tensor
D^σ_{μν} = A_{μν} B^{σ} .
Through reduction according to indices ν and σ (i.e., putting ν = σ), the co-variant four vector
D_μ = D^{ν}_{μν} = A_{μν} B^{ν} is generated.
These we denote as the inner product of the tensor A_{μν} and B^{σ}. Similarly we get from the tensors A_{μν} and B^{στ} through outer multiplication and two-fold reduction the inner product A_{μν} B^{μν}. Through outer multiplication and one-fold reduction we get out of A_{μν} and B^{στ}, the mixed tensor of the second rank D^{τ}_{μ} = A_{μν} B^{τν}. We can fitly call this operation a mixed one; for it is outer with reference to the indices μ and τ and inner with respect to the indices ν and σ.
