Page:The principle of relativity (1920).djvu/183

 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.