Page:The principle of relativity (1920).djvu/179

 rank. Not every such tensor can be built from two four-vectors, (according to 8). But it is easy to show that any 16 quantities A^{μν}, can be represented as the sum of A^μ B^ν of properly chosen four pairs of four-vectors. From it, we can prove in the simplest way all laws which hold true for the tensor of the second rank defined through (9), by proving it only for the special tensor of the type (8).

Contravariant Tensor of any rank:—If is clear that corresponding to (8) and (9), we can define contravariant tensors of the 3rd and higher ranks, with 4^3, etc. components. Thus it is clear from (8) and (9) that in this sense, we can look upon contravariant four-vectors, as contravariant tensors of the first rank. Co-variant tensor.

If on the other hand, we take the 16 products A_{μν} of the components of two co-variant four-vectors A_{μ} and B_{ν},

(10) A_{μν} = A_{μ} B_{ν}.

for them holds the transformation law

(11) A_{στ´} = [part]x_{μ}/[part]x_{σ´} · [part]x_{ν}/[part]x_{τ´} A_{μν}.

By means of these transformation laws, the co-variant tensor of the second rank is defined. All re-marks which we have already made concerning the contravariant tensors, hold also for co-variant tensors.

Remark:—

It is convenient to treat the scalar Invariant either as a contravariant or a co-variant tensor of zero rank.