Thus we get the desired proof for the four-vector, ψ ∂φ/∂xμ and hence for any four-vectors Aμ as shown above.
With the help of the extension of the four-vector, we can easily define “extension” of a co-variant tensor of any rank. This is a generalisation of the extension of the four-vector. We confine ourselves to the case of the extension of the tensors of the 2nd rank for which the law of formation can be clearly seen.
As already remarked every co-variant tensor of the 2nd rank can be represented as a sum of the tensors of the type Aμ Bν.
It would therefore be sufficient to deduce the expression of extension, for one such special tensor. According to (26) we have the expressions
are tensors. Through outer multiplication of the first with Bν and the 2nd with Aμ we get tensors of the third rank. Their addition gives the tensor of the third rank
"(27)"
where Aμν is put = Aμ Bν. The right hand side of (27) is linear and homogeneous with reference to Aμν, and its first differential co-efficient, so that this law of formation leads to a tensor not only in the case of a tensor of the type Aμ Bν but also in the case of a summation for all such tensors, i.e., in the case of any co-variant tensor of the second rank. We call Aμνσ the extension of the tensor Aμν. It is clear that (26) and (24) are only special cases of (27) (extension of the tensors of the first and zero rank). In general we can get all special laws of formation of tensors from (27) combined with tensor multiplication.