$$ \frac{\partial}{\partial x_{\nu}} (\psi \frac{\partial \phi}{\partial
x_{\mu}}) - \begin{Bmatrix}\mu\nu\\\tau\end{Bmatrix} (\psi
\frac{\partial \phi}{\partial x_{\tau}}) $$
Thus we get the desired proof for the four-vector, ψ ∂φ/∂_x__{μ} and
hence for any four-vectors A_{μ} 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 A_{μ} B_{ν}.
It would therefore be sufficient to deduce the expression of extension,
for one such special tensor. According to (26) we have the expressions
$$ \frac{\partial A_{\mu}}{\partial x_{\sigma}} -
\begin{Bmatrix}\sigma\mu\\\tau\end{Bmatrix} A_{\tau} $$
$$ \frac{\partial B_{\nu}}{\partial x_{\sigma}} -
\begin{Bmatrix}\sigma\mu\\\tau\end{Bmatrix} B_{\tau} $$
are tensors. Through outer multiplication of the first with B_{ν} and
the 2nd with A_{μ} we get tensors of the third rank. Their addition
gives the tensor of the third rank
(27)
$$ A_{\mu\nu\sigma} = \frac{\partial A_{\mu\nu}}{\partial x_{\sigma}} -
\begin{Bmatrix}\sigma\mu\\\tau\end{Bmatrix} A_{\tau\nu} -
\begin{Bmatrix}\sigma\nu\\\tau\end{Bmatrix} A_{\mu\tau} $$
where A_{μν} is put = A_{μ} B_{ν}. The right hand side of (27) is linear
and homogeneous with reference to A_{μν}, 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 A_{μ} B_{ν} 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 A_{μνσ} the extension of the tensor
A_{μν}. 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.
Some special cases of Particular Importance.
_A few auxiliary lemmas concerning the fundamental tensor._ We shall
first deduce some of the lemmas much used afterwards. According to the
law of differentiation of determinants, we have
(28) _dg_ = _g_^{μν} _g dg__{μν} = -_g__{μν} _gdg_^{μν}.