(B)
$$ \frac{1}{2} \frac{\partial g^{\tau\nu}}{\partial x_{\nu}} A_{\tau} +
\frac{1}{2} \frac{\partial g^{\mu\tau}}{\partial x_{\mu}} A_{\tau} +
\frac{1}{\sqrt{-g}} \frac{\partial \sqrt{-g}}{\partial x_{\alpha}}
g^{\mu\alpha} A_{\tau} $$
Both the first members of the expression (B), and the second member of
the expression (A) cancel each other, since the naming of the
summation-indices is immaterial. The last member of (B) can then be
united with first of (A). If we put
_g_^{μν} A_{μ} = A^{ν},
where A^{ν} as well as A_{μ} are vectors which can be arbitrarily
chosen, we obtain finally
$$ \Phi = \frac{1}{\sqrt{-g}} \frac{\partial}{\partial x_{\nu}}
(\sqrt{-g} A^{\nu}) $$
This scalar is the _Divergence_ of the contravariant four-vector A^{ν}.
_Rotation of the (covariant) four-vector._
The second member in (26) is symmetrical in the indices μ, and ν. Hence
A_{μν} - A_{νμ} is an antisymmetrical tensor built up in a very simple
manner. We obtain
∂A_{μ} ∂A_{ν}
(36) B_{μν} = -------------- - ------------
∂_x__{ν} ∂_{_x_μ}
_Antisymmetrical Extension of a Six-vector._
If we apply the operation (27) on an antisymmetrical tensor of the
second rank A_{μ{ν²}} and form all the equations arising from the cyclic
interchange of the indices μ, ν, σ, and add all them, we obtain a tensor
of the third rank
(37) B_{μνσ} = A_{μνσ} + A_{νσμ} + A_{σμν}
∂A_{μν} ∂A_{νσ} ∂A_{σμ}
= ------------ + ------------- + ------------
∂_x__{σ} ∂_x__{μ} ∂_x__{ν}
from which it is easy to see that the tensor is antisymmetrical.
_Divergence of the Six-vector._
If (27) is multiplied by _g_^{μα} _g_^{νβ} (mixed multiplication), then
a tensor is obtained. The first member of the right hand side of (27)
can be written in the form
$$ \frac{\partial}{\partial x_{\sigma}} (g^{\mu\alpha}
g^{\nu\beta} A_{\mu\nu}) - g^{\mu\alpha} \frac{\partial
g^{\nu\beta}}{\partial x_{\sigma}} A_{\mu\nu} - g^{\nu\beta}
\frac{\partial g^{\mu\alpha}}{\partial x_{\sigma}} A_{\mu\nu} $$
If we replace _g_^{μα} _g_^{νβ} A_{μνσ} by A_{σ}^{αβ}, _g_^{μα} _g_^{νβ}
A_{μν} by A^{αβ} and replace in the transformed first member
∂_g_^{νβ}/∂_x__{σ} and ∂_g_^{μα}/∂_x__{σ}
with the help of (34), then from the right-hand side of (27) there
arises an expression with seven terms, of which four cancel. There
remains
(38) $$ A^{\alpha\beta}_{\sigma} = \frac{\partial
A^{\alpha\beta}}{\partial x_{\sigma}} + \begin{Bmatrix}\sigma & &
\kappa\ \alpha end{Bmatrix} A^{\kappa\beta} + \begin{Bmatrix}\sigma & &
\kappa\ \beta&\end{Bmatrix} A^{\alpha\kappa} $$
This is the expression for the extension of a contravariant tensor of
the second rank; extensions can also be formed for corresponding
contravariant tensors of higher and lower ranks.
We remark that in the same way, we can also form the extension of a
mixed tensor A_{μ}^{α}
(39) $$ A^{\alpha}_{\mu\sigma} = \frac{\partial
A^{\alpha}_{\mu}}{\partial x_{\sigma}} - \begin{Bmatrix}\sigma & &
\mu\ \tau&\end{Bmatrix} A^{\alpha}_{\tau} + \begin{Bmatrix}\sigma & &
\tau\ \alpha&\end{Bmatrix} A^{\tau}_{\mu} $$
By the reduction of (38) with reference to the indices β and σ(inner
multiplication with δ_{β}^{σ}), we get a contravariant four-vector
$$ A^{\alpha} = \frac{\partial A^{\alpha\beta}}{\partial x_{\beta}} +
\begin{Bmatrix}\beta & & \kappa\ \beta&\end{Bmatrix} A^{\alpha\kappa} +
\begin{Bmatrix}\beta & & \kappa\ \alpha&\end{Bmatrix} A^{\kappa\beta} $$