$$ \frac{\partial g^{\mu\nu}}{\partial x_{\sigma}} = - ( g^{\mu\tau} \begin{Bmatrix}\tau & & \sigma\ \nu&\end{Bmatrix} + g^{\nu\tau} \begin{Bmatrix}\tau & & \sigma\ \mu&\end{Bmatrix} ) $$ By substituting the right-hand side of (34) in (29), we get (29a) $$ \frac{1}{\sqrt{-g}} \frac{\partial \sqrt{-g}}{\partial x_{\sigma}} = \begin{Bmatrix}\mu \sigma\\\mu\end{Bmatrix} $$ _Divergence of the contravariant four-vector._ Let us multiply (26) with the contravariant fundamental tensor _g_^{μν} (inner multiplication), then by a transformation of the first member, the right-hand side takes the form (A) $$ \frac{\partial}{\partial x_{\nu}} (g^{\mu\nu} A_{\mu}) - A_{\mu} \frac{\partial g^{\mu\nu}}{\partial x_{\nu}} - \frac{1}{2} g^{\tau\alpha} (\frac{\partial g_{\mu\alpha}}{\partial x_{\nu}} + \frac{\partial g_{ u\alpha}}{\partial x_{\mu}} - \frac{\partial g_{\mu\nu}}{\partial x_{\alpha}}) g^{\mu\nu} A_{\tau} $$ According to (31) and (29), the last member can take the form (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