Formulae analogous to~\Eq{(34.3)} can be obtained for the second derivatives of a tensor~$A_{...\mu..}$ instead of for a vector~$A_{\mu}$. The result is easily found to be \[ A_{...\mu..\nu\sigma} - A_{...\mu..\sigma\nu} = \sum B_{\mu\sigma\nu}^{\epsilon} A_{...\epsilon..}, \Tag{(34.8)} \] the summation being taken over all the suffixes~$\mu$ of the original tensor. The corresponding formulae for contravariant tensors follow at once, since the $g^{\mu\nu}$ behave as constants in covariant differentiation, and suffixes may be raised on both sides of~\Eq{(34.8)}. \PageSep{74} \Section{35.}{Miscellaneous formulae} The following are needed for subsequent use--- \\ Since \begin{gather*} g_{\mu\nu} g^{\mu\alpha} = \text{$0$ or $1$,} \\ g^{\mu\alpha} \, dg_{\mu\nu} + g_{\mu\nu}\, dg^{\mu\alpha} = 0. \end{gather*} Hence \begin{align*} g^{\mu\alpha}\, g^{\nu\beta} \, dg_{\mu\nu} &= -g_{\mu\nu}\, g^{\nu\beta}\, dg^{\mu\alpha} = -g_{\mu}^{\beta}\, dg^{\mu\alpha} \\ &= -dg^{\alpha\beta}. \Tag{(35.11)} \end{align*} Similarly \[ dg_{\alpha\beta} = -g_{\mu\alpha} g_{\nu\beta}\, dg^{\mu\nu}. \Tag{(35.12)} \] Multiplying by~$A^{\alpha\beta}$, we have by the rule for lowering suffixes \begin{align*} A^{\alpha\beta}\, dg_{\alpha\beta} &= -(g_{\mu\alpha} g_{\nu\beta}A^{\alpha\beta})\, dg^{\mu\nu} \\ &= -A_{\mu\nu}\, dg^{\mu\nu} = -A_{\alpha\beta}\, dg^{\alpha\beta}. \Tag{(35.2)} \end{align*} For any tensor~$B_{\alpha\beta}$ other than the fundamental tensor the corresponding formula would be \[ A^{\alpha\beta}\, dB_{\alpha\beta} = A_{\alpha\beta}\, dB^{\alpha\beta} \] by~\Eq{(26.3)}. The exception for $B_{\alpha\beta} = g_{\alpha\beta}$ arises because a change~$dg_{\alpha\beta}$ has an additional indirect effect through altering the operation of raising and lowering suffixes. Again $dg$~is formed by taking the differential of each~$g_{\mu\nu}$ and multiplying by its co-factor~$g · g^{\mu\nu}$ in the determinant. Thus \[ \frac{dg}{g} = g^{\mu\nu}\, dg_{\mu\nu} = -g_{\mu\nu}\, dg^{\mu\nu}. \Tag{(35.3)} \] The contracted $3$-index symbol \index{Three-index symbol!contracted}% \begin{align*} \{\mu\sigma, \sigma\} &= \tfrac{1}{2} g^{\sigma\lambda} \left\{ \frac{\dd g_{\mu\lambda}}{\dd x_{\sigma}} + \frac{\dd g_{\sigma\lambda}}{\dd x_{\mu}} - \frac{\dd g_{\mu\sigma}}{\dd x_{\lambda}}\right\} \\ &= \tfrac{1}{2} g^{\sigma\lambda}\, \frac{\dd g_{\sigma\lambda}}{\dd x_{\mu}}. \end{align*} The other two terms cancel by interchange of the dummy suffixes $\sigma$ and~$\lambda$. Hence by~\Eq{(35.3)} \begin{align*} \{\mu\sigma, \sigma\} &= \frac{1}{2g}\, \frac{\dd g}{\dd x_{\mu}} \\ &= \frac{\dd}{\dd x_{\mu}} \log \sqrt{-g}. \Tag{(35.4)} \end{align*} We use $\sqrt{-g}$ because $g$~is always negative for real coordinates.