The above formulae are primarily definitions; but we have to prove that the quantities on the right are actually tensors. This is done by an obvious generalisation of the method of the preceding section. Thus if in place of~\Eq{(29.1)} we use \[ \frac{d}{ds} \left(A_{\mu\nu}\, \frac{dx_{\mu}}{ds}\, \frac{dx_{\nu}}{ds}\right) \text{ is invariant along a geodesic,} \] we obtain \[ \frac{\dd A_{\mu\nu}}{\dd x_{\sigma}}\, \frac{dx_{\sigma}}{ds}\, \frac{dx_{\mu}}{ds}\, \frac{dx_{\nu}}{ds} + A_{\mu\nu}\, \frac{dx_{\nu}}{ds}\, \frac{d^{2}x_{\mu}}{ds^{2}} + A_{\mu\nu}\, \frac{dx_{\mu}}{ds}\, \frac{d^{2}x_{\nu}}{ds^{2}}. \] Then substituting for the second derivatives from~\Eq{(28.5)} the expression reduces to \[ A_{\mu\nu\sigma}\, \frac{dx_{\mu}}{ds}\, \frac{dx_{\nu}}{ds}\, \frac{dx_{\sigma}}{ds} \text{ is invariant,} \] showing that $A_{\mu\nu\sigma}$~is a tensor. The formulae \Eq{(30.1)}~and \Eq{(30.2)} are obtained by raising the suffixes $\nu$ and~$\mu$, the details of the work being the same as in deducing \Eq{(29.4)} from~\Eq{(29.3)}. Consider the expression \[ B_{\mu\sigma} C_{\nu} + B_{\mu} C_{\nu\sigma}, \] the $\sigma$ denoting covariant differentiation. By~\Eq{(29.3)} this is equal to \begin{multline*} \left(\frac{\dd B_{\mu}}{\dd x_{\sigma}} - \{\mu\sigma, \alpha\}\, B_{\alpha}\right) C_{\nu} + B_{\mu} \left(\frac{\dd C_{\nu}}{\dd x_{\sigma}} - \{\nu\sigma, \alpha\}\, C_{\alpha}\right) \\ = \frac{\dd}{\dd x_{\sigma}}\, (B_{\mu} C_{\nu}) - \{\mu\sigma, \alpha\}\, (B_{\alpha} C_{\nu}) - \{\nu\sigma, \alpha\}\, (B_{\mu} C_{\alpha}). \end{multline*} \PageSep{63} But comparing with~\Eq{(30.3)} we see that this is the covariant derivative of the \index{Covariant derivative of vector!of invariant}% \index{Covariant derivative of vector!utility of}% tensor of the second rank~$(B_{\mu}C_{\nu})$. Hence \[ (B_{\mu} C_{\nu})_{\sigma} = B_{\mu\sigma} C_{\nu} + B_{\mu} C_{\nu\sigma}. \Tag{(30.5)} \] Thus in covariant differentiation of a product the distributive rule used in ordinary differentiation holds good. Applying~\Eq{(30.3)} to the fundamental tensor, we have \begin{align*} g_{\mu\nu\sigma} &= \frac{\dd g_{\mu\nu}}{\dd x_{\sigma}} - \{\mu\sigma, \nu\}\, g_{\alpha\nu} - \{\nu\sigma, \alpha\}\, g_{\mu\alpha} \\ &= \frac{\dd g_{\mu\nu}}{\dd x_{\sigma}} - [\mu\sigma, \nu] - [\nu\sigma, \mu] \\ &= 0 \text{ by \Eq{(27.5)}.} \end{align*} Hence the covariant derivatives of the fundamental tensors vanish identically, and the fundamental tensors can be treated as \emph{constants} in covariant differentiation. It is thus immaterial whether a suffix is raised before or after the differentiation, as our definitions have already postulated.