The second covariant derivative of~$A_{\mu}$ is found by inserting in~\Eq{(30.3)} the value of~$A_{\mu\nu}$ from~\Eq{(29.3)}. This gives %[** TN: Rebreaking] \begin{align*} A_{\mu\nu\sigma} &= \frac{\dd}{\dd x_{\sigma}} \left(\frac{\dd A_{\mu}}{\dd x_{\nu}} - \{\mu\nu, \alpha\}\, A_{\alpha}\right) \\ &\qquad\qquad\begin{aligned} &- \{\mu\sigma, \alpha\} \left(\frac{\dd A_{\alpha}}{\dd x_{\nu}} - \{\alpha\nu, \epsilon\}\, A_{\epsilon}\right) \\ &- \{\nu\sigma, \alpha\} \left(\frac{\dd A_{\mu}}{\dd x_{\alpha}} - \{\mu\alpha, \epsilon\}\, A_{\epsilon}\right) \end{aligned}\displaybreak[0] \\ &= \frac{\dd^{2} A_{\mu}}{\dd x_{\sigma}\, \dd x_{\nu}} \begin{aligned}[t] &- \{\mu\nu, \alpha\}\, \frac{\dd A_{\alpha}}{\dd x_{\sigma}} - \{\mu\sigma, \alpha\}\, \frac{\dd A_{\alpha}}{\dd x_{\nu}} \\ &- \{\nu\sigma, \alpha\}\, \frac{\dd A_{\mu}}{\dd x_{\alpha}} + \{\nu\sigma, \alpha\}\, \{\mu\alpha, \epsilon\}\, A_{\epsilon} \end{aligned} \\ &+ \{\mu\sigma, \alpha\}\, \{\alpha\nu, \epsilon\}\, A_{\epsilon} - A_{\alpha}\, \frac{\dd}{\dd x_{\sigma}} \{\mu\nu, \alpha\}. \Tag{(34.1)} \end{align*} The first five terms are unaltered when $\nu$ and~$\sigma$ are interchanged. The last two terms may be written, by changing the dummy suffix~$\alpha$ to~$\epsilon$ in the last term, \[ A_{\epsilon}\left(\{\mu\sigma, \alpha\}\, \{\alpha\nu, \epsilon\} - \frac{\dd}{\dd x_{\sigma}} \{\mu\nu, \epsilon\}\right). \] \PageSep{72} Hence %[** TN: Re-breaking] \begin{align*} A_{\mu\nu\sigma} - A_{\mu\sigma\nu} &= A_{\epsilon}\biggl(\{\mu\sigma, \alpha\} \{\alpha\nu, \epsilon\} - \frac{\dd}{\dd x_{\sigma}} \{\mu\nu, \epsilon\} \\ &\qquad - \{\mu\nu, \alpha\} \{\alpha\sigma, \epsilon\} - \frac{\dd}{\dd x_{\nu}} \{\mu\sigma, \epsilon\}\biggr). \Tag{(34.2)} \end{align*} The rigorous quotient theorem shows that the co-factor of~$A_{\epsilon}$ must be a tensor. Accordingly we write \[ A_{\mu\nu\sigma} - A_{\mu\sigma\nu} = A_{\epsilon} B_{\mu\nu\sigma}^{\epsilon}, \Tag{(34.3)} \] where \[ B_{\mu\nu\sigma}^{\epsilon} = \{\mu\sigma, \alpha\} \{\alpha\nu, \epsilon\} - \{\mu\nu, \alpha\} \{\alpha\sigma, \epsilon\} - \frac{\dd}{\dd x_{\nu}} \{\mu\sigma, \epsilon\} - \frac{\dd}{\dd x_{\sigma}} \{\mu\nu, \epsilon\}. \Tag{(34.4)} \] This is called the Riemann-Christoffel tensor. It is only when this tensor \index{B@$B_{\mu\nu\sigma}^{\epsilon}$ (Riemann-Christoffel tensor)}% \index{Riemann-Christoffel tensor}% vanishes that the order of covariant differentiation is permutable.