A possible pitfall in differentiating a summed expression should be noticed. The result of differentiating $a_{\mu\nu} x_{\mu} x_{\nu}$ with respect to~$x_{\nu}$ is not~$a_{\mu\nu} x_{\mu}$ but $(a_{\mu\nu} + a_{\nu\mu}) x_{\mu}$. The method of performing such differentiations may be illustrated by the following example. Let \[ h_{\nu\tau} = a_{\mu\nu} a_{\sigma\tau} x_{\mu} x_{\sigma}, \] where $a_{\mu\nu}$~represents constant coefficients. Then \begin{align*} \frac{\dd h_{\nu\tau}}{\dd x_{\alpha}} &= a_{\mu\nu} a_{\sigma\tau} \left(\frac{\dd x_{\mu}}{\dd x_{\alpha}}\, x_{\sigma} + \frac{\dd x_{\sigma}}{\dd x_{\alpha}}\, x_{\mu}\right) \\ &= a_{\mu\nu} a_{\sigma\tau} (g_{\alpha}^{\mu} x_{\sigma} + g_{\alpha}^{\sigma} x_{\mu}) \quad\text{by \Eq{(22.3)}.} \end{align*} \PageSep{75} \index{Differentiation of summed expression}% Repeating the process, \begin{align*} \frac{\dd^{2} h_{\nu\tau}}{\dd x_{\alpha}\, \dd x_{\beta}} &= a_{\mu\nu} a_{\sigma\tau} (g_{\alpha}^{\mu} g_{\beta}^{\sigma} + g_{\alpha}^{\sigma} g_{\beta}^{\mu}) \\ &= a_{\alpha\nu} a_{\beta\tau} + a_{\beta\nu} a_{\alpha\tau}. \end{align*} Hence changing dummy suffixes \[ \frac{\dd^{2}}{\dd x_{\mu}\, \dd x_{\sigma}} (a_{\mu\nu} a_{\sigma\tau} x_{\mu} x_{\sigma}) = a_{\mu\nu} a_{\sigma\tau} + a_{\sigma\nu} a_{\mu\tau}. \Tag{(35.5)} \] Similarly if $a_{\mu\nu\sigma}$~is symmetrical in its suffixes \[ \frac{\dd^{3}}{\dd x_{\mu}\, \dd x_{\nu}\, \dd x_{\sigma}} (a_{\mu\nu\sigma} x_{\mu} x_{\nu} x_{\sigma}) = 6a_{\mu\nu\sigma}. \Tag{(35.6)} \] The pitfall arises from repeating a suffix three times in one term. In these formulae the summation applies to the repetition within the bracket, and not to the differentiation. \Subsection{Summary.} Tensors are quantities obeying certain transformation laws. Their importance lies in the fact that if a tensor equation is found to hold for one system of coordinates, it continues to hold when any transformation of coordinates is made. New tensors are recognised either by investigating their transformation laws directly or by the property that the sum, difference, product or quotient of tensors is a tensor. This is a generalisation of the method of dimensions in physics.