If $I$~is an invariant, $IA_{\mu}$~is a covariant vector; hence its covariant derivative is \begin{align*} (IA_{\mu})_{\nu} &= \frac{\dd}{\dd x_{\nu}}(IA_{\mu}) - \{\mu\nu, \alpha\}\, IA_{\alpha} \\ &= A_{\mu}\, \frac{\dd I}{\dd x_{\nu}} + IA_{\mu\nu}. \end{align*} But by the rule for differentiating a product~\Eq{(30.5)} \[ (IA_{\mu})_{\nu} = I_{\nu} A_{\mu} + IA_{\mu\nu}, \] so that \[ I_{\nu} = \frac{\dd I}{\dd x_{\nu}}. \] Hence the covariant derivative of an invariant is the same as its ordinary derivative. It is, of course, impossible to reserve the notation~$A_{\mu\nu}$ exclusively for the covariant derivative of~$A_{\mu}$, and the concluding suffix does not denote differentiation unless expressly stated. In case of doubt we may indicate the covariant and contravariant derivatives by $(A_{\mu})_{\nu}$ and~$(A_{\mu})^{\nu}$. The utility of the covariant derivative arises largely from the fact that, when the~$g_{\mu\nu}$ are constants, the $3$-index symbols vanish and the covariant derivative reduces to the ordinary derivative. Now in general our physical equations have been stated for the case of Galilean coordinates in which the~$g_{\mu\nu}$ are constants; and we may in Galilean equations replace the ordinary derivative by the covariant derivative without altering anything. This is a necessary step in reducing such equations to the general tensor form which holds true for all coordinate-systems. As an illustration suppose we wish to find the general equation of propagation \PageSep{64} \index{Propagation with unit velocity}% of a potential with the velocity of light. In Galilean coordinates the equation is of the well-known form \[ \Wave\phi = \frac{\dd^{2} \phi}{\dd t^{2}} - \frac{\dd^{2} \phi}{\dd x^{2}} - \frac{\dd^{2} \phi}{\dd y^{2}} - \frac{\dd^{2} \phi}{\dd z^{2}} = 0. \Tag{(30.6)} \]