Trying to understand index notation in the context of spacetime. If I have $x^{\mu}$
  and then set $\mu=\phi$
  (for example), is it acceptable to then write $x^{\phi}$
  or should I just write $\phi$
 ? For example, can I write $$\frac{\partial g_{\mu\nu}}{\partial x^{\mu}}$$
 as $$\frac{\partial g_{\phi\nu}}{\partial x^{\phi}},$$
 or should I stick with$$\frac{\partial g_{\phi\nu}}{\partial\phi}?$$
 Thanks.