I am taking an introductory course in general relativity, and we are following the Carroll book called spacetime and geometry. I've found that he writes his tensor equations without specifying at which point the tensors are to be evaluated. For example, he writes $g_{\mu \nu}$ and not $g_{\mu \nu}(x)$ or $g_{\mu \nu}(x_0)$. I find myself constantly making mistakes because I don't know how to apply the equations without knowing how they are evaluated.

My question: How do I know how to evaluate tensors, in particular the metric tensor or the Christoffel symbol when applying the quations? Often it does not become clear to me from the context.

Thank you for your time.

Kind regards,
Marius