Confused about the metric tensor

[GarageDweller]

Now let's say I have the metric for some curved two surface

ds^2=G(u,v)du^2+P(u,v)dv^2 ( the G and P functions being the 00 and 11 components, assuming the metric is diagonal)

Now my question is, since the metric defines the scalar product of two vectors, let's say
(1,0) and (0,1), for simplicity, which values do I take for u and v in G(u,v) and P(u,v)?

 

[Matterwave]

u and v would be the coordinates of wherever you are evaluating the scalar product.

In other words, the coordinates of where the vectors are located.