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## Main Question or Discussion Point

Now lets 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, lets 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)?

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, lets 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)?