# Metric tensor derivation

• I

Hello, I have a question regarding the first equation above.
it says dui=ai*dr=ai*aj*duj but I wonder how. (sorry I omitted vector notation because I don't know how to put them on)

if dui=ai*dr=ai*aj*duj is true, then
dr=aj*duj
|dr|*rhat=|aj|*duj*ajhat
where lim |dr|,|duj|->0

which means rhat=ajhat.

are both the directions actually the same or do we just don't care about directions because both will eventually be small increment of each component?

$$\lim_{x\to 0} x\hat{\imath}= \vec{0},\quad \lim_{y\to 0} y\hat{\jmath} = \vec{0}$$
this does not imply $\hat{\imath} = \hat{\jmath}$.