Understanding the Derivation of the Metric Tensor

[kidsasd987]

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

if duⁱ=aⁱ*dr=aⁱ*a^j*du_j is true, then
dr=a^j*du_j
|dr|*r^hat=|a^j|*du_j*a^jhat
where lim |dr|,|du_j|->0

which means r^hat=a^jhat. 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?

 

[jambaugh]

Your error is that the limit of two vectors can be the zero vector (and thus equal to each other) without them pointing in the same direction.
$$\lim_{x\to 0} x\hat{\imath}= \vec{0},\quad \lim_{y\to 0} y\hat{\jmath} = \vec{0}$$
this does not imply $\hat{\imath} = \hat{\jmath}$.