Metric tensor derivation

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tc14.png



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?
 

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  • #2
jambaugh
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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.
[tex]\lim_{x\to 0} x\hat{\imath}= \vec{0},\quad \lim_{y\to 0} y\hat{\jmath} = \vec{0}[/tex]
this does not imply [itex]\hat{\imath} = \hat{\jmath}[/itex].
 

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