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I Metric tensor derivation

  1. Apr 17, 2017 #1
    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?
     
  2. jcsd
  3. Apr 18, 2017 #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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