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Manifold bases

  1. Oct 3, 2007 #1
    1. The problem statement, all variables and given/known data
    I am trying to show that
    [tex] \vec{e'}_a = \frac{\partial x^b}{\partial x'^a} \vec{e}_b [/tex]

    where the e's are bases on a manifold and the primes mean a change of coordinates
    I can get that [tex] \frac{\partial x^a}{ \partial x'^b} dx'^b \vec{e}_a = dx'^a \vec{e'}_a [/tex] from the invariance of ds but what should I do next?

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Oct 4, 2007 #2

    George Jones

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    Staff Emeritus
    Science Advisor
    Gold Member

    You're dealing with coodinates bases, or else your expression above isn't true. Thus

    [tex]\vec{e}_b = \frac{\partial}{\partial x^b}[/tex]

    [tex]\vec{e'}_a = \frac{\partial}{\partial x'^a}.[/tex]
  4. Oct 6, 2007 #3
    Yes, I am dealing with coordinate bases.

    How can you set a basis vector equal to a partial derivative operator?
    Last edited: Oct 6, 2007
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