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Show that Area form is independent of parameterization?

  1. Nov 21, 2011 #1
    In differential geometry, how can one show that the area form: √(EG-F2)du[itex]\wedge[/itex]dv is independent of the choice of local parameterization?
    Here E,F,G are the coefficients of first fundamental form. Please someone gives me some ideas.
     
  2. jcsd
  3. Nov 22, 2011 #2

    Bacle2

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    Science Advisor

    Maybe if you give us the definition of E, G, F and/or how they are calculated, that
    would help.

    And, BTW : why doesn't the independence of parametrization hold for standard
    integrals in standard integration in ℝn , where the area is scaled by
    the determinant of the Jacobian J(f) of the change of variables?
     
  4. Nov 22, 2011 #3
    Actually, what I am trying to do is to change from one parameterization to another and calculate the formula by definition to see if they give the same result under different parameterization, but I am not sure I am doing something valid. For instance, E=[itex]\phi_{u}\cdot\phi_{u}[/itex] where[itex]\phi[/itex] is a local parameterization from a open set in R^2 to the surface concerned and [itex]\phi_{u}=\partial \phi / \partial u\circ\phi^{-1}[/itex] so can I write [itex]\phi_{u}=\phi_{v}\partial v/\partial u [/itex]for another parameterization? In that case I can express the thing with determinants of Jacobian and they eventually cancel out to give same result.
     
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