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Proof of the Jacobian

  1. Dec 10, 2005 #1
    Can someone explain to me this part of the proof of the jacobian?

    I dont know what they're talking about...I can follow the rest (the cross product bla bla bla bla bla) but I dont know how they're getting these two vectors...I figured it has something to do with partial differentials but Im still confused. If anyone could provide any insight Id be appreciative. :eek:
     
  2. jcsd
  3. Dec 10, 2005 #2

    Galileo

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    The notation is a bit screwy, but here's what I think they're doing.

    So suppose the surface is parametrised by [itex]\vec T(u,v)[/itex].
    Take a small rectangle in the domain with dimensions [itex]\Delta u, \Delta v[/itex], the bottom left corner being the point [itex](u_0,v_0)[/itex].
    The image of this rectangle is a patch of area, which can be approximated by the parallelogram formed by the vectors:

    [tex]\vec T(u_0+\Delta u,v_0)-\vec T(u_0,v_0)[/tex]
    and
    [tex]\vec T(u_,v_0+\Delta v)-\vec T(u_0,v_0)[/tex] (a picture helps here).

    These vectors are in turn approximated by
    [tex]\frac{\partial}{\partial u}\vec T(u_0,v_0)\Delta u[/tex]
    and
    [tex]\frac{\partial}{\partial v}\vec T(u_0,v_0)\Delta v[/tex]
    respectively.

    So area patch is about [itex]|\vec T_u \times \vec T_v|\Delta u \Delta v[/itex] and you can figure out the rest.

    Hope that helps.
     
    Last edited: Dec 10, 2005
  4. Dec 10, 2005 #3

    Hurkyl

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    It might help (me, anyways) if you would say what you're trying to prove. The Jacobian is a number associated with a matrix; it doesn't make any more sense to ask about a proof of the Jacobian than it does to ask about a proof of the number 2.
     
  5. Dec 11, 2005 #4
    Oh, I see it better now. THanks a lot, just wanted to say that before I go to bed. If i need further clarification Ill post the fool proof. THakns a lot guys
     
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