Change of Coordinate for V.Field in Mfld.

  1. WWGD

    WWGD 1,323
    Science Advisor
    Gold Member

    Hi, again:
    Just a quick question; I have "notation indigestion", i.e., I have been trying

    to figure way too many technicalities recently; I would appreciate a quick yes/no:


    Say X_p is a V.Field defined at p in a C^k manifold; k>0 . Say (U,Phi) and (U',Phi')

    are both charts containing p . Just wondering if the rule for coordinate change

    of X from chart-to-chart is; is it given by

    d(Phi o Phi'^-1) and d(Phi' o Phi^-1) ?

    I mean, I know it involves the chain rule, but I wonder if it is the chain rule

    applied to the two formulas above.

    Thanks.
     
  2. jcsd
  3. CompuChip

    CompuChip 4,298
    Science Advisor
    Homework Helper

    Think about the geometrical picture. You have patches U and U' on the manifold, which are mapped to some open subsets V and V' of Euclidean space.
    Suppose you want the coordinate change from V to V'. Suppose you have a coordinate [itex]\vec v \in V[/itex]. Then you can go to [itex]\Phi^{-1}(\vec v) \in U[/tex]. Assuming that this point is also in U', you can go to V': [itex]\Phi'( \Phi^{-1}(\vec v) ) \in V'[/itex]. You can write this as [itex]\kappa(\vec v)[/itex], where depending on your convention for composition of functions,
    [tex]\kappa = \Phi^{-1} \circ \Phi'[/tex]
    or
    [tex]\kappa = \Phi' \circ \Phi^{-1}[/tex]

    You can also go from V' to V by a similar reasoning (and in fact you will find that the coordinate change you get is [itex]\kappa^{-1}[/itex]).
     
  4. WWGD

    WWGD 1,323
    Science Advisor
    Gold Member

    Thanks, Compuchip; I don't know if I misunderstood ( or misunderestimated :smile:)
    your reply:

    But I think that both k, k' as you described them ( I am sorry, I can't make the

    'quote' function work well in here ) give me a coordinate change between the patches

    V,V' =Phi(U) and Phi'(U') respectively , as you described, but I don't see that this

    gives me a way of changing the coordinate representation of the V.Field X_p , which

    lives in U/\U' (Sorry, I am still learning Tex.).



    I know that we get the coordinate rep. ( in terms of the basis for T_pM , in

    each of the charts U,U' ) by pulling back (Thru Phi, Phi' respectively), the basis

    of T_Phi(p)R^n and T_Phi'(p) R^n respectively, to get different bases for T_pM.

    Does your k , k' give me a way of going from one basis representation of T_pR^n

    to another basis rep. of T_pR^n ?


    Thanks, and sorry for writing in ASCII. Hopefully this summer I will have time to

    learn Latex.
     
  5. It's just the chain rule from good old calculus.
     
  6. WWGD

    WWGD 1,323
    Science Advisor
    Gold Member

    Yes, Zhentil,thanks, I understand that. I was looking for what specific map
    we apply the chain rule to:

    Is it to the composition of chart maps (Phi o Phi' ^-1) ;with (U,Phi) and (U',Phi')

    overlapping charts for p ?(or, of course, the inverse of the map above, if we want to

    change in the opposite direction).
     
  7. WWGD

    WWGD 1,323
    Science Advisor
    Gold Member

    This is my issue: If M,N, are open subsets of R^n, this is easy. Now, if M,N are not
    open subsets of R^n, then , we get a basis representation for X_p as above in U,
    by pulling back the basis vectors in T_Phi(p)R^n , by the chart map Phi^-1 , and
    we get the basis rep. in the chart U' , by pulling back (using Phi'^-1) ,the basis
    vectors in T_Phi'p R^n.

    Then, to change , I think we need to push forward the vector field X_p ( in whichever
    basis) to R^n , then use the fuch forward to map this image into the other basis in R^n,
    and then pull back again. I get the idea, I think, but this is becoming a "tress v. forest"
    thing , here, and I lost track of just how to figure out the change of expression, and I
    was hoping that if someone was familiar with it, they could give me the "forest" --
    to push the analogy to its limit.
     
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