Change of Coordinate for V.Field in Mfld.

In summary: Thank you!In summary, the rule for coordinate change of a V.Field X_p from chart-to-chart is given by d(Phi o Phi'^-1) and d(Phi' o Phi^-1).
  • #1
WWGD
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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.
 
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  • #2
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]).
 
  • #3
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.
 
  • #4
It's just the chain rule from good old calculus.
 
  • #5
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).
 
  • #6
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.
 

1. What is a change of coordinate for a vector field in a manifold?

A change of coordinate for a vector field in a manifold refers to the process of transforming the representation of a vector field from one coordinate system to another. This is necessary when working with vector fields in curved spaces, such as manifolds, where the coordinates may vary from point to point.

2. Why is a change of coordinate necessary for vector fields in a manifold?

A change of coordinate is necessary because vector fields in a manifold are defined in terms of a local coordinate system, and the coordinates may vary from point to point. Without a change of coordinate, it would be difficult to compare or combine vector fields in different regions of the manifold.

3. How is a change of coordinate performed for a vector field in a manifold?

A change of coordinate for a vector field in a manifold is performed by using a coordinate transformation, which relates the coordinates in one system to the coordinates in another system. This transformation is typically done using a set of equations, such as the Jacobian matrix, which allows for the conversion of vector field components from one coordinate system to another.

4. What are the benefits of using a change of coordinate for vector fields in a manifold?

Using a change of coordinate for vector fields in a manifold allows for a more flexible and efficient way of working with vector fields in curved spaces. It allows for the comparison and combination of vector fields in different regions of the manifold, and also simplifies calculations by using a consistent coordinate system.

5. Are there any limitations to using a change of coordinate for vector fields in a manifold?

There are some limitations to using a change of coordinate for vector fields in a manifold. For example, the transformation may not be possible if the coordinate systems are not compatible, or if the transformation equations are not known. Additionally, a change of coordinate may introduce errors or distortions in the vector field representation.

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