Show that Area form is independent of parameterization?

  • Thread starter phyalan
  • Start date
  • #1
22
0
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.
 

Answers and Replies

  • #2
Bacle2
Science Advisor
1,089
10
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.
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?
 
  • #3
22
0
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.
 

Related Threads on Show that Area form is independent of parameterization?

  • Last Post
Replies
3
Views
4K
Replies
2
Views
2K
Replies
4
Views
1K
Replies
7
Views
839
Replies
2
Views
2K
Replies
1
Views
7K
  • Last Post
Replies
1
Views
3K
Replies
6
Views
861
Replies
1
Views
2K
Top