# Show that Area form is independent of parameterization?

• phyalan
In summary, the question is how to show that the area form in differential geometry, √(EG-F2)du\wedgedv, is independent of the choice of local parameterization, where E, F, and G are coefficients of the first fundamental form. The poster is asking for ideas on how to approach this problem, and also wondering why this independence does not hold for standard integrals in ℝn. They are attempting to change from one parameterization to another and calculate the formula by definition to see if they get the same result. This involves using the determinants of Jacobian matrices, which eventually cancel out to give the same result.

#### phyalan

In differential geometry, how can one show that the area form: √(EG-F2)du$\wedge$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.

phyalan said:
In differential geometry, how can one show that the area form: √(EG-F2)du$\wedge$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?

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=$\phi_{u}\cdot\phi_{u}$ where$\phi$ is a local parameterization from a open set in R^2 to the surface concerned and $\phi_{u}=\partial \phi / \partial u\circ\phi^{-1}$ so can I write $\phi_{u}=\phi_{v}\partial v/\partial u$for another parameterization? In that case I can express the thing with determinants of Jacobian and they eventually cancel out to give same result.