Mathematical physics line/surface integral problem

[flaxstrax]

Homework Statement

i have to show that the length of line integral squared on surface (ds^2) , which has vectorequation r=r(u,v) (r here is a vector), can be given as ds^2=Edu^2+2Fdudv+Gdv^2 (Find functions E,G,F)

Homework Equations

I should use diferential dr=du(t)e1+dv(t)e2 (dr, e1 and e2 here are vectors )
ds=|dr/dt|dr (dr here is vector)
dS=dS/n (dS/n is a vector )

The Attempt at a Solution

I'm pretty much stuck at the beginning , i don't see how those relevant equations help me here ?
Probably i should do something with dS/n.
Sorry for bad english and possibly translation, point out mistakes so i can learn.

 

[mighty2000]

Thank you for your post. To solve this problem, we need to use the fundamental theorem of line integrals, which states that the line integral of a vector field F along a curve C is equal to the surface integral of the curl of F over any surface S bounded by C. In this case, our surface is defined by the vector equation r=r(u,v), where r is a vector and u and v are parameters. To find the functions E, G, and F, we need to find the components of the vector r and then use the fundamental theorem of line integrals to find the surface integral of the curl of r over the surface S.

To find the components of r, we can use the differential dr=du(t)e1+dv(t)e2, where e1 and e2 are unit vectors in the u and v directions, respectively. This will give us the components of r in terms of u and v. Then, we can use the surface integral of the curl of r to find the functions E, G, and F. The surface integral is given by dS=|dr/dt|dr, where dS is a vector and dr is the differential of the vector r. By substituting the components of r into this equation, we can solve for E, G, and F.

I hope this helps you to solve the problem. If you need further assistance, please let me know. Good luck with your research!