Multivariable Calc Problem (Surface Area/Integral)

[nietzsche]

Homework Statement

Find the surface area of the ellipsoid (x/a)^2 + (y/a)^2 + (z/b)^2 = 1.

Homework Equations

If G(u,v) is a map from R2 to R3 that parametrizes the surface, then the area of the surface is equal to the double integral over the domain of G of the norm of the cross product of ∂_uG and ∂_vG.

The Attempt at a Solution

Well, I can see that we can parametrize the surface with a G whose domain is the circle x^2 + y^2 = a^2. I've tried using polar coordinates, I've tried first transforming it into a sphere and then into polar coordinates, but I just can't seem to get an integral that I can work with.

If someone could perhaps give me a push in the right direction (maybe a parametrization to try out), I would really appreciate it.

 

[sshzp4]

You need to understand elliptic integrals. Wolfram site and wikipedia entry on the elliptic have sufficient information for a solution and you can get some general elliptic manipulation strategies + history from http://everything2.com/title/elliptic+integral+standard+forms".

To derive the actual series solution of the elliptic you will have to derive the series solution to the integral of sin^(2n) theta between 0 to pi/2, transform the elliptic integral to a series representation in terms of sin^(2n) theta, then use the previously derived expression, finally sum. This might take some time the first time. However, most people just use the incomplete elliptic integral of the first kind solution directly. That is also very reasonable.

Edit: If I remember correctly, Einstein played with this problem (with his first wife), while in school. I don't think he ended up making any particularly noteworthy contributions. Most of the current formulation is still from Legendre, Weistrauss, Gauss, Jacobi. Everyone still relies on either the series form or the numerical arithmetic geometric mean technique (its a cool algorithm for calculating values to arbitrary precision) for quoting solutions to the elliptic. The series form is used extensively in the derivation of analytical solutions in engineering (I have seen a lot appear in solids).

 

[nietzsche]

Thanks, I'll have a look!