Multivariable Calc Problem (Surface Area/Integral)

In summary, using elliptic integrals to solve for a surface area is difficult and takes some time to get used to.
  • #1
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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.
 
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  • #2
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).
 
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  • #3
Thanks, I'll have a look!
 

FAQ: Multivariable Calc Problem (Surface Area/Integral)

What is multivariable calculus?

Multivariable calculus is the branch of mathematics that deals with functions of more than one variable. It involves the study of concepts such as partial derivatives, multiple integrals, and vector calculus.

What is a surface area in multivariable calculus?

In multivariable calculus, surface area refers to the measure of the total area of the surface of a three-dimensional object. It is calculated by taking the integral of the function representing the surface over a specified region.

How do you find the surface area of a three-dimensional object?

To find the surface area of a three-dimensional object, you first need to represent the surface as a function of two variables. Then, you can use a double integral to calculate the surface area over a given region.

What is an integral in multivariable calculus?

An integral in multivariable calculus is a mathematical concept that represents the accumulation of a function over a specified region. It is used to calculate quantities such as area, volume, and surface area.

How is multivariable calculus used in real life?

Multivariable calculus is used in various fields such as physics, engineering, economics, and computer science. It is used to model and analyze complex systems and phenomena, such as fluid dynamics, electricity and magnetism, optimization problems, and financial markets. It is also used in creating computer graphics and animations.

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