Jacobian determinant in multiple integration

[Bipolarity]

In what kind of math course would one learn the proof of the theorem that introduces the Jacobian to computing multiple integrals under various transformations?

My calculus textbook has this theorem, and uses it to derive the triple integral formulas for cylindrical and spherical coordinates, and the double integral formulas for polar coordinates. It also uses it to show how many integration problems can be solved much more easily by applying transformations and using the Jacobian.

But it does not give a proof. In what kind of textbook would I find a proof of this theorem?
Thanks.

BiP

 

[micromass]

"Calculus on manifolds" by Spivak contains a proof of the theorem.

 

[rollingstein]

Mary Boas gives one too.

 

[micromass]

Mary Boas gives one too.

Not in my version of Boas.

 

[rollingstein]

Not in my version of Boas.

I might be wrong.

 

[MechanicalEngr]

Here's my own proof I did in a vector analysis course I took. Cheers.