Double integral change of variable

[philnow]

Homework Statement

Hey all. The problem is to solve the double integral xy da where the constraints C is x^2 + y^2 = 1, with the change of variables x = u^2 - v^2 and y = 2uv

The problem is applying the change of variables to the constraint unit circle. After the algebra I end up with (u²+v²)^2 = 1. What shape does this represent? How can I find the constraints for U and V given this... odd... equation?

 

[LCKurtz]

I suppose you mean the domain to by $x^2 + y^2 \leq 1$, the interior of the circle. That substitution seems strange to me for this problem too. Your equation (u²+v²)² = 1 is equivalent to (u²+v²) = 1, and the interior of the xy circle maps to the interior of the uv circle. Throw in the Jacobian and it seems like the transformed integral is worse than the original. I am a bit curious where the problem came from and why that substitution is suggested.

 

[Dick]

If (u^2+v^2)^2=1 then u^2+v^2=1. It looks to me like the u,v domain is still the unit disk. In terms of complex numbers it's just the mapping f(z)=z^2. But you don't have to know that.