# Prove the change of variables formula for double integrals

1. Apr 12, 2010

### alanthreonus

1. The problem statement, all variables and given/known data
Use Green's Theorem to prove for the case f(x,y) = 1

$$\int\int_R dxdy = \int\int_S |\partial(x,y)/\partial(u,v)|dudv$$

EDIT: R is the region in the xy-plane that corresponds to the region S in the uv-plane under the transformation given by x = g(u,v), y = h(u,v), and the expression on the right is the Jacobian.

2. Relevant equations
$$A = \oint _C xdy = -\oint _C ydx = 1/2\oint _C xdy - ydx$$

3. The attempt at a solution
My textbook says that the left side of the equation I'm trying to prove is A(R), so I can apply the first part of the equation in 2., but I don't understand what it's talking about because the equation in 2. deals with line integrals, and I'm trying to prove an equation with double integrals.

Last edited: Apr 12, 2010
2. Apr 12, 2010

### Staff: Mentor

What is R? S?
How are u and v related to x and y?
Isn't the expression in absolute values on the right side the determinant of the Jacobian?

3. Apr 12, 2010

### alanthreonus

R is the region in the xy-plane that corresponds to the region S in the uv-plane under the transformation given by x = g(u,v), y = h(u,v), and, yes, that is the Jacobian.