Prove the change of variables formula for double integrals

[alanthreonus]

Homework Statement

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.

Homework Equations

A = \oint _C xdy = -\oint _C ydx = 1/2\oint _C xdy - ydx

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.

 

[Mark44]

Homework Statement

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

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?

Homework Equations

A = \oint _C xdy = -\oint _C ydx = 1/2\oint _C xdy - ydx

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.

 

[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.