Prove the change of variables formula for double integrals

alanthreonus
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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.
 
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alanthreonus said:

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?
alanthreonus said:

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.
 
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.
 
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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