Chapter 13.4, Problem 31E

Use Green’s Theorem to prove the change of variables formula for a double integral (Formula 15.9.9) for the case where f(x, y) = 1:

${\displaystyle \underset{R}{\iint }dxdy}={\displaystyle \underset{S}{\iint }\left|\frac{\partial \left(x,y\right)}{\partial \left(u,v\right)}\right|dudv}$

Here 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).

[Hint: Note that the left side is A(R) and apply the first part of Equation 5. Convert the line integral over ∂R to a line integral over ∂S and apply Green’s Theorem in the uv-plane.]