Chapter 16.4, 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:

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

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.]