(923, #31) Use Green's theorem to prove the change of variables formula for a double integral for the case where

Here R is the region in the xy-plane that corresponds to the region S in the uv-plane under the transformation given by

[Hint : Note that the left side is A ( R ) and apply the first part of equation 6. Convert the line integral over R to a line integral over S and apply Green's theorem in the uv-plane.]

Solution: From the first part of equation ( 6 ), we know .

Set Orient positively. Then

With Green's theorem, this becomes

Applying the Chain Rule we obtain

The last equation is a consequence of the mixed partials being equal.

Since x = g ( u, v ) and y = h ( u, v ) , the last integral is equal to

By C.A.H.