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:
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.]
To prove: The formula of double integral .
The transformations are,
Write the expression for Green’s theorem.
Write the expression for area of region R .
Equate the equations (2) and (3).
Consider the region is positively oriented with surface .
Substitute for in equation (4),
Find the value of by the equation (1).
Substitute for x and for in equation (5),
Apply Green’s theorem .
Apply chain rule of integration
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