Chapter 16.4, Problem 31E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

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 determine

**To prove**: The formula of double integral

Explanation

**Given data:**

The transformations are,

**Formula used:**

Write the expression for Greenâ€™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

Find the value of

Substitute

Apply Greenâ€™s theorem

Apply chain rule of integration

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