   Chapter 16.4, Problem 31E

Chapter
Section
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: ∬ R d x   d y = ∬ S | ∂ ( x ,   y ) ∂ ( u ,   v ) |   d u   d v 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 Rdxdy=S|(x,y)(u,v)|dudv .

Explanation

Given data:

The transformations are,

x=g(u,v)

y=h(u,v) (1)

Formula used:

Write the expression for Green’s theorem.

A(R)=Rxdy (2)

Write the expression for area of region R (A(R)) .

A(R)=Rdxdy (3)

Equate the equations (2) and (3).

Rdxdy=Rxdy (4)

Consider the region is positively oriented with surface S .

Substitute S for R in equation (4),

Rdxdy=Sxdy (5)

Find the value of dy by the equation (1).

dydudv=ddudv(h(u,v))dydudv=1dv(hu)+1du(hv)dy=dudvdv(hu)+dudvdu(hv)dy=hudu+hvdv

Substitute g(u,v) for x and hudu+hvdv for dy in equation (5),

Rdxdy=Sg(u,v)(hudu+hvdv)=Sg(u,v)hudu+g(u,v)hvdv=S(g(u,v)hu)du+(g(u,v)hv)dv

Apply Green’s theorem SPdx+Qdy=S(QxPy)dA .

Rdxdy=S(u(g(u,v)hv)v(g(u,v)hu))dA

Apply chain rule of integration

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