Page 95 - Vector Analysis

§3.5 The Change of Variables Formula 91

Therefore, with dxxj denoting dx1 ¨ ¨ ¨ dxj´1dxj+1 ¨ ¨ ¨ dxn, the Fubini theorem and the Piola
identity imply that

ż [ ˝ ] dx = n żR żR ¨ ¨ ¨ żR B (g ˝ ψ) JAj1 dxj dxxj
(f ψ)J (x) B xj
D ÿ ´R ´R ´R

j=1

= n żR żR ¨ ¨ ¨ żR [ ˝ ψ )JAj1]ˇˇˇxxjj =R dxxj .
(g =´R
ÿ ´R ´R ´R

j=1

Since ψ = Id outside B(0, r), we find that J = 1 and Aj1 = δ1j on B D; thus by the definition
of g,

ż[ ] żR żR żR ż

(f ˝ ψ)J (x) dx = ¨ ¨ ¨ g(R, x2, ¨ ¨ ¨ , xn) dyx1 = f (x) dx . ˝

D ´R ´R ´R D

ż1

Example 3.33. Suppose that f : [0, 1] Ñ R is Riemann integrable and (1 ´ x)f (x) dx =

ż1żx 0

5. We would like to evaluate the iterated integral f (x ´ y) dydx.

00

It is nature to consider the change of variables (u, v) = (x ´ y, x) or (u, v) = (x ´ y, y).

Suppose the later case. Then (x, y) = g(u, v) = (u + v, v); thus

Jg (u, v) = ˇˇ1 1ˇˇ = 1 .
ˇˇ0 1ˇˇ

Moreover, the region of integration is the triangle A with vertices (0, 0), (1, 0), (1, 1), and
three sides y = 0, x = 1, x = y correspond to u = 0, u + v = 1 and v = 0. Therefore, if
E denotes the triangle enclosed by u = 0, v = 0 and u + v = 1 on the (u, v)-plane, then
g(E) = A, and

ż1żx ż ż

f (x ´ y) dydx = f (x ´ y) d(x, y) = f (x ´ y) d(x, y)

00 A g(E)

ż( ) ż 1 ż 1´u

= f g1(u, v) ´ g2(u, v) |Jg(u, v)| d(u, v) = f (u) dvdu

E 00

ż1

= (1 ´ u)f (u) du = 5 .

0

Example 3.34. Let A be the triangular region with vertices (0, 0), (4, 0), (4, 2), and f :
A Ñ R be given by

f (x, y) = yax ´ 2y .