Page 93 - Vector Analysis

§3.5 The Change of Variables Formula 89

3.5 The Change of Variables Formula

Fubini theorem can be used to find the integral of a (Riemann integrable) function over a
rectangular domain if the iterated integrals can be evaluated. However, like the integral of
a function of one variable, in many cases we need to make use of several change of variables
in order to transform the integral to another integral that is easier to be evaluated. In this
section, we establish the change of variables formula for the integral of functions of several
variables.

Theorem 3.31 (Change of Variables Formula). Let U Ď Rn be an open set with volume,
and ψ : U Ñ Rn be an one-to-one C 1-mapping with C 1-inverse; that is, ψ´1 : ψ(U ) Ñ U
is also continuously differentiable. Assume that the Jacobian of ψ, J = det([Dψ]), does not
vanish in U. If f : ψ(U) Ñ R is Riemann integrable, then (f ˝ ψ)J is Riemann integrable
over U, and

ż f (y) dy = ż ˝ ψ)(x)ˇˇJ(x)ˇˇ dx = ż ˝ ˇ B (ψ1, ¨ ¨ ¨ , ψn) ˇ dx .
ψ)(x)ˇ ¨ ¨ ¨ , ˇ
ψ(U ) (f (f
ˇ B (x1, xn) ˇ
U U

The proof of Theorem 3.31 is very lengthy and requires a bit more knowledge about the
integration, so we only present the proof of a much simpler case.

Theorem 3.32. Let D Ď Rn be an open rectangle, and ψ : Rn Ñ Rn be an one-to-one C 2

mapping such that ψ = Id outside B(0, r) for some r ą 0; that is, ψ(x) = x if |x| ě r.

Assume that the Jacobian of ψ, J = det(∇ψ), does not vanish in Rn. If f : D Ñ R is of
class C 1 and is compactly supported in D; that is, cl(␣x )
P D ˇ f (x) ‰ 0( Ď D , then
ˇ

żż

f (y) dy = (f ˝ ψ)(x)J(x) dx .

D ψ´1(D)

Proof. W.L.O.G. we can assume that D = [´R, R]n is a cube and B(0, r)ĂĂD (or equiva-
lently, 0 ă r ă R). Then ψ´1(D) = D since ψ = Id outside B(0, R). Define

ż y1
g(y1, ¨ ¨ ¨ , yn) = f (z, y2, ¨ ¨ ¨ , yn) dz ,

´R