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7
Integration on Manifolds
The integral of an n-form on an n-manifold is defined by piecing together integrals over sets in Rn using
a partition of unity subordinate to an atlas. The change of variables theorem guarantees that the integral
is well defined, independent of the choice of atlas and partition of unity. Two basic theorems of integral
calculus, the change of variables theorem and Stokes’ theorem, are discussed in detail along with some
applications.
The Definition of the Integral
The aim of this section is to define the integral of an n-form on an oriented n-manifold M and prove a few
of its basic properties. We begin with a summary of the basic results in Rn.
Integration on Rn. Suppose f : Rn → R is continuous and pact support. Then fdx1 ...dxn is
defined to be the Riemann integral over any rectangle containing the support of f.
Definition. Let U ⊂ Rn be open and ω∈Ωn(U) pact support. If, relative to the standard
basis of Rn,
1
ω(x)= ω(x)dxi1 ∧···∧dxin = ω(x)dx1 ∧···∧dxn,
n! i1...in 1...n
where ponents of ω are given by
ωi1...in (x)=ω(x)(ei1 ,...,ein ),
then we define
1 n
ω= ω1...n(x)dx ···dx .
U Rn
400 7. Integration on Manifolds
Recall that if ζ is any integrable function and f : Rn → Rn is any diffeomorphism, the change of
variables theorem states that ζ◦ f is integrable and
ζ(x1,...,xn)dx1 ···dxn
Rn
1 n 1 n 1 n
= |JΩf(x ,...,x )|(ζ◦ f)(x ,...,x )dx ···dx , ()
Rn
1 n n
where Ω= dx ∧···∧dx is the standard volume form on R and JΩf is the Jacobian determinant of f
relative to Ω. This change of variables theorem can be rephrased in terms of pull backs in the following form.
Theorem (Change of Variables in Rn). Let U and V be open subsets of Rn and suppose f : U → V
is an orientation-preserving diffeomorphism. If ω∈Ωn(V ) pact support, then f ∗ω∈Ωn(U) has
compact support as well and
f ∗ω= ω()
U V
1 n ∗ 1 n
Proof. If ω= ω1...ndx ∧···∧dx , then f ω=(ω1...