1258 CHAPTER 36. INTEGRATION ON MANIFOLDS
where here the sum is taken over all possible increasing lists of n indices, I, from {1, · · · , p}and x = hiu. Thus there are
(pn
)terms in the sum. In this formula,
∂(xi1 ···xin)∂(u1···un)
is defined to
be the determinant of the following matrix.∂xi1∂u1
· · · ∂xi1∂un
......
∂xin
∂u1· · · ∂xin
∂un
.
Note that if p = n there is only one term in the sum, the absolute value of the deter-minant of Dx(u). Define a positive linear functional, Λ on Cc (Γ) as follows: First let{ψ i} be a C∞ partition of unity subordinate to the open sets, {Wi} . Thus ψ i ∈C∞
c (Wi) and∑i ψ i (x) = 1 for all x ∈ Γ. Then
Λ f ≡∞
∑i=1
∫giΓi
f ψ i (hi (u))Ji (u)du. (36.2.1)
Is this well defined?
Lemma 36.2.3 The functional defined in 36.2.1 does not depend on the choice of atlas orthe partition of unity.
Proof: In 36.2.1, let {ψ i} be a C∞ partition of unity which is associated with the atlas(Γi,gi) and let {η i} be a C∞ partition of unity associated in the same manner with the atlas(Γ′i,g′i). In the following argument, the local finiteness of the Γi implies that all sums are
finite. Using the change of variables formula with u =(
gi ◦h′j)
v
∞
∑i=1
∫giΓi
ψ i f (hi (u))Ji (u)du = (36.2.2)
∞
∑i=1
∞
∑j=1
∫giΓi
η jψ i f (hi (u))Ji (u)du =∞
∑i=1
∞
∑j=1
∫g′j(
Γi∩Γ′j
) ·
η j(h′j (v)
)ψ i(h′j (v)
)f(h′j (v)
)Ji (u)
∣∣∣∣∣∂(u1 · · ·un
)∂ (v1 · · ·vn)
∣∣∣∣∣dv
=∞
∑i=1
∞
∑j=1
∫g′j(
Γi∩Γ′j
)η j(h′j (v)
)ψ i(h′j (v)
)f(h′j (v)
)J j (v)dv. (36.2.3)
Thusthe definition of Λ f using (Γi,gi)≡
∞
∑i=1
∫giΓi
ψ i f (hi (u))Ji (u)du =
∞
∑i=1
∞
∑j=1
∫g′j(
Γi∩Γ′j
)η j(h′j (v)
)ψ i(h′j (v)
)f(h′j (v)
)J j (v)dv