Nonlinear Equations - UFRJ

50 [CH. 4: DIFFERENTIAL FORMS
be measurable. Then whenever the left integral exists,
∫
∫ ∫
f(p)dE(p) = dB(x) (det Dπ(p)Dπ(p) ∗ ) −1/2 f(p)dE x (p).
E
B
E x
with E x = π −1 (x).
Lemma 4.8. In the conditions of Theorem 4.7, there is a locally
finite open covering U = {U α } of B, and a family of smooth functions
ψ α ≥ 0 with domain B vanishing in B \ U α such that
1. Each U α ∈ U is such that there is a local trivialization Φ with
domain Φ −1 (U α ).
2.
∑
ψ α (x) ≡ 1.
α
The family {ψ α } is said to be a partition of unity for π : E → B.
Proof of theorem 4.7. Let ψ α be the partition of unity from Lemma 4.8.
By replacing f by f(ψ α ◦π) and then adding for all α, we can assume
without loss of generality that f vanishes outside the domain π −1 (U)
of a local trivialization.
Now,
∫
∫
f(p)dE(p) = f(p)dE(p)
E
=
=
∫
∫
π −1 (U)
Φ(π −1 (U))
U
∫
dB(x)
det DΦ −1 (x, y)f(Φ −1 (x, y))dB(x)dF (y)
F
det DΦ −1 (x, y)f(Φ −1 (x, y))dF (y)
using Fubini’s theorem. Note that Φ |Fx → F is a diffeomorphism, so
the inner integral can be replaced by
∫
det DΦ |Fx det DΦ −1 (p)f(p)dF x (p).
F x