1068 CHAPTER 30. INTEGRATION OF DIFFERENTIAL FORMS
=∫
intRi(Ui∩V j)η j(R−1
i (u))
ψ i(R−1
i (u))
aI(R−1
i (u)) ∂ (xi1 · · ·xin)
∂ (u1 · · ·un)du
=∞
∑k=1
∫Ri◦S−1
j (Bk)η j(R−1
i (u))
ψ i(R−1
i (u))
aI(R−1
i (u)) ∂ (xi1 · · ·xin)
∂ (u1 · · ·un)du
=∞
∑k=1
∫Bk
η j
(S−1
j (v))
ψ i
(S−1
j (v))
aI
(S−1
j (v))
∂ (xi1 · · ·xin)
∂ (v1 · · ·vn)dv
=∫
intS j(Ui∩V j)η j
(S−1
j (v))
ψ i
(S−1
j (v))
aI
(S−1
j (v))
∂ (xi1 · · ·xin)
∂ (v1 · · ·vn)dv
=∫
S j(Ui∩V j)η j
(S−1
j (v))
ψ i
(S−1
j (v))
aI
(S−1
j (v))
∂ (xi1 · · ·xin)
∂ (v1 · · ·vn)dv (30.3.9)
The equality of 30.3.8 and 30.3.9 was the goal. With this, the definition of∫
ω using theatlas (Ui,Ri) and partition of unity {ψ i}
pi=1 given in 30.3.5 is
p
∑i=1
∑I
∫RiUi
ψ i(R−1
i (u))
aI(R−1
i (u)) ∂ (xi1 · · ·xin)
∂ (u1 · · ·un)du
=q
∑j=1
p
∑i=1
∑I
∫Ri(Ui∩V j)
η j(R−1
i (u))
ψ i(R−1
i (u))
aI(R−1
i (u)) ∂ (xi1 · · ·xin)
∂ (u1 · · ·un)du
and from 30.3.8 - 30.3.9, this equals
=q
∑j=1
p
∑i=1
∑I
∫S j(Ui∩V j)
η j
(S−1
j (v))
ψ i
(S−1
j (v))
aI
(S−1
j (v))
∂ (xi1 · · ·xin)
∂ (v1 · · ·vn)dv
=q
∑j=1
∑I
∫S j(V j)
η j
(S−1
j (v))
aI
(S−1
j (v))
∂ (xi1 · · ·xin)
∂ (v1 · · ·vn)dv
which is the definition of∫
ω using the other atlas and partition of unity. This proves thetheorem.
30.3.1 The Derivative Of A Differential FormThe derivative of a differential form is defined next.
Definition 30.3.8 Let ω = ∑I aI (x)dxi1 ∧ ·· ·∧dxin−1 be a differential form of order n−1where aI has weak partial derivatives. Then define dω , a differential form of order n byreplacing aI (x) with
daI (x)≡m
∑k=1
∂aI (x)∂xk
dxk (30.3.10)
and putting a wedge after the dxk. Therefore,
dω ≡∑I
m
∑k=1
∂aI (x)∂xk
dxk ∧dxi1 ∧·· ·∧dxin−1 . (30.3.11)