30.3. INTEGRATION OF DIFFERENTIAL FORMS ON MANIFOLDS 1065

Lemma 30.3.1 Let g : U →V be C2 where U and V are open subsets of Rn. Then

n

∑j=1

(cof(Dg))i j, j = 0,

where here (Dg)i j ≡ gi, j ≡ ∂gi∂x j

.

Also recall the interesting relation of the degree to integration in Corollary 29.11.3

Corollary 30.3.2 Let f ∈ Lploc (R

n) for p≥ 1 and let h be Lipschitz where ∂U has measurezero for U a bounded open set and h(∂U)C has finitely many components. Then everythingis measurable which needs to be and∫

f (y)d (y,U ,h)dy =∫

UdetDh(x) f (h(x))dx.

(Recall that if y /∈ h(U) , then d (y,U ,h) = 0.)

Recall Proposition 23.6.4.

Proposition 30.3.3 Let Ω be an open connected bounded set in Rn such that Rn \ ∂Ω

consists of two, three if n = 1, connected components. Let f ∈ C(Ω;Rn

)be continuous

and one to one. Then f(Ω) is the bounded component of Rn \ f(∂Ω) and for y ∈ f(Ω) ,d (f,Ω,y) either equals 1 or −1.

Also recall the following fundamental lemma on partitions of unity in Corollary 15.5.9.

Lemma 30.3.4 Let K be a compact set in Rn and let {Ui}∞

i=1 be an open cover of K. Thenthere exist functions, ψk ∈C∞

c (Ui) such that ψ i ≺Ui and for all x ∈ K,

∞

∑i=1

ψ i (x) = 1.

If K is a compact subset of U1 (Ui)there exist such functions such that also ψ1 (x) = 1(ψ i (x) = 1) for all x ∈ K.

With the above, what follows is the definition of what a differential form is and how tointegrate one.

Definition 30.3.5 Let I denote an ordered list of n indices taken from the set, {1, · · · ,m}.Thus I = (i1, · · · , in). It is an ordered list because the order matters. A differential form oforder n in Rm is a formal expression,

ω = ∑I

aI (x)dxI

where aI is at least Borel measurable dxI is short for the expression

dxi1 ∧·· ·∧dxin ,