Multivariable Advanced Calculus

11.5. INTEGRATION OF DIFFERENTIAL FORMS ON MANIFOLDS 305Lemma 11.5.1 Let g : U → V be C 2 where U and V are open subsets of R n . Thenwhere here (Dg) ij≡ g i,j ≡ ∂gi∂x j.Recall Proposition 10.6.2.n∑(cof (Dg)) ij,j= 0,j=1Proposition 11.5.2 Let Ω be an open connected bounded set in R n such that R n \∂Ω consists of two, three if n = 1, connected components. Let f ∈ C ( Ω; R n) be continuousand one to one. Then f (Ω) is the bounded component of R n \ f (∂Ω) and fory ∈ f (Ω) , d (f, Ω, y) either equals 1 or −1.Also recall the following fundamental lemma on partitions of unity in Lemma 9.5.15and Corollary 9.5.14.Lemma 11.5.3 Let K be a compact set in R n and let {U i } m i=1be an open cover ofK. Then there exist functions, ψ k ∈ Cc ∞ (U i ) such that ψ i ≺ U i and for all x ∈ K,m∑ψ i (x) = 1.i=1If K is a compact subset of U 1 (U i )there exist such functions such that also ψ 1 (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 howto integrate one.Definition 11.5.4 Let I denote an ordered list of n indices taken from the set,{1, · · · , m}. Thus I = (i 1 , · · · , i n ). It is an ordered list because the order matters. Adifferential form of order n in R m is a formal expression,ω = ∑ Ia I (x) dx Iwhere a I is at least Borel measurable dx I is short for the expressiondx i1 ∧ · · · ∧ dx in ,and the sum is taken over all ordered lists of indices taken from the set, {1, · · · , m}.For Ω an orientable n dimensional manifold with boundary, let {(U i , R i )} be an orientedatlas for Ω. Each U i is the intersection of an open set in R m , O i , with Ω and so thereexists a C ∞ partition of unity subordinate to the open cover, {O i } which sums to 1 onΩ. Thus ψ i ∈ C ∞ c (O i ), has values in [0, 1] and satisfies ∑ i ψ i (x) = 1 for all x ∈ Ω.DefineNote∫Ωω ≡p∑ ∑∫i=1I(ψ i R−1iR iU i(u) ) (a I R−1i (u) ) ∂ (x i1 · · · x in )du (11.12)∂ (u 1 · · · u n )∂ (x i1 · · · x in )∂ (u 1 · · · u n ) ,given by 11.8 is not defined on R i (U i ∩ L) but this is assumed a set of measure zero soit is not important in the integral.