1070 CHAPTER 30. INTEGRATION OF DIFFERENTIAL FORMS

where A1l is the 1lth cofactor for the determinant

∂(xkε ,xi1ε · · ·xin−1ε

)∂ (u1, · · · ,un)

which is determined by a particular I. I am suppressing the ε for the sake of notation. Thenthe above reduces to

= ∑I

p

∑j=1

∫R j(U j)

n

∑l=1

A1l

m

∑k=1

∂

(ψ jaI

)∂xk

(R−1

jε (u))

∂xkε

∂uldu+ e(ε)

= ∑I

p

∑j=1

n

∑l=1

∫R j(U j)

A1l∂

∂ul

(ψ jaI ◦R−1

jε

)(u)du+ e(ε) (30.4.13)

(Note l goes up to n not m.) Recall R j (U j) is relatively open in Rn≤. Consider the integral

where l > 1. Integrate first with respect to ul . In this case the boundary term vanishesbecause of ψ j and you get

−∫

R j(U j)A1l,l

(ψ jaI ◦R−1

jε

)(u)du (30.4.14)

Next consider the case where l = 1. Integrating first with respect to u1, the term reduces to∫R jV j

ψ jaI ◦R−1jε (0,u2, · · · ,un)A11du1−

∫R j(U j)

A11,1

(ψ jaI ◦R−1

jε

)(u)du (30.4.15)

where R jVj is an open set in Rn−1 consisting of{(u2, · · · ,un) ∈ Rn−1 : (0,u2, · · · ,un) ∈ R j (U j)

}and du1 represents du2du3 · · ·dun on R jVj for short. Thus Vj is just the part of ∂Ω whichis in U j and the mappings S−1

j given on R jVj = R j (U j ∩∂Ω) by

S−1j (u2, · · · ,un)≡ R−1

j (0,u2, · · · ,un)

are such that{(S j,Vj)

}is an atlas for ∂Ω. Then if 30.4.14 and 30.4.15 are placed in

30.4.13, then it follows from Lemma 30.3.1 that this reduces to

∑I

p

∑j=1

∫R jV j

ψ jaI ◦R−1jε (0,u2, · · · ,un)A11du1 + e(ε)

Now as before, there exists a subsequence, still denoted as ε such that each ∂xsε/∂urconverges pointwise to ∂xs/∂ur and then using that these are bounded in every Lp, one canuse the Vitali convergence theorem to pass to a limit obtaining finally

∑I

p

∑j=1

∫R jV j

ψ jaI ◦R−1j (0,u2, · · · ,un)A11du1

= ∑I

p

∑j=1

∫S jV j

ψ jaI ◦S−1j (u2, · · · ,un)A11du1