30.4. STOKE’S THEOREM AND THE ORIENTATION OF ∂Ω 1073

define an atlas for ∂Ω,{

Vj,S j}

where Vj ≡ ∂Ω∩U j and S j is just the restriction of R j toVj. Then this is an oriented atlas for ∂Ω and∫

∂Ω

ω =∫

Ω

dω

where the two integrals are taken with respect to the given oriented atlass.

What if aI is only the restriction to Ω of a function in W 1,p (Rm) , p > 1? Would thesame formula still hold? Let φ ε be a mollifier and let aIε ≡ aI ∗φ ε . Then Stoke’s theoremapplies to the mollified situation and it follows∫

Ω

dωε

= ∑I

m

∑k=1

p

∑j=1

∫R j(U j)

∂

(ψ jaIε

)∂xk

(R−1

j (u))

∂(xk,xi1 · · ·xin−1

)∂ (u1, · · · ,un)

du

= ∑I

p

∑j=1

∫S jV j

ψ jaIε ◦S−1j (u2, · · · ,un)

∂(xi1 · · ·xin−1

)∂ (u2, · · · ,un)

(0,u2, · · · ,un)du1

≡∫

∂Ω

ωε

Now if you let ε → 0, it follows from the definition of convolution that

∂

(ψ jaIε

)∂xk

→∂

(ψ jaI

)∂xk

in Lp (Rm)

and so there is a subsequence such that for each k,

∂

(ψ jaIε

)∂xk

(x)→∂

(ψ jaI

)∂xk

(x)

pointwise a.e. Since R−1j ,R j are Lipschitz, they take sets of measure zero to sets of measure

zero. Hence∂

(ψ jaIε

)∂xk

◦R−1j →

∂

(ψ jaI

)∂xk

◦R−1j

pointwise a.e. on R j (U j) . Similar considerations apply to aIε . Using the Vitali conver-gence theorem in

∫Ω

dωε ,∫

Ωωε , it is possible to pass to the limit. This is because the

integrands are bounded in Lp and so they are uniformly integrable. This proves the follow-ing corollary.

Corollary 30.4.2 Let Ω be an oriented Lipschitz manifold and let

ω = ∑I

aI (x)dxi1 ∧·· ·∧dxin−1 .