1076 CHAPTER 30. INTEGRATION OF DIFFERENTIAL FORMS

However,

det(DΣ j) = 1 = det(

DΣ−1j

)and det(Ri) = det(R∗i ) = 1 by assumption. Therefore, if u ∈

(R j ◦R−1

i

)(A) , the above

degree is 1 and if u is not in this set, the above degree is 0 or undefined if u is on(R j ◦R−1

i

)(∂A). By Definition 30.1.5 Ω is indeed an oriented manifold.

30.5.2 Green’s Theorem

The general Green’s theorem is the following. It follows from Corollary 30.4.2.

Theorem 30.5.1 Let Ω be a bounded open set having Lipschitz boundary as describedabove. Also let

ω = ∑I

aI (x)dxi1 ∧·· ·∧dxin−1

be a differential form where aI is assumed to be the restriction to Ω of a function inW 1,p (Rn) , p > 1. Then ∫

∂Ω

ω =∫

Ω

dω

It can be shown that, since the boundary is Lipschitz, it would have sufficed to assumeu ∈W 1,p (Ω) and then it is automatically the restriction of one in W 1,p (Rn) . However,these terms have not all been defined and the necessary results are not proved till the topicof Sobolev spaces is discussed.

Another thing to notice is that, while the above result is pretty general, including theusual calculus result in the plane as a special case, it does not have the generality of the bestresults in the plane which involve only a rectifiable simple closed curve. The issue whether∂Ω is an oriented manifold was dealt with in the general Stokes theorem described above.

Next is a general version of the divergence theorem which comes from choosing thedifferential form in an auspicious manner.

30.6 The Divergence TheoremFrom Green’s theorem, one can quickly obtain a general Divergence theorem for Ω asdescribed above in Section 30.5.1. First note that from the above description of the R j,

∂(xk,xi1 , · · ·xin−1

)∂ (u1, · · · ,un)

= sgn(k, i1 · · · , in−1) .

So let F(x) be a Lipschitz vector field. Say F = (F1, · · · ,Fn) . Consider the differential form

ω (x)≡n

∑k=1

Fk (x)(−1)k−1 dx1∧·· ·∧ d̂xk ∧·· ·∧dxn