1078 CHAPTER 30. INTEGRATION OF DIFFERENTIAL FORMS

because it is the expansion of ∣∣∣∣∣∣∣∣∣x1,i x1,2 · · · x1,nx2,i x2,2 · · · x2,n

......

. . ....

xn,i xn,2 · · · xn,n

∣∣∣∣∣∣∣∣∣ ,a determinant with two equal columns. Thus this vector is at least in some sense normal toΩ. Since it works in the divergence theorem, it is called the exterior normal.

One could normalize the vector of 30.6.22 by dividing by its magnitude. Then it wouldbe the unit exterior normal n. Letting J (u1) be its usual Euclidean norm, this equals

J (u1)2 =

n

∑k=1

(∂ (x1, · · · x̂k · · · ,xn)

∂ (u2, · · · ,un)

)2

and by the Binet Cauchy theorem this equals

det(DR−1

s (u)∗DR−1s (u)

)1/2

Thus the expression in 30.6.21 reduces to∫Bs

(ψsF◦R−1

s (u1))·n(R−1

s (u1))

J (u1)du1.

By the area formula, Theorem 29.5.3, this reduces to∫∂Ω∩Ws

ψsF ·ndH n−1 =∫

∂Ω

ψsF ·ndH n−1

It follows upon summing over s and using that the ψs add to 1,∫∂Ω

F ·ndH n−1 =∫

Ω

p

∑s=1

div(ψsF)dx

=∫

Ω

p

∑s=1

ψs,kFk +ψs div(F)dx =∫

Ω

Fk

(p

∑s=1

ψs

),k

+ψs div(F)dx

=∫

Ω

div(F)dx

This proves the following general divergence theorem.

Theorem 30.6.1 Let Ω be a bounded open set having Lipschitz boundary as describedabove. Also let F be a vector field with the property that for each component function ofF, Fk is the restriction to Ω of a function in W 1,p (Rn), p > 1. Then there exists a normalvector n which is defined a.e. on ∂Ω such that∫

∂Ω

F ·ndH n−1 =∫

Ω

div(F)dx