486 CHAPTER 18. DIFFERENTIAL FORMS

This divergence theorem is discussed more later when also the unit vector N whose jth

component is N j is described as an exterior unit normal provided∂(x1,··· ,xp)∂ (u1,··· ,uk)

≥ 0. Note how

the (−1) j+1 on the a j is responsible for the (−1) j+l instead of (−1)1+l in the descriptionof N j.

18.6 Examples of r([a,b])I want to point out that there are many examples of r([a,b]) which fit into the aboveintegration by parts idea of Stokes theorem in which r is Lipschitz, in order to tie this moreto the way we usually think of these theorems in Calculus. Consider the following picturein which a closed ball B of radius 1 is inscribed into the box Q ≡ [−1,1] ≡∏

ki=1 [−1,1].

Let P be the projection map onto this ball B.

Then it is geometrically obvious that the projection map P satisfies P(Q\B) = ∂B a setof H k measure zero and that P : ∂ [−1,1]→ ∂B is one to one and onto on ∂ [−1,1]. Nowlet r : B→Rp for p≥ k be Lipschitz and one to one. Then r ◦P : [−1,1]→ r (B) satisfiesall the necessary conditions for an application of Stokes theorem including the geometricdescriptions just given. What kinds of sets are in r (B) for B a closed ball and r Lipschitzand one to one? I think you can see that this would include virtually everything of interest.You could stretch B in various directions, pinch it, bend it, etc. Roughly speaking, imaginea ball of soft clay and doing what a child would do to it before he tears it into little pieces,throws them around the room and stomps them into the carpet. The result would be one ofthe possible sets r (B). Since r is a homeomorphism, the interior of B corresponds to therelative interior of r (B), points x ∈ r (B) = r◦P([−1,1]) which are not in r (∂B). Theboundary faces of [−1,1] and r◦P restricted to these faces will parametrize finitely manydisjoint pieces of r (∂B).

Of course you could also consider chains of such boxes as described earlier in the casethat r is C1. However, when you can allow r to be Lipschitz, it is clear that the theory issufficiently general to include most things which would be of interest in any applicationfrom a single box. Next is a discussion of orientation placed here to make an analogy withthe case of line integrals and oriented curves.

18.7 Orientation and DegreeHere I will consider orientation briefly. As in the case of a curve, it reduces to considera-tions of r−1◦r̂.

Proposition 18.7.1 Suppose r ([a,b]) = r̂([â, b̂

]), two sets in Rp and both r, r̂ are

one to one and Lipschitz, [a,b] ,[â, b̂

]being two parameter domains in Rk,k ≤ p. Then

for

ω = a(x)dxi1 ∧·· ·∧dxik

Assume also that r−1◦r̂ is Lipschitz and det(D(r−1◦r̂

)(t))≥ 0 a.e. This is a statement

about orientation. It follows then that∫r ω =

∫r̂ ω.