478 CHAPTER 18. DIFFERENTIAL FORMS

Then from Lemma 18.1.2 again,

= dα (x)∧β (x)+(−1)k∑

IaI (x)dxI ∧∑

Jd (bJ (x))∧dxJ

= dα (x)∧β (x)+(−1)kα (x)∧dβ (x)

One of the important properties of the exterior derivative is that d2 = 0. Let

ω = ∑I

aI (x)dxI , dω = ∑I

p

∑r=1

aI,xr dxr ∧dxI

Then by definition, d2ω = ∑I ∑pr=1 ∑

ps=1 aI,xrxs dxs∧dxr ∧dxI

= ∑I

∑r<s

aI,xrxsdxs∧dxr ∧dxI +∑I

∑s<r

aI,xrxsdxs∧dxr ∧dxI

= ∑I

∑r<s

aI,xrxsdxs∧dxr ∧dxI +∑I

∑r<s

aI,xsxr dxr ∧dxs∧dxI

In keeping with the convention that we assume aI are as smooth as desired, we can concludethat the mixed partial derivatives are equal and so the above reduces to

∑I

∑r<s

aI,xrxs dxs∧dxr ∧dxI +∑I

∑r<s

aI,xrxs dxr ∧dxs∧dxI

= ∑I

∑r<s

aI,xrxs dxs∧dxr ∧dxI−∑I

∑r<s

aI,xrxs dxs∧dxr ∧dxI = 0 ■

It might be interesting to note that if one means weak derivatives, then the mixed partialderivatives are always equal.

18.3 Stokes TheoremNow that the algebra of differential forms has been presented, it is time for the main topicStokes theorem. Recall [a,b] is defined as ∏

kl=1 [al ,bl ] . Let x ∈Rp, p≥ k,r : [a,b]→Rp,

and let ω a k−1 form be given as follows

ω = ∑I∈J

α I (x)dxi1 ∧·· ·∧dxik−1

where here I = (i1, ..., ik−1) is an increasing list of k−1 indices from (1,2, ..., p) and J willdenote the set of all such increasing lists of indices. Thus there are C (p,k−1) elements inthe set J. Assume that α I is C1 but here x= r (u) is C2. After this case is done, it is easyto generalize.

This is a generalization of line integrals which is about integration over curves in space.Recall there was a parameter domain [a,b] and a map r : [a,b]→ Rp and there were twoorientations or directions over the curve which was the set of points r ([a,b]). This conceptof orientation is dealt with in multiple dimensions by the use of differential forms and theconcept of determinants from linear algebra. The interval [a,b] is replaced by [a,b] . Stokestheorem then is a statement about r ([a,b]) and the boundary of r ([a,b]) just as it is in thecase of a line integral.

Stokes theorem relates the integral over some r to the integral over the “boundary” ofr. This is defined next.