1064 CHAPTER 30. INTEGRATION OF DIFFERENTIAL FORMS

Proof: This follows from a computation. By Corollary 5.4.5 on Page 71, det(BA) =

1m! ∑

(i1···im)∑

( j1··· jm)sgn(i1 · · · im)sgn( j1 · · · jm)(BA)i1 j1 (BA)i2 j2 · · ·(BA)im jm

1m! ∑

(i1···im)∑

( j1··· jm)sgn(i1 · · · im)sgn( j1 · · · jm) ·

n

∑r1=1

Bi1r1Ar1 j1

n

∑r2=1

Bi2r2 Ar2 j2 · · ·n

∑rm=1

BimrmArm jm

Now denote by Ik one subsets of {1, · · · ,n} having m elements. Thus there are C (n,m) ofthese. Then the above equals

=C(n,m)

∑k=1

∑{r1,··· ,rm}=Ik

1m! ∑

(i1···im)∑

( j1··· jm)sgn(i1 · · · im)sgn( j1 · · · jm) ·

Bi1r1Ar1 j1Bi2r2Ar2 j2 · · ·BimrmArm jm

=C(n,m)

∑k=1

∑{r1,··· ,rm}=Ik

1m! ∑

(i1···im)sgn(i1 · · · im)Bi1r1 Bi2r2 · · ·Bimrm ·

∑( j1··· jm)

sgn( j1 · · · jm)Ar1 j1Ar2 j2 · · ·Arm jm

=C(n,m)

∑k=1

∑{r1,··· ,rm}=Ik

1m!

sgn(r1 · · ·rm)2 det(Bk)det(Ak)

=C(n,m)

∑k=1

det(Bk)det(Ak)

since there are m! ways of arranging the indices {r1, · · · ,rm}.

30.3 Integration Of Differential Forms On ManifoldsThis section presents the integration of differential forms on manifolds. This topic is ahigher dimensional version of what is done in calculus in finding the work done by a forcefield on an object which moves over some path. There you evaluated line integrals. Dif-ferential forms are just a higher dimensional version of this idea and it turns out they arewhat it makes sense to integrate on manifolds. The following lemma, on Page 429 used inestablishing the definition of the degree and in giving a proof of the Brouwer fixed pointtheorem is also a fundamental result in discussing the integration of differential forms.