480 CHAPTER 18. DIFFERENTIAL FORMS

Then expanding the determinant in 18.3 along the first row, it equals

= ∑I∈J

p

∑j=1

∫[a,b]

∂α I

∂x j(r (u))

expanding determinantk

∑l=1

∂x j

∂ulAI

1l du

where AI1l is the (1, l)th cofactor for the determinant of 18.4.

AI1l = (−1)1+l ∂

(xi1 , · · · ,xik−1

)∂ (u1, · · · , ûl · · · ,uk)

, I = (i1, · · · , ik−1) (18.5)

Then this equals

= ∑I∈J

k

∑l=1

∫[a,b]

p

∑j=1

∂α I

∂x j(r (u))

∂x j

∂ulAI

1ldu= ∑I∈J

k

∑l=1

∫[a,b]

∂α I (r (u))

∂ulAI

1ldu

Now

∑l

∂α I (r (u))

∂ulAI

1l = ∑l

∂

∂ul

(α I (r (u))AI

1l)−∑

lα I (r (u))AI

1l,l

= ∑l

∂

∂ul

(α I (r (u))AI

1l)

By Lemma 7.11.2, that cofactor identity depending on equality of mixed partials. There-fore, from 18.3 and Fubini’s theorem,∫

rdω = ∑

I∈J

k

∑l=1

∫[a,b]

∂

∂ul

(α I (r (u))AI

1l)

= ∑I∈J

k

∑l=1

∫[a,b]l

∫[al ,bl ]

∂

∂ul

((α I (r (u))AI

1l))

duldul

=k

∑l=1

∫[a,b]l

∑I∈J

((α I ◦r)AI

1l)(ul (bl))−

((α I ◦r)AI

1l)(ul (al))dul (18.6)

where here [a,b]l means the [al ,bl ] is missing in the product [a,b] and ul (bl) is givenby the formula (u1, ...,ul−1,bl ,ul+1, ...,uk) with ul (al) defined similarly. The term AI

1l is

the cofactor in 18.5 (−1)1+l ∂

(xi1 ,··· ,xik−1

)∂ (u1,··· ,ûl ··· ,uk)

. The term∫[a,b]l

((α I ◦r)AI

1l

)(ul (bl))dul is

an integration over the variables corresponding to a face of [a,b] and so it is a kind ofboundary term. By Definition 18.3.1 or simply making a definition that this is what wemean by the integral over the boundary, this is

∫∂r ω . Thus, this proves Stokes’ theorem.

Theorem 18.3.3 Let ω = ∑I α I (x)dxi1 ∧ ·· · ∧ dxik−1 be a k− 1 form. Let r :[a,b]→ Rp, p≥ k be in C2 ([a,b] ;Rp) . Then

∫∂r ω =

∫r dω .

Note that there is no assumption that Dr has nonzero determinant. Everything above isvalid under an assumption that r is only C2. There was a reason why in calculus smoothcurves had a parametrization with nonvanishing derivative. If the derivative vanishes, this