18.3. STOKES THEOREM 479
Definition 18.3.1 Denote as Λ(k) the set of finite sums of differential forms of or-der k. I will now describe the boundary ∂ : Λ(k)→ Λ(k−1) by first defining it on r andthen, if desired, one would know it on all of Λ(k). If k = 1, then ∂r ≡ r (b)− r (a) . Ingeneral, for r : [a,b]→ Rp and l ≤ k,
∂lr (u1, · · · , ûl , · · · ,uk)
≡ r (u1, · · · ,bl ,ul+1, · · · ,uk)−r (u1, · · · ,al ,ul+1, · · · ,uk)≡ rbl −ral .
Note that rbl ,ral are defined on
[a1,b1]×·· ·× [al−1,bl−1]× [al+1,bl+1]×·· ·× [ak,bk]≡ [a,b]l .
Specifically, if ω = ∑I aI (x)dxI where I denotes ordered lists of indices of length k− 1taken from {1, ..., p}∫
∂lrω ≡
∫[a,b]l
∑I
aI(rbl (u)
)det
(drI
bl(u)
du
)dmk−1
−∫[a,b]l
∑I
aI(rbl (u)
)det
(drI
al(u)
du
)dmk−1
where u comes from [a,b]l . Here rbl is r with bl in the lth position, similar for ral . Thus
det(
drIbl(u)
du
),det
(drI
al(u)
du
)are (−1)1+l AJ
1l where AJ1l is the (1, l)th cofactor of det
(DrJ
)where J has length k and the matrix DrJ is the matrix of 18.4, having the top row(
x j,u1 x j,u2 · · · x j,uk
)Then
∫∂r ω ≡ ∑l
∫∂lr
ω .
With this preparation, Stoke’s theorem follows from a computation.
dω ≡∑I∈J
p
∑j=1
∂α I
∂x j(x)dx j ∧dxi1 ∧·· ·∧dxik−1 , I = (i1, · · · , ik−1)
Definition 18.3.2 As discussed earlier, define∫r
dω ≡∑I∈J
p
∑j=1
∫[a,b]
∂α I
∂x j(r (u))
∂(x j,xi1 · · ·xik−1
)∂ (u1, · · · ,uk)
du (18.3)
By definition,∂
(x j ,xi1 ···xik−1
)∂ (u1,··· ,uk)
is the determinant ofx j,u1 x j,u2 · · · x j,ukxi1,u1 xi1,u2 · · · xi1,uk
......
...xik−1,u1 xik−1,u2 · · · xik−1,uk
(18.4)
Note how this matrix is just the matrix of Df where
f (u) =(x j (u) ,xi1 (u) , ...,xik−1 (u)
)T.