18.5. WHAT DOES IT MEAN? 483
Now a.e. point of [a,b]l is a Lebesgue point and so we can pass to a limit and obtainpointwise a.e. convergence of rh
ui(ul (bl)) to rui (ul (bl)) .
Some comment on this might be useful because these Lebesgue points are not always atthe center of the box of sides of length 2h determined by the limits of the iterated integrals.A given point ul (bl) ∈ B(ul (bl) ,3h) where this ball is taken with respect to ∥·∥
∞. Thus,
from Lebesgue’s fundamental theorem of calculus, we have on this face at Lebesgue pointsthe following converges to 0 as h→ 0 which is what was desired(
1+2h
bi−ai
)(32
)k−1
·(1
3h
)k−1 ∫B(ul(bl),3h)
∥rui (ul (bl))−rui (t1, ..., ti, ...bl , ...tk)∥dtl
Indeed,( 3
2
)k−1 ( 13h
)k−1=( 1
2h
)k−1and the integral in the above is taken over the larger set
B(ul (bl) ,3h).As to points of [a,b] , the pointwise convergence of rh
u j(u) to ru j (u) follows from
similar reasoning but is a little less involved. Now rh is clearly C1 and so we have Stokestheorem for rh.
k
∑l=1
∫[a,b]l
∑I∈J
((α I ◦rh
)AIh
1l
)(ul (bl))−
((α I ◦r)AIh
1l
)(ul (al))dul
=∫r
dω ≡∑I∈J
p
∑j=1
∫[a,b]
∂α I
∂x j
(rh (u)
) ∂(x j,xi1 · · ·xik−1
)∂ (u1, · · · ,uk)
du
where the superscript indicates that all is defined in terms of rh. Then from the dominatedconvergence theorem, it follows that we can pass to a limit and obtain Stokes theoremwhere the boundary terms are defined from Rademacher’s theorem on the faces of [a,b].
Theorem 18.4.1 Let r : [a,b]⊆Rk→Rp, p≥ k be Lipschitz and also suppose thatα I ∈ C1 (r ([a,b])) for I ⊆ J, the set of increasing lists of k− 1 indices from (1,2, ..., p).Then one obtains Stokes theorem∫
r(∂ [a,b])ω ≡
k
∑l=1
∫[a,b]l
∑I∈J
((α I ◦r)AI
1l)(ul (bl))−
((α I ◦r)AI
1l)(ul (al))dul
=∫r
dω ≡∑I∈J
p
∑j=1
∫[a,b]
∂α I
∂x j(r (u))
∂(x j,xi1 · · ·xik−1
)∂ (u1, · · · ,uk)
du
where the partial derivatives of AI1l = (−1)1+l ∂
(xi1 ,··· ,xik−1
)∂ (u1,··· ,ûl ··· ,uk)
are defined in terms of Rad-emacher’s theorem applied to the k−1 dimensional faces of [a,b].
18.5 What Does it Mean?Stokes theorem is a statement about integration by parts. However, one can give geometricmeaning to what it says. These considerations will come from the area formula which Iwill use as needed.