29.7. THE DIVERGENCE THEOREM 1029

Now by Lemma 29.1.1 and the equality of mp−1 and H p−1 on Rp−1, the above integralequals ∫

∂U∩Qi

ψ i f (y)ni (y) ·wdH p−1 =∫

∂Uψ i f (y)ni (y) ·wdH p−1.

Similar arguments apply to the other terms and therefore,

limt→0

∫U

f (x+ tw)− f (x)t

dmp =N

∑i=1

∫∂U

ψ i f (y)ni (y) ·wdH p−1

=∫

∂Uf (y)

N

∑i=1

ψ i (y)ni (y) ·wdH p−1 =∫

∂Uf (y)n(y) ·wdH p−1 (29.7.35)

Then let n(y)≡ ∑Ni=1 ψ i (y)ni (y) .

I need to show first there is no other n which satisfies 29.7.35 and then I need to showthat |n(y)| = 1. Note that it is clear |n(y)| ≤ 1 because each ni is a unit vector and this isjust a convex combination of these. Suppose then that n1 ∈ L∞

(∂U,H p−1

)also works in

29.7.35. Then for all f ∈C1c (Rp) ,∫

∂Uf (y)n(y) ·wdH p−1 =

∫∂U

f (y)n1 (y) ·wdH p−1.

Suppose h ∈C (∂U) . Then by the Tietze extension theorem, there exists f ∈Cc (Rp) suchthat the restriction of f to ∂U equals h. Now by Lemma 29.7.5 applied to a boundedopen set containing the support of f , there exists a sequence { fm} of functions in C1

c (Rp)converging uniformly to f . Therefore,∫

∂Uh(y)n(y) ·wdH p−1 = lim

m→∞

∫∂U

fm (y)n(y) ·wdH p−1

= limm→∞

∫∂U

fm (y)n1 (y) ·wdH p−1 =∫

∂Uh(y)n1 (y) ·wdH p−1.

Now H p−1 is a Radon measure on ∂U and so the continuous functions on ∂U are densein L1

(∂U,H p−1

). It follows n ·w = n1 ·w a.e. Now let {wm}∞

m=1 be a countable densesubset of the unit sphere. From what was just shown, n ·wm= n1 ·wm except for a set ofmeasure zero, Nm. Letting N = ∪mNm, it follows that for y /∈ N,n(y) ·wm= n1 (y) ·wm forall m. Since the set is dense, it follows n(y) ·w = n1 (y) ·w for all y /∈ N and for all w aunit vector. Therefore, n(y) = n1 (y) for all y /∈N and this shows n is unique. In particular,although it appears to depend on the partition of unity {ψ i} from its definition, this is notthe case.

It only remains to verify |n(y)|= 1 a.e. I will do this by showing how to compute n. Inparticular, I will show that n = ni a.e. on ∂U ∩Qi. Let W ⊆W ⊆Qi∩∂U where W is openin ∂U. Let O be an open set such that O∩ ∂U = W and O ⊆ Qi. Using Corollary 16.1.2there exists a C∞ partition of unity {ψm} such that ψ i = 1 on O. Therefore, if m ̸= i,ψm = 0on O. Then if f ∈C1

c (O) , ∫W

f w ·ndH p−1 =∫

∂Uf w ·ndH p−1

=∫

U∇ f ·wdmp =

∫U

∇(ψ i f ) ·wdmp