36.2. INTEGRATION ON MANIFOLDS 1257

Definition 36.2.1 Let U ⊆ Rn be an open set and let h : U → Rm. Then for r ∈ [0,1),h ∈ Ck,r (U) for k a nonnegative integer means that Dα h exists for all |α| ≤ k and eachDα h is Holder continuous with exponent r. That is

|Dα h(x)−Dα h(y)| ≤ K |x−y|r .

Also h ∈Ck,r(U)

if it is the restriction of a function of Ck,r (Rn) to U.

Definition 36.2.2 Let Γ be a closed subset of Rp where p≥ n. Suppose Γ = ∪∞i=1Γi where

Γi = Γ∩Wi for Wi a bounded open set. Suppose also {Wi}∞

i=1 is locally finite. This meansevery bounded open set intersects only finitely many. Also suppose there are open boundedsets, Ui having Lipschitz boundaries and functions hi : Ui→ Γi which are one to one, onto,and in Cm,1 (Ui) . Suppose also there exist functions, gi : Wi→Ui such that gi is Cm,1 (Wi) ,and gi ◦ hi = id on Ui while hi ◦ gi = id on Γi. The collection of sets, Γ j and mappings,g j,{(

Γ j,g j)}

is called an atlas and an individual entry in the atlas is called a chart. Thus(Γ j,g j

)is a chart. Then Γ as just described is called a Cm,1 manifold. The number, m is

just a nonnegative integer. When m = 0 this would be called a Lipschitz manifold, the leastsmooth of the manifolds discussed here.

For example, take p = n+1 and let

hi (u) = (u1, · · · ,ui,φ i (u) ,ui+1, · · · ,un)T

for u =(u1, · · · ,ui,ui+1, · · · ,un)T ∈Ui for φ i ∈Cm,1 (Ui) and gi : Ui×R→Ui given by

gi (u1, · · · ,ui,y,ui+1, · · · ,un)≡ u

for i = 1,2, · · · , p. Then for u ∈Ui, the definition gives

gi ◦hi (u) = gi (u1, · · · ,ui,φ i (u) ,ui+1, · · · ,un) = u

and for Γi ≡ hi (Ui) and (u1, · · · ,ui,φ i (u) ,ui+1, · · · ,un)T ∈ Γi,

hi ◦gi (u1, · · · ,ui,φ i (u) ,ui+1, · · · ,un)

= hi (u) = (u1, · · · ,ui,φ i (u) ,ui+1, · · · ,un)T .

This example can be used to describe the boundary of a bounded open set and since φ i ∈Cm,1 (Ui) , such an open set is said to have a Cm,1 boundary. Note also that in this example,Ui could be taken to be Rn or if Ui is given, both hi and and gi can be taken as restrictionsof functions defined on all of Rn and Rp respectively.

The symbol, I will refer to an increasing list of n indices taken from {1, · · · , p} . Denoteby Λ(p,n) the set of all such increasing lists of n indices.

Let

Ji (u)≡

 ∑I∈Λ(p,n)

(∂(xi1 · · ·xin

)∂ (u1 · · ·un)

)21/2