Chapter 30

Integration Of Differential Forms30.1 Manifolds

Manifolds are sets which resemble Rn locally. To make the concept of a manifold moreprecise, here is a definition.

Definition 30.1.1 Let Ω ⊆ Rm. A set, U, is open in Ω if it is the intersection of an openset from Rm with Ω. Equivalently, a set, U is open in Ω if for every point, x ∈U, thereexists δ > 0 such that if |x−y| < δ and y ∈ Ω, then y ∈U. A set, H, is closed in Ω if itis the intersection of a closed set from Rm with Ω. Equivalently, a set, H, is closed in Ω ifwhenever, y is a limit point of H and y ∈Ω, it follows y ∈ H.

Recall the following definition.

Definition 30.1.2 Let V ⊆ Rn. Ck(V ;Rm

)is the set of functions which are restrictions to

V of some function defined on Rn which has k continuous derivatives and compact support.When k = 0, it means the restriction to V of continuous functions with compact support.

Definition 30.1.3 A closed and bounded subset of Rm, Ω, will be called an n dimen-sional manifold with boundary, n ≥ 1, if there are finitely many sets, Ui, open in Ω andcontinuous one to one functions, Ri ∈ C0

(Ui,Rn

)such that RiUi is relatively open in

Rn≤ ≡ {u ∈ Rn : u1 ≤ 0} , R−1

i is continuous. These mappings, Ri, together with their do-mains, Ui, are called charts and the totality of all the charts, (Ui,Ri) just described is calledan atlas for the manifold. Define

int(Ω)≡ {x ∈Ω : for some i,Rix ∈ Rn<}

where Rn< ≡ {u ∈ Rn : u1 < 0}. Also define

∂Ω≡ {x ∈Ω : for some i,Rix ∈ Rn0}

whereRn

0 ≡ {u ∈ Rn : u1 = 0}

and ∂Ω is called the boundary of Ω. Note that if n = 1, Rn0 is just the single point 0. By

convention, we will consider the boundary of such a 0 dimensional manifold to be empty.

This definition is a little too restrictive. In general the collection of sets, Ui is not finite.However, in the case where Ω is closed and bounded, compactness of Ω can be used toget a finite covering and since this is the case of most interest here, the assumption that thecollection of sets, Ui, is finite is made. However, most of what is presented here can begeneralized to the case of a locally finite atlas.

Theorem 30.1.4 Let ∂Ω and int(Ω) be as defined above. Then int(Ω) is open in Ω and∂Ω is closed in Ω. Furthermore, ∂Ω∩ int(Ω) = /0, Ω = ∂Ω∪ int(Ω), and for n≥ 2, ∂Ω isan n− 1 dimensional manifold for which ∂ (∂Ω) = /0. The property of being in int(Ω) or∂Ω does not depend on the choice of atlas.

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