1256 CHAPTER 36. INTEGRATION ON MANIFOLDS

If x is such that ∑∞j=1 φ j(x) = 0, then x /∈ ∪∞

i=1Ui because φ i equals one on Ui. Conse-quently γ (y) = 0 for all y near x thanks to the fact that ∪∞

i=1Ui is closed and so ψ i(y) = 0for all y near x. Hence ψ i is infinitely differentiable at such x. If ∑

∞j=1 φ j(x) ̸= 0, this

situation persists near x because each φ j is continuous and so ψ i is infinitely differentiableat such points also thanks to Lemma 36.1.3. Therefore ψ i is infinitely differentiable. Ifx ∈ K, then γ (x) = 1 and so ∑

∞j=1 ψ j(x) = 1. Clearly 0 ≤ ψ i (x) ≤ 1 and spt(ψ j) ⊆ Vj.

This proves the theorem.The method of proof of this lemma easily implies the following useful corollary.

Corollary 36.1.7 If H is a compact subset of Vi for some Vi there exists a partition of unitysuch that ψ i (x) = 1 for all x ∈ H in addition to the conclusion of Lemma 36.1.6.

Proof: Keep Vi the same but replace Vj with Ṽj ≡Vj \H. Now in the proof above, ap-plied to this modified collection of open sets, if j ̸= i,φ j (x)= 0 whenever x∈H. Therefore,ψ i (x) = 1 on H.

Theorem 36.1.8 Let H be any closed set and let C be any open cover of H. Then thereexist functions {ψ i}

∞

i=1 such that spt(ψ i) is contained in some set of C and ψ i is infinitelydifferentiable having values in [0,1] such that on H, ∑

∞i=1 ψ i (x) = 1. Furthermore, the

function, f (x) ≡ ∑∞i=1 ψ i (x) is infinitely differentiable on Rn. Also, spt(ψ i) ⊆ Ui where

Ui is a bounded open set with the property that {Ui}∞

i=1 is locally finite and each Ui iscontained in some set of C.

Proof: By Lemma 36.1.2 there exists an open cover of H composed of bounded opensets, Ui such that each Ui is a subset of some set of C and the collection, {Ui}∞

i=1 is locallyfinite. Then the result follows from Lemma 36.1.6 and Lemma 36.1.3.

Corollary 36.1.9 Let H be any closed set and let {Vi}mi=1 be a finite open cover of H. Then

there exist functions {φ i}mi=1 such that spt(φ i)⊆Vi and φ i is infinitely differentiable having

values in [0,1] such that on H, ∑mi=1 φ i (x) = 1.

Proof: By Theorem 36.1.8 there exists a set of functions, {ψ i}∞

i=1 having the propertieslisted in this theorem relative to the open covering, {Vi}m

i=1 . Let φ 1 (x) equal the sum of all

ψ j (x) such that spt(

ψ j

)⊆V1. Next let φ 2 (x) equal the sum of all ψ j (x) which have not

already been included and for which spt(

ψ j

)⊆ V2. Continue in this manner. Since the

open sets, {Ui}∞

i=1 mentioned in Theorem 36.1.8 are locally finite, it follows from Lemma36.1.3 that each φ i is infinitely differentiable having support in Vi. This proves the corollary.

36.2 Integration On ManifoldsManifolds are things which locally appear to be Rn for some n. The extent to which theyhave such a local appearance varies according to various analytical characteristics whichthe manifold possesses.