Page 129 - Vector Analysis

§4.3 The Surface Integrals 125

We note the if tζjujPJ is a partition-of-unity of A, then the property of local finiteness of
tspt(ζj)ujPJ ensures that for each point x P A has a neighborhood on which all but finitely
many λj’s are zero.

Lemma 4.55. Let A Ď Rn be a subset, tUiuiPI be an open cover of A, and tVjujPJ be a
collection of open sets such that each Vj is a subset of some Ui; that is, for each j P J ,
Vj Ď Ui for some i P I. If tVjujPJ has a subordinate C k-partition-of-unity of A, so has
tUiuiPI .

Proof. Let tζjujPJ be a partition-of-unity of A subordinate to tVjujPJ , and f : J Ñ I

be a map such that Vj Ď Uf(j) (we note that such f in general is not unique). Define

χi : Rn Ñ [0, 1] by ÿ (4.16)

χi(x) = ζj(x) .

jPf ´1(i)

Then clearly spt(χi) Ď Ui and ř χi(x) = 1 for all x P A. Moreover, since the sum (4.16)

iPI

is a finite sum, χi is of class C k for all i P I since ζj if of class C k for all j P J . Now

we show that ␣spt(χi)(iPI is locally finite. Let x P Rn be given. By the local finiteness of

␣spt(ζj )(jPJ there exists r ą 0 such that #␣j P J ˇ B(x, r) X spt(ζj ) ‰ H( ă 8. By the
ˇ

fact that f ´1(i1) X f ´1(i2) = H if i1 ‰ i2 (that is, each j P J belongs to f ´1(i) for exactly

one i P I) and that

y P B(x, r) X spt(χi) ô y P B(x, r) X spt(ζj) for some j P f ´1(i) ,

we must have

#␣i P I ˇ B(x, r) X spt(χi) ‰ H( ď #␣j P J ˇ B(x, r) X spt(ζj ) ‰ H( ă 8. ˝
ˇ ˇ

Theorem 4.56. Let Σ Ď R3 be a regular C k-surface. Then every open cover of Σ has a
subordinate C k-partition-of-unity of Σ.

Proof. Let tOiuiPI be a given open cover of Σ. Let t Uj, φjujPJ be a collection of C k-charts
of Σ such that t UjujPJ is a locally finite open cover of Σ and for each j P J , U j Ď Oi for
some i P I. By Lemma 4.55, it suffices to find a C k-partition-of-unity of Σ subordinate to

t UjujPJ .
W.L.O.G., we can assume that Uj and Vj ” φ(Uj) is bounded for all j P J . Define

ψj = φ´j 1. Then tVj, ψjujPJ is a collection of local parametrization of Σ. Choose a collection
of open sets tWjujPJ such that Wj Ď Vj for all j P J and ␣ψj(Wj)(jPJ is still an open cover