Page 130 - Vector Analysis

126 CHAPTER 4. Vector Calculus

of Σ. For each j P J, let ␣Bk(j )(Nj be a collection of open balls satisfying Wj Ď Nj

k=1 Ť Bk(j)

k=1
and cl(Bk(j)) Ď Vj for all k P t1, ¨ ¨ ¨ , Nju. For j P J and k P t1, ¨ ¨ ¨ , Nju, with cj,k and rj,k

denoting the center and the radius of B( k(j), respectively, let
)
$ 1 if x P Bk(j) ,
& exp }x ´ cj,k}2R2 ´ rj2,k
µ(j,k)(x) =
if x R Bk(j) ,
%0

Nj

and then define χj : R2 Ñ R by χj(x) = ř µ(j,k)(x). Then χj ą 0 in Wj, and χj = 0

Nj k=1

outside Ť Bk(j). Further define

k=1

# (χj ˝ φj)(x) if x P Uj ,
0 if x P UjA .
λj(x) =

Then λj ą 0 on ψj(Wj) which implies that ř λj ą 0. Finally, we define ζj = λj .
ř λj
jPJ
jPJ

Then tζjujPJ is a C k-partition-of-unity subordinate to the open cover t UjujPJ . ˝

Definition 4.57 (Piecewise Regular Surface). A surface Σ Ď R3 is said to be piecewise

k

regular if there are finite many curves C1, ¨ ¨ ¨ , Ck such that Σz Ť Ci is a disjoint union of

regular surfaces. i=1

Definition 4.58. Let Σ Ď R3 be a piecewise regular surface such that Σ is the disjoint

union of regular surfaces Σi, where i P I for some finite index set I. For a continuous

ż

function f : Σ Ñ R, the surface integral of f over Σ, still denoted by f dS, is defined by

żż Σ

ÿ

f dS = f dS .

Σ iPI Σi

Definition 4.59. Let RΣ be the collection of piecewise regular surfaces in R3. The surface

element is a set function S : RΣ Ñ R that satisfies the following properties:

1. S (Σ) ą 0 for all Σ P RΣ.

2. If Σ is the union of finitely many regular surfaces Σ1, ¨ ¨ ¨ , Σk that do not overlap
except at their boundaries, then

S (Σ) = S (Σ1) + ¨ ¨ ¨ + S (Σk) .

3. The value of S agrees with the area on planar surfaces; that is,

S (P) = A(P) for all planar surfaces P.