Page 128 - Vector Analysis

124 CHAPTER 4. Vector Calculus

if spt(ζj) Ď ψi(Vi). Similarly, for a bounded continuous function f defined on Σ, the surface
integral of f over Σ can be defined by

żż ż (ζjf ) ˝ ψi?gi dS .

ÿÿ ÿ (4.15)

f dS = (ζjf ) dS =

Σ jPJ Σ jPJ choose one i such that Vi

spt(ζj ) Ď ψi(Vi)

Remark 4.52. Defining the surface integrals of a function as above, a question arises natu-
rally: is the surface integral given by (4.15) independent of the choice of the parametrization
and the partition-of-unity? In other words, if a regular C k-surface Σ admits two collections
of local parametrization t Ui, φiuiPI and tVj, ψjujPJ , and tζiuiPI and tλjujPJ are C k-partition-
of-unity subordinate to tUiuiPI and tVjujPJ , respectively. Is it true that

ÿ ÿ ż (ζif ) ˝ φi?gi dS = ÿ ÿ ż (λjf ) ˝ ψj?gj dS ,

iPI choose one i such that Ui jPJ choose one j such that Vj

spt(ζj ) Ď φi(Ui) spt(λk ) Ď ψj (Vj )

where gi and gj are the first fundamental form associated with the parametrization t Ui, φiu
and tVj, ψju, respectively.

The answer to the question above is affirmative, and the surface integral given by (4.15)
is indeed independent of the choice of parametrization of the surface and the partition-of-
unity; however, we will not prove this and only treat this as a known fact.

Now we focus on the existence of a collection of functions tζjujPJ discussed above.

Definition 4.53. A collection of subsets of Rn is said to be locally finite if for every point
x P Rn there exists r ą 0 such that B(x, r), the ball centered at x with radius r, intersects
at most finitely many sets in this collection.

Definition 4.54 (Partition of Unity). Let A Ď Rn be a subset. A collection of functions
tζjujPJ is said to be a partition-of-unity of A if

1. 0 ď ζj ď 1 for all j P J .

2. The collection of sets ␣spt(ζj)(jPJ is locally finite.

3. ř ζj(x) = 1 for all x P A.

jPJ

Let t UjujPJ be an open cover of A; that is, Uj is open for all j P J and A Ď Ť Uj .

jPJ

A partition-of-unity tζjujPJ of A is said to be subordinate to t UjujPJ (or tUjujPJ has a

subordinate partition-of-unity of A) if spt(ζj) Ď Uj for all j P J .