Page 134 - Vector Analysis

130 CHAPTER 4. Vector Calculus

␣φi(Ui), φi´1( is called a local parametrization of M. The collection of charts Φ = t Ui, φiuiPI
is called an atlas.

Two charts t Ui, φiu and t Uj, φju are said to be C r-compatible or have C r-overlap if

the coordinate change φj ˝ φ´i 1 : φi(Ui X Uj) Ñ φj(Ui X Uj)

is of class C r. An atlas Φ on M is called C r if every pair of its charts is C r-compatible.

A maximal C r-atlas α on M is called a differentiable structure, and the pair tM, αu is

called a manifold of class C r.

A function f : M Ñ R is said to be of class C r if f ˝ φi´1 : Ui Ñ R is of class C r for all
charts tUi, φiu.

In particular, a regular C 1-curve C Ď R3 is a one-dimensional C 1-manifold, and a regular
C 1-surface Σ Ď R3 is a two-dimensional C 1-manifold.

Definition 4.67 (Metric). Let Σ Ď Rn be a (n´1)-dimensional manifold. The metric tensor

associated with the local parametrization tV, ψu (at p P Σ) is the matrix g = [gαβ](n´1)ˆ(n´1)

given by n

gαβ = ψ,α ¨ψ,β = ÿ B ψi B ψi in V .

i=1 B yα B yβ

Proposition 4.68. Let Σ Ď Rn be a (n ´ 1)-dimensional manifold, and g = [gαβ](n´1)ˆ(n´1)

be the metric tensor associated with the local parametrization tV, ψu (at p P Σ). Then the

metric tensor g is positive definite; that is,

n´1 @v = n´1 Bψ ‰ 0.
B yγ
ÿ gαβvαvβ ą 0 ÿ vγ

α,β=1 γ=1

Definition 4.69 (The first fundamental form). Let Σ Ď Rn be a (n ´ 1)-dimensional mani-
fold, and g = [gαβ](n´1)ˆ(n´1) be the metric tensor associated with the local parametrization
tV, ψu (at p P Σ). The first fundamental form associated with the local parametrization
tV, ψu (at p P Σ) is the scalar function g = det(g).

Definition 4.70 (Surface integrals). Let M be an (n ´ 1)-dimensional C 1-manifold, tUiuiPI

be a collection of charts of M and tζiuiPI is a partition-of-unity of M subordinate to tUiuiPI.

The “surface integral” (or simply integral) of a scalar function f : M Ñ R over M, denoted

ż

by f dS, is defined by

M

żż [ ) ˝ φ´1 ]?gi dx ,
(ζif
ÿ
f dS =

M iPI φi(Ui)