Page 114 - Vector Analysis

110 CHAPTER 4. Vector Calculus

Definition 4.32. A connected domain D is said to be simply connected if every simple
closed curve can be continuously shrunk to a point in D without any part ever passing out
of D.

Theorem 4.33. Let D Ď R2 be simply connected, and F = (M, N ) : D Ñ R2 be of class
C 1. If My = Nx, then F is conservative.

The theorem above can be proved using Theorem 4.30 and Green’s theorem (Theorem
4.90), and is left till Section 4.8 (where Green’s theorem is introduced).

4.3 The Surface Integrals

4.3.1 Surfaces

Definition 4.34. A subset Σ Ď R3 is called a surface if for each p P Σ, there exist an open
neighborhood U Ď Σ of p, an open set V Ď R2, and a continuous map φ : U Ñ V such
that φ : U Ñ V is one-to-one, onto, and its inverse ψ = φ´1 is also continuous. Such a
pair tU, φu is called a coordinate chart (or simply chart) at p, and tV, ψu is called a (local)
parametrization at p.

Remark 4.35. In some literatures the surface is defined in the following equivalent but
reversed way: A subset Σ Ď R3 is a surface if for each p P Σ, there exists a neighborhood
U Ď R3 of p and a map ψ : V Ñ U XΣ of an open set V Ď R2 onto U XΣ Ď R3 such that ψ is
a homeomorphism; that is, ψ has an inverse φ = ψ´1 : U X Σ Ñ V which is continuous. The
mapping ψ is called a parametrization or a system of (local) coordinates in (a neighborhood
of) p.

Definition 4.36 (Regular surfaces). A surface Σ Ď R3 is said to be regular if for each

p P Σ, there exists a differentiable local parametrization tV, ψu of Σ at p such that Dψ(q),

the derivative of ψ at q, has full rank for all q P V) ; that is, Dψ(q) : R2 Ñ R3 is one-to-one
(
for all q P V. The range of the map Dψ ψ´1(p) is called the tangent plane of Σ at p,

and is denoted by TpΣ.

In the following, we always assume that Dψ(q) has full rank for all q P V if tV, ψu
is a local parametrization of a regular surface Σ Ď R3.