Page 133 - Vector Analysis

§4.5 Manifolds, Charts, Atlas and Differentiable Structure 129

Then the unit normal vector field on ψ(V) compatible with the parametrization tV, ψu is

(N ˝ ψ)(u, v) = ψ,1 ˆψ,2 = av2 + (4 2 cos(u/2))2 ˆ
(2 + v sin u ) sin u,
ˆ}ψ(,1vˆcψo,s2u}R+3 + 2v
2 sin u
2 2
´ v sin u + (2 + v u) u cos u, ´(2 + v cos u) cos u) ,
cos sin
2 22 2 2

but N does not have a continuous extension on M since if Nr is a continuous extension of
N; that is, Nr is a unit normal vector field on M and N = Nr on ψ(V), then

(0, 0, ´1) = lim (N ˝ ψ)(u, 0) = Nr (2, 0, 0) = lim (N ˝ ψ)(u, 0) = (0, 0, 1)
uÑ0+ uÑ2π´

which is a contradiction.
Another way of seeing that M is not oriented is the following. Let r(t) = G(t, 0) =

(´2 sin t, 2 cos t, 0), and C = r([0, 2π]) Ď M be a closed curve on M. If there is a continuous
unit normal vector field Nr on M, then Nr is also continuous on C. However, Nr is never
continuous on C since by moving N continuously along C, starting from r(0) and moving
along C in the direction r 1 and back to r(0) = r(2π), we obtain a different vector which
implies that Nr ˝ r is not continuous at r(0) = r(2π) = (2, 0, 0).

Definition 4.64. An open set Ω Ď R3 is said to be of class C k if the boundary B Ω is a
regular C k-surface.

Theorem 4.65. Let Ω Ď R3 be a bounded open set of class C 1. Then B Ω is oriented.

4.5 Manifolds, Charts, Atlas and Differentiable Struc-
ture

In the following, we introduce a more abstract concept, the so-called manifolds, which is a
generalization of regular surfaces.

Definition 4.66. A topological space M is called an n-dimensional manifold if it is
locally homeomorphic to Rn; that is, there is an open cover U = tUiuiPI of M such that
for each i P I there is a map φi : Ui Ñ Rn which maps Ui homeomorphically onto an open
subset of Rn. The pair t Ui, φiu is called a chart (or coordinate system) with domain Ui, and