Page 131 - Vector Analysis

§4.4 Oriented Surfaces 127

4.4 Oriented Surfaces

In the study of surfaces, orientability is a property that measures whether it is possible to
make a consistent choice of surface normal vector at every point. A choice of surface normal
allows one to use the right-hand rule to define a “counter-clockwise” direction of loops in
the surface that is required in the presentation of the Stokes theorem (Theorem 4.86), a
main result in vector calculus which will be introduced later in Section 4.7.2.

Definition 4.60. A regular surface Σ Ď R3 is said to be oriented if there exists a contin-
uous vector-valued function N : Σ Ñ R3 such that }N}R3 = 1 and for all p P Σ, N(p) ¨ v = 0
for all v P TpΣ. Such a vector-field N is called a unit normal of Σ.

Suppose that Σ Ď R3 is a connected regular surface. Since at each p P Σ the tangent
plane TpΣ of Σ at p has two normal directions, Σ has at most two continuous unit normal
vector fields. If in addition that Σ is oriented, there are exactly two continuous unit normal
vector fields of Σ, and one is the opposite of the other. The two unit normal vector fields
define two sides of the surface.

Suppose further that this oriented surface Σ is the boundary of an open set Ω Ď R3
(for example, a sphere is the boundary of a ball), then one of the unit normal vector fields
N : B Ω Ñ R3 has the property that p + tN(p) R Ω for all small but positive t. Such
a normal is called the outward-pointing unit normal of B Ω, and the opposite of the
outward-pointing unit normal of B Ω is called the inward-pointing unit normal of B Ω.

Example 4.61. Consider the unit sphere S2 = ␣(x, y, z) P R3 ˇ x2 + y2 + z2 = 1(. Then
ˇ

N : S2 Ñ R3 defined by N(p) = p, where the right-hand side is treated as the vector

p ´ 0, is a continuous unit normal vector field on Σ; thus S2 is an oriented surface. Let

B(0, 1) = ␣(x, y, z) P R3 ˇ x2 + y2 + z2 ă 1( be the unit ball in R3. Then N is the outward-
ˇ

pointing unit normal of BB(0, 1).

Let Σ Ď R3 be a regular surface, p P Σ, and tV, ψu be a local parametrization of Σ at p.
Since ψ,1 and ψ,2 are linearly independent, ψ,1 ˆψ,2 ‰ 0; thus the vector n given by

n = ψ,1 ˆψ,2 ˝ ψ´1
}ψ,1 ˆψ,2 }R3

is a unit normal vector field on ψ(V). As a consequence, a regular C 1-surface that can be
parameterized by one single parametrization tV, ψu; that is, Σ = ψ(V), is always oriented.