440 CHAPTER 23. STOKES AND GREEN’S THEOREMS
orientable surface.
23.3.2 The Mobeus BandIt turns out there are more general formulations of Stoke’s theorem than what is presentedabove. However, it is always necessary for the surface S to be orientable. This meansit is possible to obtain a vector field of unit normals to the surface which is a continuousfunction of position on S.
An example of a surface which is not orientable is the famous Mobeus band, obtainedby taking a long rectangular piece of paper and gluing the ends together after putting a twistin it. Here is a picture of one.
There is something quite interesting about this Mobeus band and this is that it can bewritten parametrically with a simple parameter domain. The picture above is a maple graphof the parametrically defined surface
R(θ ,v)≡
x = 4cosθ + vcos θ
2
y = 4sinθ + vcos θ
2 ,
z = vsin θ
2
θ ∈ [0,2π] ,v ∈ [−1,1] .
An obvious question is why the normal vector R,θ ×R,v/∣∣R,θ ×R,v
∣∣ is not a continuousfunction of position on S. You can see easily that it is a continuous function of both θ andv. However, the map, R is not one to one. In fact, R(0,0) =R(2π,0). Therefore, nearthis point on S, there are two different values for the above normal vector. In fact, a tediouscomputation will show that this normal vector is(
4sin 12 θ cosθ − 1
2 v,4sin 12 θ sinθ + 1
2 v,−8cos2 12 θ sin 1
2 θ −8cos3 12 θ +4cos 1
2 θ)
D
where
D =
(16sin2
(θ
2
)+
v2
2+4sin
(θ
2
)v(sinθ − cosθ)
+ 43 cos2(
θ
2
)(cos(
12
θ
)sin(
12
θ
)+ cos2
(12
θ
)− 1
2
)2)
and you can verify that the denominator will not vanish. Letting v = 0 and θ = 0 and 2π
yields the two vectors (0,0,−1) ,(0,0,1) so there is a discontinuity. This is why I wascareful to say in the statement of Stoke’s theorem given above that R is one to one.
The Mobeus band has some usefulness. In old machine shops the equipment was runby a belt which was given a twist to spread the surface wear on the belt over twice the area.
The above explanation shows that R,θ ×R,v/∣∣R,θ ×R,v
∣∣ fails to deliver an orientationfor the Mobeus band. However, this does not answer the question whether there is some