It turns out there are more general formulations of Stoke’s theorem than what is presented
above. However, it is always necessary for the surface S to be orientable. This means it is
possible to obtain a vector field of unit normals to the surface which is a continuous function
of position on S.

An example of a surface which is not orientable is the famous Mobeus band, obtained by
taking a long rectangular piece of paper and gluing the ends together after putting a twist in
it. Here is a picture of one.

PICT

There is something quite interesting about this Mobeus band and this is that it can be
written parametrically with a simple parameter domain. The picture above is a maple graph
of the parametrically defined surface

(
{ x = 4cosθ+ v cos θ2
R (θ,v) ≡ y = 4sin θ+ vcos θ2, θ ∈ [0,2π],v ∈ [− 1,1].
( z = vsin θ
2

An obvious question is why the normal vector R_{,θ}× R_{,v}∕

|R,θ ×R,v|

is not a continuous
function of position on S. You can see easily that it is a continuous function of both θ and v.
However, the map, R is not one to one. In fact, R

(0,0)

= R

(2π,0)

. Therefore, near this
point on S, there are two different values for the above normal vector. In fact, a tedious
computation will show that this normal vector is

( )
-4sin 12θcosθ-−-12v,4sin-12θsin-θ+-12v,−-8cos2-12θsin-12θ−-8cos3 12θ-+-4cos 12θ
D

where

( ( ) 2 ( )
D = 16 sin2 θ + v-+ 4 sin θ v(sin θ− cosθ)
2 2 2 )
3 2( θ)( (1 ) (1 ) 2( 1 ) 1)2
+ 4 cos 2 cos 2 θ sin 2 θ + cos 2θ − 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 was
careful 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 run by 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 orientation for
the Mobeus band. However, this does not answer the question whether there is some
orientation for it other than this one. In fact there is none. You can see this by looking at the
first of the two pictures below or by making one and tracing it with a pencil. There
is only one side to the Mobeus band. An oriented surface must have two sides,
one side identified by the given unit normal which varies continuously over the
surface and the other side identified by the negative of this normal. The second
picture below was taken by Ouyang when he was at meetings in Paris and saw it at a
museum.