Orientation and Orientability

Learning Goals: students see that things get a little weird with vector surface integrals when surfaces can have only one side.

The surface vector integral F⋅dS involves a dot product, so the direction of dS makes a ∫S difference. Like a line integral where if you go along the path in the opposite direction you end up with the opposite answer, the same is true here. What are we to make of this opposite-ness here? What is actually opposite?

Since we are measuring the flow of the vector field through the surface, the obvious question is which way through the surface is the field flowing. For instance, if we have a surface z = f(x, y) the surface has a top and a bottom, and we can ask if the flow is upward or downward. If our surface is a sphere, we can ask whether the flow is inward or outward.

In a little piece of the surface, there are two unit normal (say, upward and downward, or inward and outward). On many surfaces, we can pick one unit normal at every point in a continuous fashion. For instance, on the sphere, we can pick the outward unit normal at every point, as shown in the picture. Surface vector integrals make sense on such surfaces, which are called orientable.

On the other hand, there are other kinds of surfaces, like Möbius strips, where you can’t do this. Though each tiny piece of the surface has two sides, overall the surface is connected together in such a way that the top has been attached to the bottom at some point. As you try to move a unit normal around, you can end up back where you started, but on the other side—the point on the surface is the same, but the normal direction has changed.

These kinds of surfaces are called non-orientable and surface vector integrals do not make any sense for them. After all, you can’t flow from one side to the other when there is only one side!

Definition: an oriented surface is an orientable surface together with a choice of consistent unit normal on the surface.

We usually orient graphs of functions with the upward normal, and closed surfaces with the outward normal. On graphs, this makes the orientation consistent with the right hand rule—as you walk around the boundary of the surface with the surface on your left, you are standing on the positive side of the surface.

Definition: given an oriented surface, a parameterization for it is orientation preserving if N and the preferred unit normal are on the same side of the surface, and orientation reversing otherwise.

For example, in our standard parameterization of the sphere, we found that N is the vector R2 sin(ϕ) (sin(ϕ)cos(θ), sin(ϕ)sin(θ), cos(ϕ)). Since R2 sin(ϕ) is positive, this vector points outward away from the center of the sphere at each point (except the north and south poles, where it is zero—not smooth!—but we don’t worry about isolated points because they don’t add much to the integral). So the standard parameterization is orientation preserving with the normal outward choice of orientation for the sphere.