We were able to endow a curve with an orientation by using the fact that we can move along it in two opposite directions. This does not work for surfaces. Hence we have to orient them in a different way. If we pick up a sheet of paper then it has two sides, the front and the back. If the sheet is blank then it is not clear which is the front and which is the back. We need to make a choice, and lay it down one way, and say, the side facing up is the front. (Maybe you mark the front by writing a headline on it.) If we have that piece of paper, \(S\text{,}\) one side `faces up’ and the other side `faces down.’ Mathematically we can express this by attaching to every point of the sheet a vector perpendicular to \(S\) pointing `upwards’. Having done this, we can move the sheet and, looking at the vectors, it is still clear what the `front’ and the `back’ of \(S\) are. Hence the vector field chosen `marks’ the front of \(S\text{,}\) as the headline you put on a sheet of paper, and thus gives it an `orientation.’ The two possible normal fields are shown in Figure 9.10.
Figure9.10.The two possible orientations of a “sheet of paper.”
To define an orientation for a `curved’ smooth surface, \(S\text{,}\) we proceed the same way as with the sheet of paper. We attach to every point a vector perpendicular to \(S\) and normalise it to length one. Such a vector is called a unit normal vector to \(S\). As usual normal to \(S\) means normal to the tangent plane to \(S\text{.}\) At every point there are exactly two such vectors. Sometimes it is possible to choose these normal vectors in such a way that they vary continuously as we move around on the surface. As with the sheet of paper these fields can be used to give the \(S\) an orientation. Fix one of the vector field and call it the positive unit normal vector field. A surface with both possible continuous unit normal vector fields is shown in Figure 9.12.
Definition9.11.Orientation of a surface.
Suppose that \(S\) is a smooth surface in \(\mathbb R^3\text{.}\) The surface is called orientable if there exists a continuous unit normal vector field on \(S\text{.}\) We can fix one of them and call it the positive unit normal vector field. Equipped with this vector field \(S\) is called positively oriented. Equipped with the opposite orientation \(S\) is called negatively oriented.
Figure9.12.The two possible unit normal vector fields on a surface giving rise to opposite orientations.
Remark9.13.
As opposed to curves, not every surface is orientable! The Möbius strip is the standard example of non-orientable surface. It is impossible to get a continuous unit normal vector field. We can start putting unit normals at one point and move continuously along the band back to the same point. Arriving there the unit normal will point in the opposite direction to that in which we started. A Möbius strip and a (discontinuous) unit normal vector field on it is shown in Figure 9.14. The reason a Möbius strip is not orientable is that it has only “one side.”
Figure9.14.A non-orientable surface: The Möbius strip
Up to now we did not worry about how to compute unit normals to a surface, but want to do so now. There are two possibilities: using an implicit representation, or using a parametric representation of the surface. The first is much easier than the second, but we will be required to use both of them.
