Orientation of a Surface

Abstract

We know from Chap. 4 that in order to evaluate the flux of a vector field across a regular surface S, we need to choose a unit normal vector at each point of S in such a way that the resulting vector field is continuous. For instance, if we submerge a permeable sphere into a fluid and we select the field of unit normal outward vectors on the sphere, then the flux of the velocity field of the fluid across the sphere gives the amount of fluid leaving the sphere per unit time. However, if we select the field of unit normal inward vectors on the sphere, then the flux of the velocity field of the fluid across the sphere gives the amount of fluid entering the sphere per unit time (which is the negative of the flux obtained in the first case). So, it is a natural question to ask which (if not all) regular surfaces admit a continuous field of unit normal vectors. The regular surfaces admitting such a continuous vector field are called orientable surfaces. Most common surfaces, such as spheres, paraboloids, and planes, are orientable. However, there do exist surfaces that are not orientable.