Some surfaces have a rim or boundary as for instance a half sphere. We assume that the rim is the finite union of piecewise smooth curves. Given a surface \(S\) we denote its boundary, or rim by \(\partial S\text{.}\) Assume now that \(S\) is an orientable surface and fix an orientation. Denote by \(\vect n\) the positive field of unit normal vectors to \(S\text{.}\) We now want to orient the boundary \(\partial S\) consistent with the orientation of \(S\text{.}\) We do this as follows. Suppose you stand at the boundary of \(S\) with the upright position given by the positive unit normal vector to \(S\text{.}\) Then walk along \(\partial S\) in such a way that your right hand points away from the surface as shown in Figure 13.1.
Figure13.1.An oriented surface with a positively oriented boundary.
