Vector Calculus: An Introduction

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 $\mathit{n}$ 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.