Advanced Calculus fi..

Chapter 5 Vector Integral Calculus 307(a)(b)Figure 5.26 (a) Oriented smooth surface. (b) Miibius strip, a nonorientablesurface.iFigure 5.27Oriented piecewise smooth surface (cylinder).If the surface S is only piecewise smooth, as is, for example, the surface of acylinder, one cannot choose a continuously varying normal vector for all of S. In thiscase we say that S has been oriented if an orientation has been chosen in each smoothpiece of S, in such a manner that along each curve that is a common boundary oftwo pieces the positive direction relative to one piece is the opposite of the positivedirection relative to the other piece. This is illustrated for a cylindrical surface inFig. 5.27. If a piecewise smooth surface S can be so oriented, we term the surface Sorientable.It should be remarked that not every surface is orientable. Figure 5.26(b) suggestsa Mobius strip, which is nonorientable. For this surface S, one easily convincesone's self that the normal n cannot be chosen to vary continuously on S and thatthe surface has the peculiar property of having only "one side." One-sidedness andnonorientability go together for surfaces in space.