15_C_2 - coincide as point sets but their normals are...

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2. This is a trick question. Observe that the given parametrization r ( u , v) satisfies r ( u + π, v) = r ( u , - v). Therefore the surface is traced out twice as u goes from 0 to 2 π . (It is a M¨obius band. See Figure 15.28 in the text.) If 1 is the part of the surface corresponding to 0 u π , and 2 is the part corresponding to π u 2 π , then 1 and
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Unformatted text preview: coincide as point sets, but their normals are oppositely oriented: ˆ N 2 = -ˆ N 1 at corresponding points on the two surfaces. Hence ZZ 1 F • ˆ N 1 dS = -ZZ 2 F • ˆ N 2 dS , for any smooth vector field, and ZZ F • ˆ N dS = ZZ 1 F • ˆ N 1 dS + ZZ 2 F • ˆ N 2 dS = ....
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