By Barrett O'Neill (Auth.)

ISBN-10: 1483231704

ISBN-13: 9781483231709

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Additional resources for Elementary Differential Geometry

Example text

2) If F*(v p ) = 0, then vp = 0. (3 ) The Jacobian matrix of F at p has rank n (dimension of the domain E n of F). 3 is not regular. But the one-to-one condition fails at only a single point, the origin. 8 shows that is Jacobian matrix has rank 2 at p ^ 0, rank 0 at 0. A mapping that has an inverse mapping is called a diffeomorphism. A diffeomorphism is thus necessarily both one-to-one and onto, but a mapping Sec. 7] 39 MAPPINGS which is one-to-one and onto need not be a diffeomorphism (Exercise 11).

3 ] K(S) 59 = || T'(s) || = \ 2 = - 25 ^ —2 > 0. a + o c Since 71' = /c/V, we get N(s) = ( —cos- , — sin- , 0 ) \ c c / Note that regardless of what values a and b have, N always points straight in toward the axis of the cylinder on which ß lies (Fig. 8). Applying the definition of cross product to B = T X N, we get E»/ \ (b . s b s a\ B{s) = l - sin - , —- cos , - 1. \c c c c c) It remains to compute torsion. Now B\s) = ( l- c o s - , -=l s i n - , 0 ) , \c c c c / and by definition, B = — TN.

We emphasize this topic not only because of its intrinsic importance, but also because the basic method used to investigate curves has proved effective throughout differential geometry. A curve in E3 is studied by assigning at each point a certain frame—that is, set of three orthogonal unit vectors. 2). In a real sense the theory of curves in E is merely a corollary of these fundamental formulas. Later on we shall use this "method of moving frames" to study a surface in E3. The general idea is to think of a surface as a kind of two-dimensional curve and follow the Frenet approach as closely as possible.