By Barrett O'Neill (Auth.)
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Extra info for Elementary Differential Geometry
F* is going to be a function that assigns to each tangent vector v to E n at p a tangent vector F#(v) to E w at F ( p ) . 5 we are consistently abbreviating v p to simply v. Now the image of a under the mapping F is the curve ß such that ß(t) = F(a(t)) = F(p + tv). We define F*(v) to be the initial velocity ß'(0) of ß (Fig. 16). Summarizing this process, we obtain the following definition. 4 Definition Let F: En —■> E m be a mapping. If v is a tangent vector to E at p, let F*(v) be the initial velocity of the curve t —> F(p + Zv) in E w .
Thus a\ = a2 = a3 = 0, as required. Now the tangent space Tp (E ) has dimension 3, since it is linearly isomorphic to E3. Thus by a well-known theorem of linear algebra, the three independent vectors d, e2, e3 form a basis for Tp (E 3 ). Hence for each vector v there are three (unique) numbers Ci, c2, c3 such that V = 51 CieiBut v#e i = (51 dei)*ej = 51 cßij = °jy and thus v = 51 (v-et-)ei. | This result (valid in any inner-product space) is one of the great laborsaving devices in mathematics. For to find the coordinates of a vector v with respect to an arbitrary basis, one must in general solve a set of nonhomogeneous linear equations, a task which even in dimension 3 is not always entirely trivial.
This parabola has the same shape as the parabola y = KOX2/2 in the xy plane, and is completely determined by the curvature KO of ß at s = 0. Finally, the torsion r 0 , which appears in the last and smallest term of 0, controls the motion of ß orthogonal to its osculating plane at 0 ( 0 ) , as shown in Fig. 10. On the basis of this discussion, it is a reasonable guess t h a t if a unitspeed curve has curvature identically zero, then it is a straight line. 3, since K = || Tf || = || ß" ||, so t h a t K = 0 if and only if 0 " = 0.