Basic Elements of Differential Geometry and Topology by S.P. Novikov, A.T. Fomenko

By S.P. Novikov, A.T. Fomenko

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REMARK. If there exist any two vectors v = vet) and w = wet), then in Euclidean geometry there holds the formula: dldt (vw) = ~w + vW . In application to a curve parametrized by the natural parameter I = t, r =ret) = x(t)el + y(t)e2, our lemma suggests: v = drldl. COROLLARY. The velocity vector vet) and the acceleration vector wet) = dvldl are orthogonal if the parameter is nanual: t =I (the arc length). DEFINITION 1. The curvature of a flat curve is a magnitude of the acceleration vector k = Iw(t)1 provided that t =I (the natural parameter).

THEOREM 1. Given the parametric equation r = r(l) of a curve, in tenns of the narwal parameter I, thefollowing Frenetformu1ae hold: 7dv dn df = kn = w, = lev, 49 FLAT CURVES where n =1:1 is the unit normal vector. Proof. Since n is a unit vector, nn = I, and the vectors, n and v are onhogonal, according to Lemma I, we have: a) dnldl1. n (Lemma I), b) dnldl= av (n 1. v and the dimension = 2). Given Ivl = I, we have lal = Idnldll. What is the value of a? Since vn = 0, we have: o= d d (vn) = ddv n + v°dtJ d = k + a(vv) = k + a = 0, (nn Whence a = I, vv = 1).

Here we have used the notation: "Sp" (= Spur) = the trace of the matrix. It is sometimes denoted by"Tr" (= Trace). Furthermore, the question has arisen why for the natural parameter 1 t we have the equality: the absolute value of the acceleration = ~y -xyl = l[vw]1. The deri:vation of this equaltiy is as follows. The acceleration is equal to w = (x',';) and (ij - y}e3 is the vector product of the velocity vectors v x' el + ye2 by the acceleration w. lwllsin ,I = Iwllsin ,I = (Ivl = I). = = Since w J..

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Basic Elements of Differential Geometry and Topology by S.P. Novikov, A.T. Fomenko
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