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

One carrier arithmetic has rendered the 'Et moi, ..., si j'avait su remark en revenir, je n'y serais element aile.' human race. It has placed logic again Jules Verne the place it belongs, at the topmost shelf subsequent to the dusty canister labelled 'discarded n- sense'. The sequence is divergent; for this reason we are able to do whatever with it. Eric T. Bell O. Heaviside Matht"natics is a device for idea. A hugely invaluable device in a global the place either suggestions and non linearities abound. equally, every kind of components of arithmetic seNe as instruments for different elements and for different sciences. employing an easy rewriting rule to the quote at the correct above one unearths such statements as: 'One provider topology has rendered mathematical physics .. .'; 'One provider common sense has rendered com puter technology .. .'; 'One carrier classification thought has rendered arithmetic .. .'. All arguably real. And all statements available this fashion shape a part of the raison d'etre of this sequence.

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Dldt (v. v) = vV + vV = 2vV = dldt (Ivf) = 0, therefore v • ~ = 0, which proves the lemma. 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.

V)If2, is the velocity vector. We shall write V, = dr/dt indicating explicitly, thereby, the parameter with respect to which the tangent vector is calculated. We shall often find it convenient to consider curves parametrized by the Mtwal (length) parameter. x = x(/), In this case V = y = y(/). Vi = (dx/d/)el + (dyld/)e2 Ivl = 1. If the curve was parametrized by an arbitrary parameter t, x =x(t), y =yet), we have the relation dl = 2 + y'2) 112 dt. Two vectors (those of velocity and acceleration) will play an imponant role: (x iT ~ = v" 47 FLAT CURVES If the parameter is natural (t =l), we shall have Iv~ = 1.

Euclidean metric. 1. Let n = 2. IJ = (g~ = i¢j, while relative to polar coordinates we have: This means that for the curve r =r(t), $ =$(t) b /= f tT 2 2 ('(t) +r d$ 2 (iF) dr. a 2. n = 3. Relative to Canesian coordinates, we have gij = 5U; relative to cynlindrical I, z =r: coordinates r =yl, cI> =