Advanced Calculus fi..

Chapter 2 Differential Calculus of Functions of Several Variables 123where .r is distance along the path. One can interpret (2.85) as a vector functionwhere r is the position vector:Thenr = xi + yj.cir dy . ,v = i + -J = r(t).dt cltThe vector v is tangent to the path at the point (.r(t), y(t)). Normally, one can use sas a parameter, instead oft, and then the velocity vector becomesa litlit tangent vector, since by (2.86),[For a detailed discussion, see CLA, pp. 329-336.1The preceding analysis extends to curves in space with no significant change.A curve in space has parametric equationsand these are equivalent to a vector function r = r(t), where nowas in Fig. 2.1 I . The velocity vectorv =-idx+dv. dzL J + -kdt dt dt= rl(t)is again tangent to the path and has magnitude Ivl = dsldt, where s is distancealong the path, increasing with t. The corresponding unit tangent vector is1 dx dv. dzT =-v = -i + -J + -k.Ivl ds ds dsFigure 2.1 1Curve in space, velocity vector v. and unit tangent vector T