. Differential and integral calculus, an introductory course for colleges and engineering schools. cos a + sin a > arc PQ > 1. chord PQ As Q =P, a = 0, sin a = 0, cos a — 1, and consequently arc PQ (a) q±p chord PQ140 = 1. Q. E. D. §101 THE DERIVATIVE OF THE ARC 141 sin $ 1, which was proved in Art. 11, is obvi- The theorem lim 0=0 9 ously a special case of this theorem (a). 101. The Derivative of the Arc: Cartesian Coordinates. Let P and Q be any two points of a curve, and let their coordinatesbe x, y and x + Ax, y + s be the length of the arcAP measured from any con-venient poin
. Differential and integral calculus, an introductory course for colleges and engineering schools. cos a + sin a > arc PQ > 1. chord PQ As Q =P, a = 0, sin a = 0, cos a — 1, and consequently arc PQ (a) q±p chord PQ140 = 1. Q. E. D. §101 THE DERIVATIVE OF THE ARC 141 sin $ 1, which was proved in Art. 11, is obvi- The theorem lim 0=0 9 ously a special case of this theorem (a). 101. The Derivative of the Arc: Cartesian Coordinates. Let P and Q be any two points of a curve, and let their coordinatesbe x, y and x + Ax, y + s be the length of the arcAP measured from any con-venient point A. Then s is afunction of x, and arc PQ isthe increment of s due to theincrement Ax of x. We there-fore set arc PQ = As. Then lim -T— = Dxs, and we seek to Ax=oA£ express Dxs in terms of Dxy. Let c be the length of the chord PQ. We may suppose Q taken at the start so near P that arc PQ is everywhere concave towards its chord. Then theorem (a) of the preceding article applies, and we have As. lim —q=p c 1, and lim Ax=0 C 1. Therefore by the principle of Art. 10As — = 1 + e and As = (1 + e)c, where e is infinitesimal. From the figure c = V(A#)2 + (Ay) (1) As = (1 + e)V(Axy+(byy and ^ - (1+ *)\Jl + (||J- Passing to limits, we have the important formula I 2>^= Vl-f-(J^)2= Vl + V 142 DIFFERENTIAL CALCULUS §101 In the differential notation I has the forms (b) ds = Vdx* + dy2. Regarding x, y, and s as functions of a parameter t, we may dividethe first equation of (1) by At, and then, on taking limits, we have II Dts = V(I>tx.)2-\-(I>ty)2, sl formula which reduces to I when t = x. Writing II in the differential notation, we have , ds dt -A%+(W In this formula t may be x, y, s, or any fourth variable. ClearingIF of fractions, we have(2) ds = Vdvc2 + dy2, which is I(b) again, and is here proved to hold whatever theindependent variable may be, whether x, y, s, or any fourth vari-able. It was shown in Art. 73 that the differential of the ordinate of acurve is
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