Mathematical Methods for Physics and Engineering - Matematica.NET

VECTOR CALCULUSTherefore, remembering that u = x, from (10.12) the arc length between x = a and x = bis given by∫ b√ √drs =a du · dr ∫ b( ) 2 dydu du = 1+ dx.a dxThis result was derived using more elementary methods in chapter 2. ◭If a curve C is described by r(u) then, by considering figures 10.1 and 10.3, wesee that, at any given point on the curve, dr/du is a vector tangent to C at thatpoint, in the direction of increasing u. In the special case where the parameter uis the arc length s along the curve then dr/ds is a unit tangent vector to C and isdenoted by ˆt.The rate at which the unit tangent ˆt changes with respect to s is given byd ˆt/ds, and its magnitude is defined as the curvature κ of the curve C at a givenpoint,∣ ∣ κ =d ˆt∣∣∣ ∣ ds ∣ = d 2 ˆr ∣∣∣ds 2 .We can also define the quantity ρ =1/κ, which is called the radius of curvature.Since ˆt is of constant (unit) magnitude, it follows from (10.8) that it is perpendicularto d ˆt/ds. The unit vector in the direction perpendicular to ˆt is denotedby ˆn and is called the principal normal at the point. We therefore haved ˆt= κ ˆn. (10.13)dsThe unit vector ˆb = ˆt × ˆn, which is perpendicular to the plane containing ˆtand ˆn, is called the binormal to C. The vectors ˆt, ˆn and ˆb form a right-handedrectangular cooordinate system (or triad) at any given point on C (see figure 10.3).As s changes so that the point of interest moves along C, the triad of vectors alsochanges.The rate at which ˆb changes with respect to s is given by d ˆb/ds and is ameasure of the torsion τ of the curve at any given point. Since ˆb is of constantmagnitude, from (10.8) it is perpendicular to d ˆb/ds. We may further show thatd ˆb/ds is also perpendicular to ˆt, as follows. By definition ˆb · ˆt =0,whichondifferentiating yields0= d ( )ˆb · ˆt = d ˆbds ds · ˆt + ˆb · d ˆtds= d ˆbds · ˆt + ˆb · κ ˆn= d ˆbds · ˆt,where we have used the fact that ˆb · ˆn = 0. Hence, since d ˆb/ds is perpendicularto both ˆb and ˆt, we must have d ˆb/ds ∝ ˆn. The constant of proportionality is −τ,342