Advanced Calculus fi..

268 Advanced Calculus, Fifth EditionFigure 5.1Line integral.Figure 5.2Work = J, F . dr.Here the curve has been subdivided into n pieces of lengths Als, Azs, . . . , A,s, andthe point (x:, y:) lies on the ith piece. The limit is taken as n becomes infinite, whilethe maximum A,s approaches 0.A third example of a line integral is that of work. If a particle moves from oneend of C to the other under the influence of a force F, the work done by this forceis defined aswhere F+ denotes the component of F on the tangent T in the direction of motion. Thisintegral can be thought of as a limit of a sum as previously. However, another interpretationis possible. We first remark that the work doneby a constant force F in moving aparticle from ASB on the line segment A B is F . AB; for this scalar product is equalto IF1 . cos or - 1AB 1 . o being the angle between F and 3. and hence to the productof force component in direction of motion by the distance moved. Now the motionof the particle along C can be thought of as the sum of many small displacementsalong line segments, as suggested in Fig. 5.2. If these displacements are denoted byAir, Azr, ..., A,,r, the work done would be approximated by a sum of form