Advanced Calculus fi..

Chapter 5 Vector Integral Calculus 281one end of C to the other. For the work done is defined in mechanics as "force timesdistance" or, more precisely, as follows:work = (tangential force component) ds,(5.30)Sand ur is precisely this tangential force component.The line integral [ P dx + Q dy can be defined directly as a vector integral asfollows:where[u.dr=n+oo lirnn maxA,s+O 1=1CCU(X:,Y;).A,I. I , Iand A,r = Aixi + A,yj. Equation (5.17) can then be written in the vector form:. v' 2If C is represented in terms of a parameter t, then' ' -.'>fIf r is the position vector of a particle of mass m moving on C and u is the forceapplied, then by Newton's Second Law, t: 1 sHence, if Ivl = v,. lk = dt (lmy2) 2One thus concludes thatdl.that is, the work done equals the gain in kinetic energy. This is a basic law ofmechanics.