Advanced Calculus fi..

272 Advanced Calculus, Fifth EditionFormulas (5.4) and (5.5) reduce the integrals to ordinary definite integrals andare thus essential for computation of particular integrals. Thus let C be the pathx = 1 + t, y = t2, 0 5 t 4 1 , directed with increasing t. Then1It is logically easier to prove I1 first, for I is an immediate consequence of 11. Toprove 11, one notes that the sum C f (x:, y:)A,x can be written asNow Aix = xi -xi-] = @'(tf*)Air by the Mean Value theorem. Hence the sum canbe written aswhere F(t) = f [#(t), $(t)] and tf and tf* are both between ti-] and ti. It is easilyshown [see CLA, Section 12-25] that this sum approaches as limit the integralas required. Formula (5.5) is proved in the same way.We remark that the value of a line integral on C does not depend on the particularparametrization of C, but only on the order in which the points of C are traced. (SeeProblem 5 after Section 5.3.)In many applications the path C is not itself smooth but is composed of a finitenumber of arcs, each of which is smooth. Thus C might be a broken line. In this case,C is termed piecewise smooth. The line integral along C is simply, by definition,the sum of the integrals along the pieces. One verifies at once that (5.2), (2.3), andthe theorems I and I1 continue to hold. In (5.4) and (5.5) the functions @'(r) and@'(t) will have jump discontinuities, which will not interfere with the existence ofthe integral (cf. Section 4.1). Throughout this book all paths of integration for lineintegrals will be piecewise smooth unless otherwise speciJied.If the curve C is represented in the formthen one can regard x itself as parameter, replacing r; that is, C is given by theequationsx=x, y=g(x), a i x ~ b