Advanced Calculus fi..

31 2 Advanced Calculus, Fifth EditionOne can also define this as the limit of a sum, as for the line integral. For the surfacez = f (x, y), one findsa2 a2ax ay 4--i--j+kThis can also be written:ndu=nlsecyIdxdy= - dx dy.In. klFrom this and the corresponding expressions for surfaces x = g(y, z), y = h(x, z),one obtains the formulas1I1For the surfaces in parametric form we haveor, more concisely, by (5.82),du = f (PI x P2) du dv.The formal properties of line and surface integrals are analogous to those for lineintegrals in the plane (Section 5.3) and need no special discussion here. Furthermore,the definitions and properties of line and surface integrals carry over without changeto piecewise smooth curves and surfaces, provided that the surfaces are orientable.PROBLEMS1. Evaluate the line integrals:I'a) ~ ~ b q ~ ) z d x + ~ d wherecisthecurvex ~ + ~ d z . =cost, y = sint, z = t,COIt52n;b) c