Question

Let C be the closed, piecewise smooth curve formed by traveling in straight lines between the points (0, 0, 0), (2, 1, 5), (1, 1, 3), and back to the origin, in that order. Use Stokes' theorem to evaluate the integral: (Use symbolic notation and fractions where needed.) (xyz) dx + (3xy) dy + (x) dz = D .

Answer 1

long a 6,0,0) to (2,1,5) +12,1, 5) +1-7)(0,0,0) r(t) = (at, 4, 5t) along a dr = 6, 1,5) oskel Foder - S604 + st" + lot) dt z (Bayt y tops that 9 5+2+5 8 12 12

along § (2,05) l (113) rit) +(11,3)+(-1) ( 2,1,) 3 472-24, t+1-4, 3+15-+5+) Plt) , (2-4, 1,5-24) head on :(-1,0,-2) , Ostb! = 5(2-4)(5-24) (-1) +6221)(-2) de 1 (-(10-9€ +224) -10 14t) dr tat-10-10 tut) dł - 5 (-2tºt 137-20) at • la tet 2017 + 1 - 20 = Vent - 4+39-120 -85

along z_ ( 1,1, 3) 6(0,0,0) TIL) z +(0,0,0) +(-+)(1,1,3) r(t) 2 (-4, 1-t, 3-36) et = 61,-1,-3) Friday = (6-)3_) (1) +3(1-5°-13 +1-67636 oc =S (1-3+++) {3+3+) +-3(i+at+t”) -3437) '(348-31*șt=-132-3+48 -34% ö -3+3+ ) at - Sistem GE "4187–9) at z S*32?_64°4127-9).de z oboru ofte Haf-43 )** } -2 4-4 a Ž -276-9.) 3-5 31425 -17

S**T= (stragedy * (*** > - 12-85-47 144-17025 12 I Fdr -Y7 . . 12 E.