Question

8. (10 points) Use Stokes' Theorem to evaluate where (..) - (awy, 3) and C is the triangle with vertices (1,0,0), (0,1,0), and (0,0,1). You may orient either clockwise or counterclockwise (when looking down the positive z-axis), but you must indicate which orientation you used

Answer 1

Question use stoke's Theorem to evaluate brodo land c is the triangle where [F(x, y, z) = (xy, yz, 2x) with vertices (1,00),(0,1,0) and (0,0,1) you may orient Ć either clockwise or counterclockwise but you must indicate which oriente tion you used. I solution given (ny, z) = (xy, yz, 2x7 Triangle Z Stokes theorem А-ФВС ФА $ 5. dr = sscords S B1600) K î VXF = 2 2 x dy (0,0) Z y=1-2 g Eline. Tay yz ZX (61,0,0) A (x)= x 0-y)-1(1-0) + KC0-x) (VX FJ = (-4,-1,-x) Limits of triangle y → o to 1-2 oti Х

s is the triangular part of the plane (x + y + 2 = 1) [2=1-x-y with normal (-1,-1,-1) so that & is Positively oriented wirts -$7.47 = Sfcsxpedie G S 67. dr = f (-4,-1,- x)•(-1,-1,-1) dx.dy G DEV & fedi = ffleyi-xi-xitittimbj=16) dzdy D-v → F. di = { $(y + 2 + x) dx.dy. D-V OR (x=1 / (y=1-2) S S CI+x+y) dy dz (2=9) (y=0) (2= 1) fr. di ytay + y2 ft) (2-0) (4-0) -off-42 Hay - 1 x + x(1-1) + (1-x)2 (1-x2) dx -

dr.di - (18+ 2 x=1 (1-gtx-x² + 1tr²- 2x (a=0) (x=1) 2- 20²4 1+x²–2x fr di = ((2 - 2x + 1 + x² - 2x) dx (x=0 (x=1) fF.di = 13-22-21) 42 C (20) 2 3x 3 (x=0 APEX mfc7d7 = [(** - -72) -(0) 3 + 2 5 2 $3.48 5 6 * Answer