Question

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Let C be the closed curve defined by r(t) = cos ti+ sin tj + sin 2tk for 0 <t<27. (a) [5 pts) Show that this curve C lies on the surface s defined by 2 = 2xry. (b) (20 pts] By using Stokes' Theorem, evaluate the line integral s F. dr where F(x, y, z) = (y2 + cos z)i + (sin y +22)j + xk

Answer 1

Solutions Given Curve С I= any co 8C) = carb i + sint j + sinat ko ostat a surface si 2 since the parametric equations of curre o dore yu= sint * 24 1= sinat Now Lubstitute these equations of the curve into the right hand of z=2xy 7 2xy = 2 sint coft = 266st fint a fint coft = sinat I sim 2A=28iNA COSA any Z anyt from ♡ so every kont of the parametric curre Satisfies the equation of the surface 2 and so the cure rit lies on the surface 7-2014 HP की F (ny, 2) = (43+ (osx it (sinyt z j + xk Fidr = ? The surfaces bounded by C which is given by r(x,y) = (2,4, 244)

DATE: 1 PAGE No 20 | Fidy=-(((*XF) 45 = 31 S 4 Fodr - 3r Answer 4