Question

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 z = 2xy. (b) (20 pts) By using Stokes' Theorem, evaluate the line integral / F. dr C where F(x, y, z) = (y2 + cos x)i + (sin y + x2)j + xk

Please explain all steps. Need to understand.

Thanks

Answer 1

solution : () 1) consider the above given that r CH = costit sihtja sinath or (H = (cost, sint, sinat) C ostean af 2= 2 & 9 2 cost sint 2 (cost) (sint) = sinat "curre a lies on s defined by Z= 2xy b F(x, y, z) = (y 34 coss) i + (sing + 2² jt ock Fox, 3, 2) = (y + cast, sing+z², x) 2 3 (sinat) F11cm) - [[sint) t cos(cost), sin(sint) + 12038] ( OCH = [ -sint, cost, 2cos 24 [cost, sint, sinat

Lsineg @ Flor Hill = (sin3h +cos cost)(-sint) + (s in tsin24) (cost) + cost (2cos 2t) -sinhtasint.cos (cost) + cost sihlsint) & cost sinat +2cos & cosat 2 TT F dor - -sinky-sint cos (cos d) + cost sihasini) et cost sin} t tacost cos 24) at 11 2 -3T foto toto r Sih (o)=0 Je ** -3% Edora sin 21=0 J since = cosa