Question

Question 5 25 (5.1) Sketch some vectors in the vector field given by F(r, y) 2ri + yj. (3) (5.2) Evaluate the line integral fe F dr, where F(r, y, 2) = (x + y)i + (y- 2)j+22k and C is given by the vector function r(t) = ti + #j+Pk, 0 <t<1 (4) costrt>, 0St<1 (5.3) Given F(r, y) = ryi + yj and C: r(t)=< t + singat, t (3) (a) Find a function f such that F Vf (2) (b) Use part (5.3)(a) above to evaluate F dr (5.4) Use Green's Theorem for line integrals to find the work done by the force F(r,y)=r(ar+yi ryj in moving a particle from the origin along 1-axis to (1,0), then along the line segment to (0,1). and then back to the origin along the y-axis (3) (4) ry that lies within the cylinder r2ty= 1. (5.5) Find the arca of the part of the surface (5.6) Use Stoke's Theorem to evaluate F.dr where F(r, y.:) = yzi + 2rj + ek 16. 5. (6) and C' is the circle r +y2

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Answer 1

(x,) (2,0) Fx,,2)= (X+)i+3-2>j+zk da= 2+dt, dy- 3tt,dZ=st°at p + 3t 37 - 30 2 1343 1 C. t+ sinnt, ttlont, SF. draf 2 2

S.A) F Work dang = 1+yz/ - S 1049)dntzf - Z=エ Suspice \ 52 + 2. S 7 et t=T , y-Yno

217 2-1) F 2i2 zj+ ej c . Fkd e22)4 ( 3-je**)j+ (2z-z )k.k dxd andy SS ) = 5 S Ar