{a} Evaluate a / e\"{yalnzdz+mainzdy+ ooezdz] A where AB is a. straight path from A[o,o, 1r} to 3(2, 9\" g}. (Ill marks] (h) Determine the einmllatien of F = (y'. 1:. 3\") along the curve C. where C is the plane y + z = 2 out by the cylinder :2 + y\" = 1, oriented counter clockwise when viewed from above. Sketch everything except the vector eld. {10 marks} 3. (a) Consider the vector field, defined over a simply connected region, F = (2xz) i +(2yz?) 3 +(ax] + by=z-1) k where a and b are non-zero contants. (i) Find a and b such that the given field F is conservative. (ii) Find a scalar potential function for F. (iii) Hence, evaluate the work done by the field from (2, 1, n) to (-1, 2, n). (7 marks) (b) With the help of Green's Theorem, evaluate the following line integral by(4x2 + y? ) dx + x(2x? + 3y?) dy where C is the boundary of the ellipse x+=1. (6 marks) (c) With the help of Stokes' Theorem, evaluate the line integral OF . dF = [[ (v xF).ndo for the vector field F = (3yx] +2') i + (y?)]+(4yx?)/, where S is the surface defined by 3x + 6y +4z =12, and C is its boundary shown as below: 2 (7 marks)