It might be appreciated that you help me to figure out answers and steps too..

I don't know where to start with.. so please give me detailed steps.

. Evaluate ffs ewdxdy, where S is the region enclosed by my = 1,223; = 2,3,: = 3,3; = 4.1: using the change of variable 13,; = u, E = v. . Compute the following multiple integrals using suitable method. (a) [f x3dxdy, where R is the region bounded by :r-axis, ya'xis, 2: = 2, y = 1 + :r, and y = 3 R :c. (b) [ mdzdydz, where V is the region which is bounded above by a sphere $2 + y:3 + z = 4 and is bounded below by a plane 2: = 1. . Find gradf = Vf for f(:r,y, z) = 2:2 + y\" + 2\". Hence calculate (a) the directional derivative of f at (1, 1,1) in the direction of the unit vector %(2, 2, 1), (b) the maximum rate of change of the function at (1, l, 1) and its direction. . Let 1:5 = (5:: + 23,: + ez)i'+ (bx 3y 22)_"+ {4.1: + cy + 22)}: be a vector eld on HF, where a, b, and c are real constants. (a) Find the value; of a, b, and c such that F. is irrotational. (b) With the values of e, b, and 6 obtained in (a), determine a potential function up on R3 for which V99 = Fl 5. Compute the following line integrals using suitable method. (a) f0 13 - d1?" where 13 = 333+ 43:3,}; and C is the boundary curve of the region in the rst quadrant bounded by I-axis, y = :3, and a circle :02 + y2 = 1. (b) f0 13'. - (if? where Ii". = [25:22:2 cos(1 + 2:2 + 3y3)]i'+ [Qyz cos(1 + $2 + 3y3)]j' + [22 sin(l + 1:2 + 3%)]??? and C is the path moving from a point {0, 1,2) and (3, 4, 7) along a straight line. 6. Compute the following surface integrals using suitable method. (a)ff5(:c2 + y2)d3, where S is the part of the surface 2 = 9 y lying inside the cylinder 2:2 + y2 = 1. (b) ffs 1-5- dS, where 13'. = yf+ 5c; and S is the part of the cone 2 = 1 /$2 + y? lying inside the cylinder 32 + 3'2 = 9. (Here, 11' is upward pointing normal)