Calculating work done by greens theorem

Question

Assignment:

Q1.  Find the curl of the vector field F at the indicated point:
 
F(x, y, z) = x²zi - 2xzj + yzk @ (2 , -1, 3)

Q2.  Evaluate the following line integral using the Fundamental theorem of line Integrals:
 
∫_c [2(x + y) i + 2(x +y) j ] .dr

 C : smooth curve from (-1 ,1) to ( 3, 2)

Q3.  Use Green’s Theorem to calculate the work done by the force F in moving a particle around the closed path C:
 
F(x, y) =(e^x =3y) i + (e^y + 6x) j

C: r = 2 cos Θ

Q4.  Find the area of the surface over the part of the plane:
 
r( u, v) = ui + 2vj - vk where 0≤u≤4 and 0≤v≤2

Q5. Use the Divergence Theorem to evaluate ∫∫_s F.Nds   and find the outward flux of F through the surface of the solid bounded by the graphs of the equations:
 F( x, y, z) = x2 zi -yj +xyzk
s : x =a , x= a, y =0, y =a, z=0, z= a

Q6.  Verify Stoke’s Theorem by evaluating ∫∫_c F.Tds  as a line integral:
 
F(x, y ,z) = (-y + z)i + (x -z) j + (x -y) k
s: z= 4 -x2 -y2, 0≤Z

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