Problem 1: Evaluate the flow integral of the velocity field F = (x + y)i^ + (x2 + y2)j^ along each of the following paths from (1, 0) to (-1,0) in the xy-plane
(i) the upper half of the circle x2 + y2 = 1,
(ii) the line segment from (1, 0) to (-1, 0),
(iii) the line segment from (1, 0) to (0, -1) followed by (0, -1) to (-1, 0).
Problem 2: Find the flux of the field F given in the Exercise 1. outward across the triangle with vertices (1, 0), (0,1) and (-1, 0).
Problem 3: Find the circulation of F = 2xi^ + 2zj^ + 2yk^ around the closed path consisting of the following three curves traversed in the direction of increasing t :
(i) C1 : costi^ + sinj^ + tk^, o ≤ t ≤ Π/2,
(ii) C2 : j^ + Π/2(1 - t)k^, 0 ≤ t ≤ 1,
(iii) C3 : t^i + (1 - t)j^ 0 ≤ t ≤ 1.
Problem 4: Find the flow of the field F = ∇(xy2z3) along the line segment from (1, 1, 1) to (2, 1, -1).
Problem 5: Find a potential function f for the field
(i) F = (y sin z)i^ + (x sin z)j^ + (xy cos z)k^,
(ii) F = ey+2z (i^ + xj^ + 2xk^).
Problem 6: Evaluate the following integrals by first showing that the differential forms in the integrals are exact:
i) ∫(0,0,0) (1,2,3) 2xydx + (x2 - z2)dy - 2yzdz,
(ii) ∫(0,2,1) (1,Π/2,2) 2 cos ydx + (1/y - 2x sin y)dy + 1/zdz,
(iii) ∫(-1,-1,-1) (2,2,2) 2xdx + 2ydy + 2zdz/x2 + y2 + z2
Problem 7: Evaluate the following integrals using Green's theorem in the plane