Reference no: EM132506283

Question 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).

Question 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).

Question 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.

Question 4. Find the flow of the field F = ∇(xy²z³) along the line segment from (1, 1, 1) to (2, 1, -1).

Question 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 = e^y+2z (i^ + xj^ + 2xk^).

Question 6. Evaluate the following integrals by first showing that the differential forms in the inte¬grals 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

Question 7. Evaluate the following integrals using Green's theorem in the plane