1. Define a vector field as

and path C as

for π ≤ t ≤ 2π. Evaluate the line integral ∫_c F · dr.

(Hint: A trig identity will help simplify one of the terms in the resulting intcgrand)

2. Define a vector field as

and define the path C as the path graphed below lying in the x-y plane starting at x point (x,y,z) = (-2,0,0) and ending a (4,2,0). (The first part of the path is a quarter-circle) Parameterize path C via some r(t) and then evaluate the line integral ∫_c F · dr.

(Hint: Divide path C into two sub-paths C₁ and C₂)

C₁: a quarter circle from (-2,0,0) to (0,-2,0) , CCW

C₂: A straight line from (0,-2,0) to (4,2,0)

3. (a) Prove

has path-independent integrals.

(b) Define path C as

for 0 ≤ t ≤ π. Evaluate the line integral ∫_c F · dr. (Hint: Is there anyth'ng special about this path's endpoints?)

4. Given that

where f(r) = x²y + 3yz + 2x, evaluate the line integral ∫_c F · dr. Path C is any path from point (1,-1 1) to point (3,4,-5).

5. (a) Prove

is a conservative field (and thus has path-independent line integrals) by finding a scalar function f(r) such that F = ∇f

(b) Evaluate ∫_c F · dr where path C is parameterized as

for 0 ≤ t ≤ 0.5 by using function f(r) C 4t3 from part (a).