Reference no: EM131005070
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).
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