conservative vector fields

(a) Curl(F)=(0,0,3x^2-3y^2) ≠ (0, 0, 0), so, F is not conservative.
(b) G=grad( xyz/√(x^2+y^2+z^2) ), so, G is conservative.
The potential function of G is -xyz/√(x^2+y^2+z^2) + c
or xyz/√(x^2+y^2+z^2) + c,
according to the definition G=-grad(P) or grad(P).
(c) ∫_C F∙dr=∫[0~p/2] ( (sint)^3, (cost)^3, 8t^3 )∙(-sint, cost, 2) dt
=∫[0~π/2] [(cost)^4-(sint)^4+16t^3] dt
=∫[0~π/2] [ cos(2t)+ 16t^3] dt= (1/2)π^3
∫_C G∙dr= xyz/√(x^2+y^2+z^2) sub. (x,y,z)=(1,0,0) to (0, 1, π)
=0

2011-05-19 22:11:16 補充：
(1) Conservative force may be defined as F=grad(P) or -grad(P)
(2)Sorry! ∫[0~π/2] [ cos(2t)+ 16t^3] dt= (1/4)π^4