14.5. EXERCISES 269

, and ∇(x2 siny+ z

). Thus each of these vector fields is of the form ∇ f where f is a

function of three variables. For each f in the above, compute f (1,2,3)− f (0,0,0)and compare with your solutions to the above line integrals. You should get thesame thing from f (1,2,3)− f (0,0,0) . This is not a coincidence and will be fullydiscussed later. Such vector fields are called conservative.

8. Here is a vector field(y,x+ z2,2yz

)and here is the parametrization of a curve C.

R(t) = (cos2t,2sin2t, t) where t goes from 0 to π/4. Find∫

C F ·dR.

9. If f and g are both increasing functions, show that f ◦ g is an increasing functionalso. Assume anything you like about the domains of the functions.

10. Suppose for t ∈ [0,3] the position of an object is given by r (t) = ti+ tj + tk.Also suppose there is a force field defined on R3,F (x,y,z)≡ yzi+ xzj+ xyk. Find∫

C F · dR where C is the curve traced out by this object which has the orientation de-termined by the direction of increasing t. Repeat the problem for r (t)= ti+t2j+tk.

11. Suppose for t ∈ [0,1] the position of an object is given by r (t) = ti+ tj + tk.Also suppose there is a force field defined on R3,F (x,y,z) ≡ zi+ xzj+ xyk. Find∫

C F ·dR where C is the curve traced out by this object which has the orientation de-termined by the direction of increasing t. Repeat the problem for r (t)= ti+t2j+tk.

12. Let F (x,y,z) be a given force field and suppose it acts on an object having mass mon a curve with parametrization, (x(t) ,y(t) ,z(t)) for t ∈ [a,b]. Show directly thatthe work done equals the difference in the kinetic energy. Hint:∫ b

aF (x(t) ,y(t) ,z(t)) ·

(x′ (t) ,y′ (t) ,z′ (t)

)dt =

∫ b

am(x′′ (t) ,y′′ (t) ,z′′ (t)

)·(x′ (t) ,y′ (t) ,z′ (t)

)dt,

etc.

13. Suppose for t ∈ [0,2π] the position of an object is given by

r (t) = 2ti+ cos(t)j+ sin(t)k.

Also suppose there is a force field defined on R3,

F (x,y,z)≡ 2xyi+(x2 +2zy

)j+ y2k.

Find the work∫

C F ·dR where C is the curve traced out by this object which has theorientation determined by the direction of increasing t.

14. Here is a vector field(y,x2 + z,2yz

)and here is the parametrization of a curve C.

R(t) = (cos2t,2sin2t, t) where t goes from 0 to π/4. Find∫

C F ·dR.

15. Suppose for t ∈ [0,1] the position of an object is given by r (t) = ti+ tj + tk.Also suppose there is a force field defined on R3,F (x,y,z)≡ yzi+ xzj+ xyk. Find∫

C F · dR where C is the curve traced out by this object which has the orientation de-termined by the direction of increasing t. Repeat the problem for r (t)= ti+t2j+tk.

You should get the same answer in this case. This is because the vector field happensto be conservative. (More on this later.)