25.3. EXERCISES 519
3. Let r (t) = (t,cos(3t) ,sin(3t)) for t ∈ [0,1]. Find the length of this curve.
4. Suppose for t ∈ [0,π] the position of an object is given by r (t) = ti+ cos(2t)j+sin(2t)k. Also suppose there is a force field defined on R3,
F (x,y,z)≡ 2xyi+(x2 +2zy
)j+ y2k
Find the work ∫CF ·dR
where C is the curve traced out by this object having the orientation determined bythe direction of increasing t.
5. In the following, a force field is specified followed by the parametrization of a curve.Find the work.
(a) F = (x,y,z) ,r (t) =(t, t2, t +1
), t ∈ [0,1]
(b) F = (x− y,y+ z,z) ,r (t) = (cos(t) , t,sin(t)) , t ∈ [0,π]
(c) F =(x2,y2,z+ x
),r (t) =
(t,2t, t + t2
), t ∈ [0,1]
(d) F = (z,y,x) ,r (t) =(t2,2t, t
), t ∈ [0,1]
6. The curve consists of straight line segments which go from (0,0,0) to (1,1,1) andfinally to (1,2,3). Find the work done if the force field is
(a) F =(2xy,x2 +2y,1
)(b) F =
(yz2,xz2,2xyz+1
)(c) F = (cosx,−siny,1)
(d) F =(2xsiny,x2 cosy,1
)7. Show the vector fields in the preceding problems are respectively
∇(x2y+ y2 + z
),∇(xyz2 + z
),∇(sinx+ cosy+ z−1)
and ∇(x2 siny+ z
). Thus each of these vector fields is of the form ∇ f where f is
a function of three variables. Use Proposition 25.2.1 to evaluate each of the lineintegrals. Compare with what you get by doing it directly.
8. 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.Verify a scalar potential is φ (x,y,z) = xyz.
9. 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.
10. 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.