Exercicios resolvidos James Stewart vol. 2 7ª ed - ingles

160 . 0 CHAPTER 13 VECTOR FUNCTIONS
37. f~"' 12 (3sin 2 t cost i + 3 sin t cos 2 t j + 2 sin t cost k ) dt
= (I; 12 3 sin 2 t cost dt) i + (.ro" 12 3 sin t cos 2 t dt) j + (I; 12 2 sin t cost dt) k
= [sin 3 t) ~ 12 i + [- cos 3 t] ~ 12 j+ [sin 2 t) ~ 12 k = ( 1 - 0) i + ( 0 + 1) j + ( 1 - 0) k = i + j + k
39. f (sec 2 t i + t(e + 1) 3 j + t 2 ln t k ) dt = (f sec 2 t dt) i + (f t(t,2 + 1) 3 dt) j + (f t 2 ln t dt) k
=tan t i + t(t 2 + 1) 4 j + (tt 3 ln t- ~t 3 )k + C,
where C is a vector constant of integration. [For the z-cornponent,- integrate by pa~s with u = ln t, dv = t 2 dt.]
41. r'(t) = 2t i + 3t 2 j + Vtk =} r (t) = t2.i + e j + tt 3 1 2 k + c , where cis a constant vector.
But i + j = r (1) = i + j + i k + C. Thus C = -tk and r (t) = ·t 2 i + t 3 j + ( tt 3 1 2 - ~) k.
Fo.r Exercises 43-46, 1et u(t) = (u 1 (t), u 2 (t), u 3 (t)) and v (t) = (v 1 ( t), v2(t), v 3 ( t)). In each of these exercises, the procedure is to apply
Theorem 2 so that the corresponding properties of derivatives of real-valued funciions can be used.
d
d
43. -d [u (t) + v (t)] = -d (u1(t) +v1(t),u2(t) +v2(t),ua(t) +v 3 (t))
t t .
= (! [u1(t) +v1 (t)],! [u2(t) +v2(t)], ;t [ua(t) + va(t)])
= M(t) + vW), u~(t) + vHt), u3(t) +vW))
= M (t), u~ (t), u3(t)) + M(t), v~(t), v3(t)) = u' (t) + v' (t)
d
d
45. dt [u(t) x v(t)] = dt (u2(t)va(t) - ua(t)v2(t),ua(t)v1(t) - u1(t)v3(t),u1(t)v2(t) - u2(t)v1(t))
= (u~va(t) + u2(t)v3(t) - u3(t)v2(t) - u3(t)v2(t),
u3(t)vl(t) + ua(t)v~ (t)- uW)va(t)- u1(t)vW),
u~ (t)vz(t) .+ U1 (t)vW) - u2(t)vl (t) - uz(tM (t))
= (uW)va(t) - u3(t)v2 (t), u3(t)vl (t)- u~(t)va(t), ui(t)'!!2(t)- u2(t)vl(t))
+ (uz(t)vW) - ua(tM(t), u3(t)vi (t)- ul(t)v3(t), u1 (t)vW) :- uz(t)v~ (t))
= u' (t) x v (t) + u (t) x v' (t)
· Alternate solution: Let r(t) = u(t) x v(t). Then
r(t +h)- r (t) = [u (t +h) X v(t +h)] - (u(t) x v(t)]
= [u(t + h) x v(t + h)] - [u(t) x v(t)] + [u (t + h) x v(t)] - [u (t t h) X v (t)]
= u (t + h) x [v (t +h) - v(t)] + [u(t +h) - u (t)] X v(t)
(Be careful of the order of the cross product.) Dividing through by hand taking the limit as h--+ 0 we have
r'(t) = lim u (t +h) x [v~ +h) - v (t)] + lim [u(t +h) - u (t)] X v(t) = u(t) x v'(t) + u'(t) X v (t)
h- o . h- o h
by Exercise 13.1.49(a) and Definition 1.
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