I6-57. On circlex= cosθ, y= sinθ, dx=−sinθdθ, dy= cosθdθ
∫
CF~·d~r=∫Cyz dx+ 2xdy+y dz=∫ 2 π0−2 sin^2 θdθ+ 2 cos^2 θ dθ= 0I6-60. (b) (d)
I6-61.
(a)dx
x= dy
−y=⇒ xy=c (b) dx
2 x=dy
2 y=⇒ y=cx (c)dx
2 y=dy
2 x=⇒ x^2 −y^2 =cI6-62. (c) VectorsA~andB~ form a plane. N~ =A~×B~ is normal to plane and has
the same direction as~r−~r 0.
(d) (~r 2 −~r 1 )×(~r 3 −~r 1 ) =N~ is normal to plane and(~r−~r 1 )×N~ =~ 0 is the
equation of the line.I6-63. (b)
~r= cos 2tˆe 1 + sin 2tˆe 2
~v=d~r
dt=−2 sin 2tˆe 1 + 2 cos 2tˆe 2~a=d~v
dt=d^2 ~r
dt^2 =−4 cos 2tˆe 1 −4 sin2tˆe 2I6-65.
(a) ∂
F~
∂x= 2xe1 +yzˆe 2 + 2xy^2 z^2 ˆe 3(d) ∂(^2) F~
∂x^2
= 2ˆe 1 + 2y^2 z^2
Solutions Chapter 6