6-63. For the curves defined by the given parametric equations, find the position
vector, velocity vector and acceleration vector at the given time.
(a) x=t, y = 2t, z = 3 t, t 0 = 1
(b) x= cos 2 t, y = sin 2 t, z = 0, t 0 = 0
(c) x= cos 2 t, y = sin 2 t, z = 3t, t 0 =π6-64. Show for A=A(t), B =B(t),and C =C(t)that
d
dt[
A·(B×C)]
=A·B×dC
dt +
A·dB
dt ×
C+dA
dt ·
B×C6-65. If F = (x^2 +z)ˆe 1 +xyz eˆ 2 +x^2 y^2 z^2 ˆe 3 find the partial derivatives
(a) ∂
F
∂x, (b)∂
F
∂y, (c) ∂
F
∂z, (d)∂(^2) F
∂x^2
, (e) ∂
(^2) F
∂y^2
, (f) ∂
(^2) F
∂z^2
6-66. Find the partial derivatives
(a) ∂Φ
∂x, (b) ∂Φ
∂y, (c)∂(^2) Φ
∂x^2
, (d)∂
(^2) Φ
∂y^2
, (e) ∂
(^2) Φ
∂x ∂y
in each of the following cases.
(i) Φ = u^2 +v^2 with u=xy and v=x+y
(ii) Φ = uv with u=xy and v=x+y
(iii) Φ = v^2 + 2vwith v=x+y
6-67. Let Φ = Φ(r, θ)denote a scalar function of position in polar coordinates. If
the coordinates are changed to cartesian, where x=rcos θ y =rsin θ,
(a) Show that
∂Φ
∂y=∂Φ
∂rsin θ+∂Φ
∂θcos θ
r
∂^2 Φ
∂y^2 =∂Φ
∂rcos^2 θ
r +∂^2 Φ
∂r^2 sin(^2) θ+ 2 ∂^2 Φ
∂r ∂θ
sin θcos θ
r −^2
∂Φ
∂θ
sin θcos θ
r +
∂^2 Φ
∂θ^2
cos^2 θ
r^2
(b) Show that ∂
(^2) Φ
∂x^2 +
∂^2 Φ
∂y^2 =
∂^2 Φ
∂r^2 +
1
r
∂Φ
∂r +
1
r^2
∂^2 Φ
∂θ^22
6-68. Show the equation of the tangent plane to point (x 1 , y 1 , z 1 )on the surface of
sphere centered at (x 0 , y 0 , z 0 ), having radius a, is given by (r −r 1 )·(r 1 −r 0 ) = 0
Sketch a diagram illustrating these vectors.
6-69. For the scalar function of position F =F(u, v ),where u=u(x, y ), v =v(x, y )
calculate the quantities
∂F
∂x ,∂F
∂y ,∂^2 F
∂x^2 ,∂^2 F
∂x ∂y ,∂^2 F
∂y^2