6-46. For u =u (t) = t^2 ˆe 1 +tˆe 2 + 2tˆe 3 and v =v (t) = t^3 ˆe 1 +t^2 ˆe 2 +t^6 eˆ 3
find the derivatives
(a) d
dt(u ·v ) (b) d
dt(u ×v )6-47. If U =U(x, y ) = (2 x^2 y+y^2 x)ˆe 1 + (xy + 3x^2 y)ˆe 2 ,
then find ∂
U
∂x,∂
U
∂y,∂
(^2) U
∂x^2
,∂
(^2) U
∂y^2
, ∂
(^2) U
∂x ∂y
6-48. Consider a rigid body in pure rotation with angular velocity given by
ω =ω 1 ˆe 1 +ω 2 ˆe 2 +ω 3 ˆe 3 .. For 0 an origin on the axis of rotation and the vector
r (t) = x(t)ˆe 1 +y(t)ˆe 2 +z(t)ˆe 3 denoting the position vector of a particle P in the rigid
body, show that the components x, y, z must satisfy the differential equations
dx
dt=ω 2 z−ω 3 y, dy
dt=ω 3 x−ω 1 z, dz
dt=ω 1 y−ω 2 x6-49. For the space curve r =r (t) = t^2 ˆe 1 +tˆe 2 +t^2 ˆe 3 find
(a) dr
dtand ds
dt=∣∣
∣∣dr
dt∣∣
∣∣(b) The unit tangent vector to the curve at any time t.
6-50. For A, B, C functions of time tshow
d
dt[
A×(B×C)]
=A×(
B ×dC
dt)
+A×(
dB
dt ×
C)
+dA
dt ×(
B×C)6-51. Letting x=rcos θ, y =rsin θthe position vector r =xˆe 1 +yˆe 2 becomes a
function of rand θwhich can be denoted r =r (r, θ).
(a) Show that ∂r∂r is perpendicular to the vector ∂r∂θ and assign a physical interpreta-
tion to your results.
(b) Find unit vectors ˆer and ˆeθ in the directions ∂r∂r and ∂r∂θ and sketch these unit
vectors.
6-52. Evaluate the given line integrals along the curve y= 3 xfrom (1 ,3) to (2 ,6)
using F =F(x, y ) = xy ˆe 1 + (y−x)ˆe 2.
(a)∫CF·dr (b)∫CF ×dr6-53. For F= (xy +1) ˆe 1 +(x+z+1)ˆe 2 +(z+1) ˆe 3 ,evaluate the line integral I=
∫CF·dr,