Page 211 - Intro to Tensor Calculus

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EXERCISE 2.2
I 1. Find a set of parametric equations for the straight line which passes through the points P 1 (1, 1, 1) and
P 2 (2, 3, 4). Find the unit tangent vector to any point on this line.

2
1
1
I 2. Consider the space curve x = 1 sin t, y = t − sin 2t, z =sin t where t is a parameter. Find the unit
2 2 4
i
i
i
vectors T ,B ,N ,i =1, 2, 3 at the point where t = π.
3
2
I 3. A claim has been made that the space curve x = t, y = t ,z = t intersects the plane 11x-6y+z=6 in
three distinct points. Determine if this claim is true or false. Justify your answer and find the three points
of intersection if they exist.
I 4. Find a set of parametric equations x i = x i (s 1 ,s 2 ),i =1, 2, 3 for the plane which passes through the
points P 1 (3, 0, 0), P 2 (0, 4, 0) and P 3 (0, 0, 5). Find a unit normal to this plane.
2
I 5. For the helix x =sin ty =cos tz = t find the equation of the tangent plane to the curve at the
π
point where t = π/4. Find the equation of the tangent line to the curve at the point where t = π/4.

∂T m
I 6. Verify the derivative = Mg rm ˙x .
∂ ˙x r

d ∂T ∂g rm
n m
m
I 7. Verify the derivative = M g rm ¨x + ˙ x ˙x .
dt ∂ ˙x r ∂x n
∂T 1 ∂g mn m n
I 8. Verify the derivative = M ˙ x ˙x .
∂x r 2 ∂x r
I 9. Use the results from problems 6,7 and 8 to derive the Lagrange’s form for the equations of motion
defined by equation (2.2.23).

3
2
1
I 10. Express the generalized velocity and acceleration in cylindrical coordinates (x ,x ,x )= (r, θ, z)and
show
2
dx 1 dr 1 δV 1 d r dθ 2
1
V = = f = = − r
dt dt δt dt 2 dt
dx 2 dθ δV 2 d θ 2 dr dθ
2
2
2
V = = f = = +
dt dt δt dt 2 r dt dt
dx 3 dz 3 2
3 δV d z
3
V = = f = =
dt dt δt dt 2
Find the physical components of velocity and acceleration in cylindrical coordinates and show
2
dr d r dθ 2
V r = f r = 2 − r
dt dt dt
dθ d θ dr dθ
2
V θ =r f θ =r +2
dt dt 2 dt dt
dz 2
V z = d z
dt f z = 2
dt