Module MA1132 (Frolov), Advanced Calculus
Tutorial Sheet 2
To be solved during the tutorial session Thursday/Friday, 28/29 January 2016
1. Consider the vector function (with values in R3 )
r(t) = ln t i − t j +
t2
k
4
(a) Find the arc length of the graph of r(t) if 1 ≤ t ≤ 2.
(b) Find a positive change of parameter from t to s where s is an arc length parameter
of the curve having r(1) as its reference point.
2. Show that in cylindrical coordinates a curve given by the parametric equations r = r(t),
θ = θ(t), z = z(t) for a ≤ t ≤ b has arc length
s
2 2
Z b 2
dr
dθ
dz
+ r2
+
dt .
(1)
L=
dt
dt
dt
a
Hint: x = r cos θ, y = r sin θ.
3. Show that in spherical coordinates a curve given by the parametric equations ρ = ρ(t),
θ = θ(t), φ = φ(t) for a ≤ t ≤ b has arc length
s
2
2
Z b 2
dρ
dθ
dφ
2
L=
+ ρ2 sin φ
+ ρ2
dt .
(2)
dt
dt
dt
a
Hint: x = ρ sin φ cos θ, y = ρ sin φ sin θ, z = ρ cos φ.
4. Consider the vector function
r(t) = e−t cos t i − e−t sin t j + e−t k .
(a) Find T(t), N(t), and B(t), at t = 0.
(b) Find equations for the TN-plane at t = 0.
(c) Find equations for the NB-plane at t = 0.
(d) Find equations for the TB-plane at t = 0.
1
(3)