Module MA1132 (Frolov), Advanced Calculus
Homework Sheet 2
Each set of homework questions is worth 100 marks
Due: at the beginning of the tutorial session Thursday/Friday, 4/5 February 2016
Name:
Do not use Mathematica to answer the questions.
Use Mathematica only to check your answers.
1. Consider the vector function (with values in R3 )
r(t) = ln(3 −
√
t) i + (1 +
√
√
(3 − t)2
k
t) j +
4
(1)
(a) Find the arc length of the graph of r(t) if 1 ≤ t ≤ 4.
(b) Find a negative change of parameter from t to s where s is an arc length parameter
of the curve having r(4) as its reference point. It is sufficient to find s as a function
of t.
2. Consider parabolic coordinates (µ, ν)
x = µν ,
1
y = (µ2 − ν 2 ) .
2
(2)
(a) What are the dimensions of µ and ν?
(b) Recall that if a parabola is described by the equation y = a(x − h)2 + k, then its
vertex is at (h, k) and its focus is at (h, k + 1/(4a)).
Show that curves of constant ν form confocal parabolae that open upwards (i.e.,
towards +y), while curves of constant µ are confocal parabolae that open downwards
(i.e., towards −y). Where are the foci of all these parabolae located?
(c) Show that in parabolic coordinates a curve given by the parametric equations µ =
µ(t), ν = ν(t) for a ≤ t ≤ b has arc length
v
2 2 !
Z bu
u
dµ
dν
t(µ2 + ν 2 )
+
dt .
(3)
L=
dt
dt
a
3. Consider elliptic coordinates (u, v)
x = luv,
y 2 = l2 (u2 − 1)(1 − v 2 ) ,
u ≥ 1 , −1 ≤ v ≤ 1 , l is a dimensionfull constant
(a) What is the dimension of l?
(b) Show that curves of constant u are ellipses, while curves of constant v are hyperbolae.
1
(c) Show that in the elliptic coordinates a curve given by the parametric equations
u = u(t), v = v(t) for a ≤ t ≤ b has arc length
s
2
2
Z b
u2 − v 2 du
u2 − v 2 dv
L=l
+
dt .
(4)
u2 − 1 dt
1 − v 2 dt
a
4. Consider the vector function
r(t) = e−t i + e−t cos t j − e−t sin t k .
(5)
(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.
5. Prove the Serret-Frenet formulae
(a)
(b)
(c)
dT
ds
dB
ds
dN
ds
= κ(s)N(s)
= −τ (s)N(s), where τ is a scalar called the torsion of r(s).
= −κ(s)T(s) + τ (s)B
Bonus question (each bonus question is worth extra 25 marks)
1. The evolute of a smooth parametric curve C in 2-space is the curve formed from the
centres of curvature of C. Use Mathematica to find parametric equations of the evolute
of the curve C, and to plot C and its evolute if C is
(a) a parabola y = ax2 , a > 0.
x2
a2
+
y2
b2
= 1.
(c) a hyperbola
x2
a2
−
y2
b2
(b) an ellipse
= 1.
(d) a cycloid x = a(θ − sin θ), y = a(1 − cos θ).
2