Engineering Mathematics 3 – Continuous Assessment for Vector Calculus
Please submit your solution via the Assignments page on Canvas by 5pm, Friday 23 Feb.
(a) Consider the surface of an elliptic cone,
S =
(x, y, z) ∈ R
3
: x
2 + 4y
2 4z
2 = 0, z > 0
.
It can be represented parametrically as
S =
r(s, t) = 2s costi + s sin tj + s k : s ∈ (0, ∞), t ∈ [0, 2π)
.
Figure 1: The cone for part (a). Its cross-section is an ellipse
that is twice as long in the x-direction as it is in the y-direction.
Consider the point P0 = (8, 3, 5) on the surface. Find a normal vector (either inward or outward) to
the surface at P0 in two different ways (i) and (ii), and then solve (iii).
(i) Let f(x, y, z) = x
2 + 4y
2 4z
2 and find
N1
def = grad f(P0).
(ii) Let s0 and t0 be such that r(
s0, t0) = P0. Show that cost0 = 4/5 and sin t0 = 3/5, then find
T1
def =
r
s(s0, t0), T2
def =
r
t(s0, t0),
and finally find
N2
def = T1 × T2.
(iii) Show that N1 and N2 point in opposite directions.
(b) Suppose a particle is moving in 3D space. The particle’s position vector (r) as a function of time (t)
can be expressed parametrically in Cartesian coordinates as
r(t) = x(t)i + y(t)j + z(t)k,
or in cylindrical polar coordinates as
r(t) = ρ(t) ρ(t) + z(t)k, (1)
where ρ(t)
def
= cos θ(t)i + sin θ(t)j is a time-dependent unit vector.
(i) Let θ(t)
def = sin θ(t)i + cos θ(t)j be another time-dependent unit vector. Show that
d ρ
dt
=
dθ
dt
θ,
d
θ
dt
=
dθ
dt
ρ.
(ii) Using equation (1) and part (i), show that
dr
dt
=
dρ
dt
ρ + ρ
dθ
dt
θ +
dz
dt
k,
d
2
r
dt
2
=