Engineering Math II (002) Midterm Exam
1. Let V be a complex inner product space, and W be an n-dimensional subspace of V.
a) Let S be an orthonormal subset of a finite-dimensional inner product space V.
Show that S is linearly independent.
b) Prove the projection theorem: that is, for any v ∈ V, there exist unique w ∈ W
and z ∈ W⊥ satisfying v = w + z. Here W⊥ := {v ∈ V : ⟨v, w⟩ = 0 for any w ∈
W} is the orthogonal complement of W. Furthermore, show that w is the best
approximation of v on W.
2.
a) Find the singular value decomposition the following matrix

1

A=
0
1
−1
1
0

0

1
,
1
that is, find orthogonal matrices P, Q and a diagonal matrix ∆ satisfying A =
Q∆P ∗ .
b) Find the Moore-Penrose pseudoinverse (say A− ) of A.
c) Find the minimal solution of the consistent system Ax = (1 3 4)t .
3. Suppose that a planet P with mass m is moving at a distance r from the sun S (located
at the origin). According to Newton’s law of gravitation, the gravitational force F is
Km
given by F = − 2 r̂. On the other hand, Newton’s second law states that F = ma,
r
where a is the acceleration vector. In expressing the vector ma in terms of r̂ and
ϕ̂, that is, expressing ma in the form ma = mar r̂ + maϕ ϕ̂, where ar and aϕ are the
components of a in the direction of r̂ and ϕ̂, respectively, derive that
r̈ − r(ϕ̇)2 = −
K
, rϕ̈ + 2ϕ̇ṙ = 0,
r2
dr
dϕ
and ϕ̇ =
. (Hint: Use the chain rule.)
dt
dt
a
4. Consider the vector field F = r̂ + bz k̂.
r
where ṙ :=
a) Compute ∇ · F, the divergence of F.
I
b) Evaluate the flux
F · ds outward S, where S is the jar specified by z = ±1 and
r = 3 − z2.
S
5. Find the Fourier series of the 2π-periodic function f (x) = x(π − |x|) (−π < x ≤ π).
Use this to evaluate
∞
X
(−1)n
1
1
1
= 3 − 3 + 3 − ··· .
3
(2n + 1)
1
3
5
n=0
6. Consider the differential equation (1 − x2 )y ′′ − xy ′ + n2 y = 0 for n ∈ N ∪ {0}.
a) One can show that this equation has a polynomial solution of degree n, say Tn (x).
Find T1 (x) and T2 (x) satisfying Tn (1) = 1.
b) Show that
Z
1
−1
Tn (x)Tm (x)
√
dx = 0 for n ̸= m.
1 − x2
7. Let f (x) be a piecewise smooth and absolutely
Z ∞ integrable function on R with
1
f (ξ)e−iwξ dξ be the Fourier transform
lim f (x) = 0 and F {f (x)}(w) = √
|x|→∞
2π −∞
of f (x).
a) Find the Fourier transform of the function y = e−a|x| (a > 0).
1
(a > 0). (Hint: Use a).)
b) Find the Fourier transform of the function y = 2
x + a2
c) Describe the solution of the heat equation ut = c2 uxx with initial condition
1
u(x, 0) = 2
as an integral form.
x +1
8. Consider a vibrating string occupying the interval 0 ≤ x ≤ π satisfying
1
utt = uxx . Suppose the string is fastened at the end.
4
a) By using the separation of variables, derive two ordinary differential equations of
x and t, and determine the nontrivial solution of each equations.
b) Assume the initial displacement u(x, 0) is f (x) = sin2 x and the initial velocity
ut (x, 0) of this string is zero for 0 ≤ x ≤ π. Use the Fourier series to determine
the displacement u(x, t).
9.
a) Solve the following linear transport equation
ux + 3ut = sin (x + t), u(x, 0) = cos x.
b) Using the Laplace transform, find the current i(t) in the RLC circuit in the figure
below, assuming zero initial current and charge, and R = 2Ω, L = 1H, C = 0.5F,
E(t) = δ(t − 2).