THE UNIVERSITY OF HONG KONG
BACHELOR OF ENGINEERJNG: LEVEL II EXAMINATION
DEPARTMENT OF CIVIL ENGINEERJNG:
MATHEMATICS II (CIVB2001)
May 12, 2000 9:30 am - 12:30 pm (3 hours)
Answer FIVE questions: ALL of Section A and TWO from Section B.
All questions carry equal marks.
Use of Electronic Calculators:
The Calculator: (i) should be silent and battery-operated:
(ii) should not have any printing device, alphanumeric display,
alphanumeric keyboard, or graphic display; and
(iii) should not contain any recorded data or program.
It is the candidate's responsibility to ensure that his calculator operates satisfactorily.
Candidates must record the name and type of their calculators on the front page of their
examination scripts.
Section A
Answer ALL questions of this section.
For all questions of this section, determine whether each statement 1s true or false.
WARNING: Negative marks will be given to wrong answers.
1. (a) jcos zj :s; 1 for all complex z.
(b)
(c) f(z)=z is differentiable at z=O .
(d) If !m z > 0 , then let= I > 1 .
(e) cf .._\ce~-l)dz=i1r, where c istheunitcircle.
(f) An-- .!.e= dz = i:rr, where C is the unit circle.
'k· z
I
(g) ~-,... ze~ dz = i:rr , where C is the unit circle.
'-f·
(h) _!_=I :.o (-Ir (z-1r for all z such that jz-lj < 1 .
z
1 .
(i) res ( , i) = _ _:_ .
(z 2 +1) -
7
4
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(j) A bilinear map can be found, which maps conformally a crescent onto another
crescent with different apical angles.
2. (a) A subset of a linearly independent set dfvectors is linearly independent.
(b) In a vector space of dimension n , a set of n linearly independent vectors is a basis.
(c) In an inner product space. an orthogonal set of vectors is a basis.
(d) Let A be n x n and X n x 1 . The equation AX = 0
- has non-trivial solution
<::::::>A is invertible.
(e) Let A be n x n , and X and .r-:, be n x 1 . The equation AX = ~~ always has
a solution.
(f) Similar matrices have the same rank, the same determinant, the same trace, and the
same eigenvalues.
(g) A matrix coordinate of an isomorphism is invertible.
(h) If an eigenvalue of A is zero,_then A is not invertible.
(i) ( ~ ~) is diagonalizable over 1R .
U)
[ o~ ~ o~J is diagonalizable over JR. .
'"' ·
-' (a) res ( coshz • j ;r(l + 2k)) =- 1 for any integer k .
e= + 1
[~ ~J
1 0
(b)
[:
0
l] is similar to -1
0
(c) L"' nsl
sinnt
- - , - =e
cos/ . .
sm(smt).
n.
(d) ltl = :r -~I"' cos(2n+l)t , -rr5:.t5:.rr .
2 :r n-o (2n+l) 2
1
(e) The Fourier transform j of f(x)=--, is given by:
l+x-
.
~
f(J.)= ~
-e-I-''', -oo<J.<oo.
2
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Section B
Choose TWO questions from this section.
4. (a) Compute [ ~ by the method of residues, using the contour as shown.
" x·' +1
0
>x
(b) Given w=e'; z-a where rjJ is real and JaJ < 1.
1-az. '
i) Show that JzJ ~ 1 <=> )wl ~ 1 . and interprete this result geometrically.
ii) Show that a direction at a is rotated by the angle rjJ under this map.
5. (a) Compute f~dx by contour integration. Sketch the contour used, and explain
- x 3 +1
why a certain integral approaches zero.
(b) i) Find A" where A =(31 4)3 .
ii) Let P, .. l -- ..,
.) P"
+4
q,
{ q!IT = p, +3q, , n =0, 1.2 ....
1
Express Pn and q" in terms of p, and q"
i) Find the matrix coordinates M;· (1) and M;. (1) of the identity map on JR 3
How are they related?
ii) Are these matrices diagonalizable? Give supporting reasons for your answer.
(b) Let A be an m xn matrix with rank r, and k=dimker A. Consider the equation
AX =Y,, . Discuss briefly the nature of the solutions of this euqation as they are
related to the integers m, n, r, and k .
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7. (a) Solve the following problem ofheat flow in a uniform rod:
DE: c ~ u:c.r =ut . O~x$'.L, t~O
BC : u( (O. t )=O , u(L,t)=O
{
JC : u(x,O)=.f(x )
(b) Solve the corresponding problem with a source term.
I.e., solve the problem in which the DE is
c 2 u:u =ut +r/J (x,t),
while the BC and IC remain the same as in (a). Express the solution as a sum of the
transient and the asymptotic parts, and explain briefly what each of these parts
represents.
END OF PAPER
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