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Unit 1 Overview
Jan, 2011.
Course info
• Textbook: “Advanced Engineering Mathematics” 9th edition,
by Erwin Kreyszig.
• Lecturer: Kenneth Shum
– Office: SHB 736
– Ext: 8478
– Office hour: Mon, Tue 2:00~3:00
• Tutor: Li Huadong, Lou Wei
• Grading:
– Bi-Weekly homework (12%)
– Midterm (38%)
– Final Exam (50%)
Erwin O. Kreyszig (6/1/1922~12/12/2008)
• Before midterm: Linear algebra
• After midterm: Differential equations
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Academic Honesty
• Attention is drawn to University policy and
regulations on honesty in academic work, and
to the disciplinary guidelines and procedures
applicable to breaches of such policy and
regulations. Details may be found at
http://www.cuhk.edu.hk/policy/academichon
esty/
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System of Linear Equations
Two variables, two equations
7
6
5
4
y
3
2
1
0
-1
-2
-3
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0.2
0.4
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0.6
0.8
1
x
1.2
1.4
1.6
1.8
2
4
System of Linear Equations
Three variables, three equations
6
4
2
z
0
-2
-4
-6
-8
1
0
-1
-2
y
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-2
-1.5
-1
-0.5
0
0.5
x
5
System of Linear Equations
Multiple variables, multiple equations
How to solve?
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Determinant
• Area of parallelogram
(c,d)
(a,b)
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3x3 Determinant
• Volume of parallelepiped
(g,h,i)
(d,e,f)
(a,b,c)
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Nutrition problem
• Find a combination of food A, B, C and D in
order to satisfy the nutrition requirement
exactly.
Food A
Food B
Food C
Food D
Requirement
Protein
9
8
3
3
5
Carbohydrate
15
11
1
4
5
Vitamin A
0.02
0.003
0.01
0.006
0.01
Vitamin C
0.01
0.01
0.005
0.05
0.01
How to solve it using linear algebra?
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Electronic Circuit (Static)
• Find the current through each resistor
System of linear equations
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Electronic Circuit (dynamic)
• Find the current through each resistor
inductor
alternating
current
System of differential equations
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Spring-mass system
• Before t=0, the two springs and three masses
are at rest on a frictionless surface.
• A horizontal force cos(wt) is applied to A for
t>0.
• What is the motion of C?
A
C
B
Second-order differential equation
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System Modeling
Physical System
Reality
Physical Laws
+
Simplifying
assumptions
Mathematical
description
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Theory
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How to model a typhoon?
Lots of partial differential equations are required.
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Example: Simple Pendulum
•
•
•
•
L = length of rod
m = mass of the bob
 = angle
g = gravitational
constant

L
m
mg sin 

mg
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Example: Simple Pendulum
• arc length = s = L
• velocity = v = L d/dt
• acceleration = a
= L d2/dt2
• Apply Newton’s law
F=ma to the tangential
axis:

L
m
mg sin 

mg
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What are the assumptions?
•
•
•
•
•
•
The bob is a point mass
Mass of the rod is zero
The rod does not stretch
No air friction
The motion occurs in a 2-D plane*
Atmosphere pressure is neglected
* Foucault pendulum @ wiki
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Further simplification
• Small-angle assumption
– When  is small,  (in radian) is very close to sin .
Solutions are elliptic functions.
simplifies to
Solutions are sinusoidal functions.
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Modeling the pendulum
modeling
or
Continuous-time dynamical system
for small angle 
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Discrete-time dynamical system
• Compound interest
– r = interest rate per month
– p(t) = money in your account
– t = 0,1,2,3,4
Time is discrete
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Discrete-time dynamical system
• Logistic population growth
– n(t) = population in the t-th year
– t = 0,1,2,3,4
0.25
An example for K=1
Graph of n(1-n)
0.2
-0.05
-0.1
-0.15
negative growth
0
Slow growth
0.05
Slow growth
Proportionality constant
0.1
n*(1-n/K)
Increase in
population
fast growth
0.15
-0.2
-0.25
0
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0.2
0.4
0.6
0.8
n
1
1.2
1.4
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Sample population growth
Initialized at n(1) = 0.01
Monotonically increasing
a=0.8, K=1
Oscillating
a=2, K=1
1
1.4
1.2
0.8
1
0.6
n(t)
n(t)
0.8
0.6
0.4
0.4
0.2
0.2
0
0
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10
t
15
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0
5
10
15
20
25
t
30
35
40
45
50
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Sample population growth
Initialized at n(1) = 0.01
a=2.8, K=1 1.4
1.2
1
Chaotic
n(t)
0.8
0.6
0.4
0.2
0
0
10
20
30
40
50
t
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Rough classification
System
Static
Probabilistic
systems are
treated in
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Dynamic
Continuoustime
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Discrete-time
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Determinism
• From wikipedia: “…if you knew all of
the variables and rules you could
work out what will happen in the
future.”
• There is nothing called randomness.
• Even flipping a coin is deterministic.
– We cannot predict the result of coin
flipping because we do not know the
initial condition precisely.
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