ENGG2013Unit 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%)
• Before midterm: Linear algebra• After midterm: Differential equations
kshum 2ENGG2013
Erwin O. Kreyszig (6/1/1922~12/12/2008)
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/academichonesty/
kshum ENGG2013 3
System of Linear Equations
kshum ENGG2013 4
Two variables, two equations
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-3
-2
-1
0
1
2
3
4
5
6
7
x
y
System of Linear Equations
kshum ENGG2013 5
Three variables, three equations
-2 -1.5 -1 -0.5 0 0.5
-2-1
01-8
-6
-4
-2
0
2
4
6
xy
z
System of Linear Equations
kshum ENGG2013 6
Multiple variables, multiple equations
How to solve?
Determinant
• Area of parallelogram
kshum ENGG2013 7
(a,b)
(c,d)
3x3 Determinant• Volume of parallelepiped
kshum ENGG2013 8
(a,b,c)
(d,e,f)
(g,h,i)
Nutrition problem
• Find a combination of food A, B, C and D in order to satisfy the nutrition requirement exactly.
kshum ENGG2013 9
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?
Electronic Circuit (Static)
• Find the current through each resistor
kshum ENGG2013 10
System of linear equations
Electronic Circuit (dynamic)
• Find the current through each resistor
kshum ENGG2013 11
System of differential equations
inductor alternatingcurrent
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?
kshum ENGG2013 12
A B C
Second-order differential equation
System Modeling
kshum ENGG2013 13
Physical System
Mathematicaldescription
Physical Laws+
Simplifyingassumptions
Reality
Theory
How to model a typhoon?
kshum ENGG2013 14
Lots of partial differential equations are required.
Example: Simple Pendulum
• L = length of rod• m = mass of the bob• = angle• g = gravitational
constant
kshum ENGG2013 15
L
m
mg
mg sin
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:
kshum ENGG2013 16
L
m
mg
mg sin
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
kshum ENGG2013 17
Foucault pendulum @ wiki
Further simplification
• Small-angle assumption– When is small, (in radian) is very close to sin .
kshum ENGG2013 18
simplifies to
Solutions are elliptic functions.
Solutions are sinusoidal functions.
Modeling the pendulum
kshum ENGG2013 19
modeling
Continuous-time dynamical system
or
for small angle
Discrete-time dynamical system
• Compound interest– r = interest rate per month– p(t) = money in your account– t = 0,1,2,3,4
kshum ENGG2013 20
Time is discrete
Discrete-time dynamical system• Logistic population growth
– n(t) = population in the t-th year– t = 0,1,2,3,4
kshum ENGG2013 21
Increase in population
Proportionality constant
0 0.2 0.4 0.6 0.8 1 1.2 1.4-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
n
n*(1
-n/K
)
An example for K=1Graph of n(1-n)
Slo
w g
row
th
fast
gro
wth
Slo
w g
row
th
nega
tive
grow
th
Sample population growth
kshum ENGG2013 22
0 5 10 15 200
0.2
0.4
0.6
0.8
1
t
n(t)
a=0.8, K=1
Monotonically increasing
Initialized at n(1) = 0.01
a=2, K=1Oscillating
0 5 10 15 20 25 30 35 40 45 500
0.2
0.4
0.6
0.8
1
1.2
1.4
t
n(t)
Sample population growth
kshum ENGG2013 23
0 10 20 30 40 500
0.2
0.4
0.6
0.8
1
1.2
1.4
t
n(t)
a=2.8, K=1
Chaotic
Initialized at n(1) = 0.01
Rough classification
kshum ENGG2013 24
System
Static Dynamic
Continuous-time Discrete-time
Probabilistic systems are treated in ENGG2040
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.
kshum ENGG2013 25