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

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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/

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System of Linear Equations

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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

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Three variables, three equations

-2-1.5

-1-0.5

00.5

-2

-1

0

1-8

-6

-4

-2

0

2

4

6

xy

z

System of Linear Equations

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Multiple variables, multiple equations

How to solve?

Determinant

• Area of parallelogram

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(a,b)

(c,d)

3x3 Determinant• Volume of parallelepiped

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(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.

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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

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System of linear equations

Electronic Circuit (dynamic)

• Find the current through each resistor

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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?

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A B C

Second-order differential equation

System Modeling

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Physical System

Mathematicaldescription

Physical Laws+

Simplifyingassumptions

Reality

Theory

How to model a typhoon?

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Lots of partial differential equations are required.

Example: Simple Pendulum

• L = length of rod• m = mass of the bob• = angle• g = gravitational

constant

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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:

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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

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Foucault pendulum @ wiki

Further simplification

• Small-angle assumption– When is small, (in radian) is very close to sin .

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simplifies to

Solutions are elliptic functions.

Solutions are sinusoidal functions.

Modeling the pendulum

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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

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Time is discrete

Discrete-time dynamical system• Logistic population growth

– n(t) = population in the t-th year– t = 0,1,2,3,4

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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

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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

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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

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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.

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