1. ENGINEERING MATHEMATICS Unit 1 Overview

2. Course Description Prerequisite
This course aims at teaching students about fundamental concepts, solution methodologies, and technical applications of the following mathematical topics:
Linear algebra
Differential equations
Second order differential equations
Series
Vector calculus
Prerequisite
Calculus
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3. Textbook and Reference
“Advanced Engineering Mathematics” 9th edition, by Erwin Kreyszig
Reference
Advanced Calculus (5th Edition), by Wilfred Kaplan, Published by Addison Wesley, 2002
Erwin O. Kreyszig (6/1/1922~12/12/2008)
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4. Assessment Scheme Grading Bi-Weekly Homework (10%) Report (20%)
Class (10%)
Midterm (30%)
Final Exam (30%)
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5. Feedbacks
Students are welcome to express their comments and suggestions via:
Formal channel
Two course evaluations: First one to be conducted in the middle of the term and the second one at the end of the term.
Students are encouraged to provide specific comments and/or suggestions in addition to the numeric ratings.
Informal channel
Students are also encouraged to provide feedbacks using informal channels, such as and/or private discussing with instructor/tutors.
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6. System of Linear Equations
Two variables, two equations
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7. System of Linear Equations
Three variables, three equations
x = -2:0.1:0.5;
y = x;
[X Y] = meshgrid(x,y)
clf
mesh(X, Y, -(3*X-6*Y)/2);
hold on
mesh(X,Y,2+2*X)
mesh(X,Y,(1-Y)/3)
xlabel('x')
ylabel('y')
zlabel('z')
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8. System of Linear Equations
Multiple variables, multiple equations
How to solve?
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9. Determinant
Area of parallelogram
(c,d)
(a,b)
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10. 3x3 Determinant Volume of parallelepiped (g,h,i) (d,e,f) (a,b,c)
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11. 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 5
Carbohydrate 15 11 1 4
Vitamin A
0.02
0.003
0.01
0.006
Vitamin C
0.005
0.05
How to solve it using linear algebra?
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12. Electronic Circuit (Static)
Find the current through each resistor
System of linear equations
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13. Electronic Circuit (Dynamic)
Find the current through each resistor
alternating current
capacitor
System of differential equations
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14. 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 B C
Second-order differential equation
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15. Simplifying assumptions
System Modeling
Reality
Physical System
Physical Laws +
Simplifying assumptions
Mathematical
description
Theory
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16. How to Model a Typhoon?
Lots of partial differential equations are required.
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17. 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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18. 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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19. 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 wiki
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20. 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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21. Modeling the Pendulum modeling or Continuous-time dynamical system
for small angle 
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22. 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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23. Discrete-Time Dynamical System
Logistic population growth
n(t) = population in the t-th year
t = 0,1,2,3,4
An example for K=1
Graph of n(1-n)
Slow growth
fast growth
negative growth
Increase in population
Proportionality constant
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24. Sample Population Growth
Initialized at n(1) = 0.01
Monotonically increasing
Oscillating
a=0.8, K=1
a=2, K=1
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25. Sample Population Growth
Initialized at n(1) = 0.01
a=2.8, K=1
Chaotic
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26. Probabilistic systems are
Rough Classification
System
Static
Dynamic
Probabilistic systems are
treated in ENGG2040
Continuous-time
Discrete-time
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27. Determinism
From Wikiedia: “…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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