ENGG2013 Unit 1 Overview Jan, 2011.

Course info Textbook: “Advanced Engineering Mathematics” 9 th 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 kshum2ENGG2013 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 esty/ esty/ kshumENGG20133

System of Linear Equations kshumENGG20134 Two variables, two equations

System of Linear Equations kshumENGG20135 Three variables, three equations

System of Linear Equations kshumENGG20136 Multiple variables, multiple equations How to solve?

Determinant Area of parallelogram kshumENGG20137 (a,b) (c,d)

3x3 Determinant Volume of parallelepiped kshumENGG20138 (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. kshumENGG20139 Food AFood BFood CFood DRequirement Protein98335 Carbohydrate Vitamin A Vitamin C How to solve it using linear algebra?

Electronic Circuit (Static) Find the current through each resistor kshumENGG System of linear equations

Electronic Circuit (dynamic) Find the current through each resistor kshumENGG System of differential equations inductor alternating current

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? kshumENGG A B C Second-order differential equation

System Modeling kshumENGG Physical System Mathematical description Physical Laws + Simplifying assumptions Reality Theory

How to model a typhoon? kshumENGG Lots of partial differential equations are required.

Example: Simple Pendulum L = length of rod m = mass of the bob  = angle g = gravitational constant kshumENGG L m  mg mg sin  

Example: Simple Pendulum arc length = s = L  velocity = v = L d  /dt acceleration = a = L d 2  /dt 2 Apply Newton’s law F=ma to the tangential axis: kshumENGG 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 kshumENGG * Foucault wiki

Further simplification Small-angle assumption – When  is small,  (in radian) is very close to sin . kshumENGG simplifies to Solutions are elliptic functions. Solutions are sinusoidal functions.

Modeling the pendulum kshumENGG 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 kshumENGG Time is discrete

Discrete-time dynamical system Logistic population growth – n(t) = population in the t-th year – t = 0,1,2,3,4 kshumENGG Increase in population Proportionality constant An example for K=1 Graph of n(1-n) Slow growth fast growth Slow growth negative growth

Sample population growth kshumENGG a=0.8, K=1 Monotonically increasing Initialized at n(1) = 0.01 a=2, K=1 Oscillating

Sample population growth kshumENGG a=2.8, K=1 Chaotic Initialized at n(1) = 0.01

Rough classification kshumENGG System StaticDynamic 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. kshumENGG201325