1. ENGG2013 Unit 1 Overview Jan, 2011.

2. 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)

3. 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/ http://www.cuhk.edu.hk/policy/academichon esty/ kshumENGG20133

4. System of Linear Equations kshumENGG20134 Two variables, two equations

5. System of Linear Equations kshumENGG20135 Three variables, three equations

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

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

8. 3x3 Determinant Volume of parallelepiped kshumENGG20138 (a,b,c) (d,e,f) (g,h,i)

9. 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 Carbohydrate1511145 Vitamin A0.020.0030.010.0060.01 Vitamin C0.01 0.0050.050.01 How to solve it using linear algebra?

10. Electronic Circuit (Static) Find the current through each resistor kshumENGG201310 System of linear equations

11. Electronic Circuit (dynamic) Find the current through each resistor kshumENGG201311 System of differential equations inductor alternating current

12. 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? kshumENGG201312 A B C Second-order differential equation

13. System Modeling kshumENGG201313 Physical System Mathematical description Physical Laws + Simplifying assumptions Reality Theory

14. How to model a typhoon? kshumENGG201314 Lots of partial differential equations are required.

15. Example: Simple Pendulum L = length of rod m = mass of the bob  = angle g = gravitational constant kshumENGG201315 L m  mg mg sin  

16. 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: kshumENGG201316 L m  mg mg sin  

17. 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 kshumENGG201317 * Foucault pendulum @ wiki

18. Further simplification Small-angle assumption – When  is small,  (in radian) is very close to sin . kshumENGG201318 simplifies to Solutions are elliptic functions. Solutions are sinusoidal functions.

19. Modeling the pendulum kshumENGG201319 modeling Continuous-time dynamical system or for small angle 

20. Discrete-time dynamical system Compound interest – r = interest rate per month – p(t) = money in your account – t = 0,1,2,3,4 kshumENGG201320 Time is discrete

21. Discrete-time dynamical system Logistic population growth – n(t) = population in the t-th year – t = 0,1,2,3,4 kshumENGG201321 Increase in population Proportionality constant An example for K=1 Graph of n(1-n) Slow growth fast growth Slow growth negative growth

22. Sample population growth kshumENGG201322 a=0.8, K=1 Monotonically increasing Initialized at n(1) = 0.01 a=2, K=1 Oscillating

23. Sample population growth kshumENGG201323 a=2.8, K=1 Chaotic Initialized at n(1) = 0.01

24. Rough classification kshumENGG201324 System StaticDynamic Continuous- time Discrete-time Probabilistic systems are treated in ENGG2040

25. 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