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Advanced Engineering Mathematics Overview

Dive into linear algebra, differential equations, system modeling, and dynamic system analysis. Explore the applications in circuits, physics, and population growth.

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Advanced Engineering Mathematics Overview

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  1. ENGG2013Unit 1 Overview Jan, 2011.

  2. 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 Erwin O. Kreyszig (6/1/1922~12/12/2008) ENGG2013

  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/academichonesty/ ENGG2013

  4. System of Linear Equations Two variables, two equations ENGG2013

  5. System of Linear Equations Three variables, three equations ENGG2013

  6. System of Linear Equations Multiple variables, multiple equations How to solve? ENGG2013

  7. Determinant • Area of parallelogram (c,d) (a,b) ENGG2013

  8. 3x3 Determinant • Volume of parallelepiped (g,h,i) (d,e,f) (a,b,c) ENGG2013

  9. Nutrition problem • Find a combination of food A, B, C and D in order to satisfy the nutrition requirement exactly. How to solve it using linear algebra? ENGG2013

  10. Electronic Circuit (Static) • Find the current through each resistor System of linear equations ENGG2013

  11. Electronic Circuit (dynamic) • Find the current through each resistor alternatingcurrent inductor System of differential equations ENGG2013

  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? A C B Second-order differential equation ENGG2013

  13. System Modeling Reality Physical System Physical Laws + Simplifyingassumptions Mathematical description Theory ENGG2013

  14. How to model a typhoon? Lots of partial differential equations are required. ENGG2013

  15. Example: Simple Pendulum • L = length of rod • m = mass of the bob •  = angle • g = gravitational constant  L m mg sin   mg ENGG2013

  16. 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 ENGG2013

  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 * Foucault pendulum @ wiki ENGG2013

  18. 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. ENGG2013

  19. Modeling the pendulum modeling or Continuous-time dynamical system for small angle  ENGG2013

  20. 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 ENGG2013

  21. 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) Increase in population fast growth Slow growth Proportionality constant Slow growth negative growth ENGG2013

  22. Sample population growth Initialized at n(1) = 0.01 Monotonically increasing Oscillating a=0.8, K=1 a=2, K=1 ENGG2013

  23. Sample population growth Initialized at n(1) = 0.01 a=2.8, K=1 Chaotic ENGG2013

  24. Rough classification System Static Dynamic Probabilistic systems are treated in ENGG2040 Continuous-time Discrete-time ENGG2013

  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. ENGG2013

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