ENGINEERING MATHEMATICS Unit 1 Overview

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 ENGG2013

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) ENGG2013

Assessment Scheme Grading Bi-Weekly Homework (10%) Report (20%) Class (10%) Midterm (30%) Final Exam (30%) ENGG2013

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 email and/or private discussing with instructor/tutors. ENGG2013

System of Linear Equations Two variables, two equations ENGG2013

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') ENGG2013

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

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

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

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? ENGG2013

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

Electronic Circuit (Dynamic) Find the current through each resistor alternating current capacitor System of differential equations ENGG2013

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 ENGG2013

Simplifying assumptions System Modeling Reality Physical System Physical Laws + Simplifying assumptions Mathematical description Theory ENGG2013

How to Model a Typhoon? Lots of partial differential equations are required. ENGG2013

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

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

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

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

Modeling the Pendulum modeling or Continuous-time dynamical system for small angle  ENGG2013

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

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 ENGG2013

Sample Population Growth Initialized at n(1) = 0.01 Monotonically increasing Oscillating a=0.8, K=1 a=2, K=1 ENGG2013

Sample Population Growth Initialized at n(1) = 0.01 a=2.8, K=1 Chaotic ENGG2013

Probabilistic systems are Rough Classification System Static Dynamic Probabilistic systems are treated in ENGG2040 Continuous-time Discrete-time ENGG2013

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. ENGG2013