Engineering Mathematics II
 
 
Subject Code: EEM1026
Aim of Subject: To provide various mathematical concepts and analysis methods in matrices, ordinary and partial differential equations and statistics in the engineering context.
Learning Outcome of Subject: At the completion of the subject, students should be able to:
  • Understand the concepts of sampling theory and central limit theorem.
  • Perform statistical inference using estimation and hypothesis testing.
  • Understand the concepts of matrix and its operations.
  • Find the determinant of a square matrix and the inverse of an invertible matrix.
  • Solve a system of linear equations.
  • Understand the concepts of linear dependence and independence.
  • Understand the Fourier and Laplace transforms and their inverses.
  • Apply Fourier and Laplace transforms to solve boundary value problems.
  • Understand the concept of Z-transform and apply it to solve difference equations.
  • Solve first order ODE, in particular separable or exact differential equations using techniques such as integrating factor method.
  • Solve second order homogeneous and non-homogeneous ODEs with constant coefficients.
  • Use series solution method to solve linear ODEs.
  • Identify some special functions through their corresponding differential equations: Bessel function, Legendre polynomial etc.
  • Classify basic PDEs. In particular classify second order linear PDEs as elliptic, hyperbolic or parabolic type (wave, diffusion and Laplace equations).
  • Understand the D’Alembert formula for the wave equation.
  • Apply separation of variables method to solve linear homogeneous PDEs and transform the PDE using appropriate coordinate system that fits the boundary conditions.
Programme Outcomes:
  • Ability to acquire and apply fundamental principles of science and engineering(80 %)
  • Capability to communicate effectively(10%)
  • Ability to identify, formulate and model problems and find engineering solutions based on a systems approach(10%)
Assessment Scheme:
  • Test/Quiz/Assignment - Written test To enhance the understanding of basic concepts in lectures To consolidate the concepts and techniques in lectures through problem solving. (40%)
  • Final Exam - Written exam(60%)
Teaching and Learning Activities: 50 hours (lectures and tutorials)
Credit Hours: 3
Pre-Requisite: EEM1016: Engineering Mathematics I
References:
  • 1. Lecture Notes Series: Engineering Mathematics Volume 1 ?2nd edition, Pearson Prentice Hall, 2006. (Textbook)
  • Elementary Linear Algebra – 7th edition, Howard Anton, John-Wiley & Sons, 1994.
  • Advanced Engineering Mathematics – 8th edition, Erwin Kreyszig, John-Wiley & Sons, 1999.
  • Advanced Modern Engineering Mathematics – 3rd edition, James, G., Addison –Wesley, 1993.
  • Further Engineering Mathematics – 3rd edition, Stroud, K.A., Macmillan, 1996.
  • Partial Differential Equations and Boundary-Value Problems with Applications – 3rd edition, Pinsky, M.A., McGraw-Hill, 1998.
  • Beginning Partial Differential Equations, O’Neil, Peter V., John-Wiley, 1999.
  • Calculus – 4th edition, James Stewart, Brooks/Cole Publishing, 1999.
  • Probability and Statistics for Engineers and Scientists – 7th edition, R.E.Walpole, R.H.Myers, S.L.Myres and K. Ye, Prentice Hall, 2002.

Subject Contents

  • Statistics
    Elementary sampling theory for normal population. Central limit theorem. Statistical inference (point and interval estimation and hypothesis testing) on means, proportions and variances. Chi-squares test of goodness of fit. Examples involving engineering applications.
    Remark: Simple linear regression is moved to EEM2046/EEM2056.
     
  • Matrices and Determinants
    Matrices, some special matrices, matrix operations. Determinants and some useful theorems. Laplace’s development. Solution of system of linear equations by determinants. Linear dependence and independence, rank of a matrix. General system of linear equations, existence and properties of solution, Gaussian elimination.
     
  • Ordinary Differential Equations

    • Introduction and characteristics of a differential equation: definition, degree, order, linearity, homogeneity, concept of solution. First order equations, separable variables,exact equations, integrating factors, linear equations with applications (eg. radiocarbon dating, Newton’s law of cooling, etc.)
    Second order linear differential equations with constant coefficients.Homogeneous equations with applications such as electric circuits and mass-spring system, damping system. Non-homogeneous equations: complementary functions, particular integrals, method of undetermined coefficients. General linear second-order differential equations with variable coefficients. Elementary power series methods.
    (Remark: The D-operator method is just another method to choose the particular integral (not to determine the final form of particular integral). It is just a repetition of method of undetermined coefficients. The real usage of D-operator method (to determine the actual/final form of particular integral by considering inverse of the operator) needs at least four to six hours of lecture which is not realistic in our course structure).
     
  • Integral Transform

    • Derivation of transforms and inverses (Fourier and Laplace). Applications of these transforms in boundary and initial value problems (eg. electrical circuits with a voltage source). -transform and its application to solve finite difference equations.
     
  • Partial Differential Equations

    • Basic concepts of partial differential equations. Classification of 2nd order linear partial differential equation into basic types. The principle of linear superposition. The wave, diffusion, Laplace and Poisson's equations. Boundary and initial-value problems. D'Alembert's solution for wave equation. Method of separation of variables. Examples of partial differential equations commonly used in engineering