ENGINEERING MATHEMATICS II

Course 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.

• Matrices and Determinants
  

  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.
  

• 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.  Higher order linear differential equation.General linear second-order differential equations: with variable coefficients.  Elementary power series method.

• 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).  Z-transform and its application to solve finite difference equations.

• Partial Differential Equations

  Basic concepts of partial differential equations. Classification of second order linear partial differential equations. 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.

  
  

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 (Frobenius 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 (% of contribution)

• 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%