Engineering Mathematics III
 
 
Subject Code: EEM2036
Aim of Subject: To further strengthen the understanding and usefulness of mathematical concepts and analysis methods in algebra, vector calculus, further matrices and numerical analysis in the engineering context.
Learning Outcome of Subject: At the completion of the subject, students should be able to:
  • Understand the geometrical meanings of double and triple integrals
  • Construct and perform double and triple integration over elementary regions based on Cartesian, cylindrical, spherical and polar coordinates
  • Understand the concepts of vector fields, gradients, divergence and curl
  • Evaluate line and surface integrals using parameterization, Divergence Theorem, Stokes' Theorem and Green's Theorem
  • Apply the line and surface integrals in solving some simple engineering problems.
  • Represent inverse matrix as products of elementary matrices
  • Understand the concepts and find the eigenvalues and eigenvectors of a square matrix
  • Diagonalize a diagonalizable matrix
  • Apply the diagonalization method to solve a system of first order ordinary differential equations with constant coefficients whose coefficient matrix is diagonalizable via the integrating factor method
  • Understand the concepts of numerical methods and the derivation of the formulas
  • Analyze the accuracy of approximated solutions
  • apply numerical methods in solving the engineering problems, which are related to interpolation, numerical differentiation, numerical integration, numerical solutions of equations in one variable, numerical solutions of ordinary differential equations, numerical solutions of systems of linear equations and approximation of eigenvalues
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: EEM1026: Engineering Mathematics II
References:
  • Lecture Notes Series: Engineering Mathematics Volume 2 – 2nd edition, Pearson Prentice Hall, 2006. (Textbook)
  • Advanced Engineering Mathematics - 8th Edition, Kreyszig, John Wiley & Sons, 1999.
  • Advanced Modern Engineering Mathematics - 2nd Edition, James G., Prentice Hall, 1999.
  • Elementary Numerical Analysis – 3rd Edition, Atkinson, K., Han, W., John Wiley & Sons, 2003.
  • Numerical Analysis - 8th Edition, Burden, Richard L., Faires, J. Douglas, Brooks Cole, 2004.
  • Elementary Linear Algebra - 8th Edition, Anton, Howard, John Wiley & Sons, 2000.
  • Calculus - 7th Edition, Anton, H., Bivens, I. and Davis, S., John Wiley & Sons, 2002.
  • Calculus - 5th Edition, Stewart J., International Thomson Publishing, 2003.
  • Vector Calculus, Colley, Susan Jane, Prentice Hall, 1998.
  • Schaum’s outline series - Theory and Problems of Vector Analysis, Spiegel, Murray R., Mcgraw-Hill, 1974.
  • Thomas’ Calculus - 10th edition, Finney, Weir & Giordano, Addison-Wesley, 2001

Subject Contents

  • Multiple Integrals and Vector Calculus
    Iterated integrals, multiple integrals over elementary regions. Change of variables, Jacobians.Cylindrical and spherical polar coordinate systems.
    Vector field, gradient and directional derivative, divergence, curl. Conservative field, potential function.Line integrals, work done as line integrals. Surface integrals, flux. Divergence Theorem, Stokes' theorem.
    Applications in engineering.
    (Remarks: The “Cylindrical and spherical polar coordinate systems” are taught in multiple integrals but the terms are inserted here to make the syllabus more complete.)
     
  • Further Matrices

    • Matrix inverse by elementary matrices, adjoint, and partitioning methods. Characteristic polynomial, characteristic equation, eigenvalues and eigenvectors. Diagonalization of matrices, application to system of first order linear differential equations. Applications in physics and engineering problems.
     
  • Numerical Methods

    • Finite difference. Interpolation and its applications as curve fitting method in analyzing experimental data. Numerical differentiation and integration.Analysis of tabular information in engineering applications. Newton Raphson method for roots of equations. Roots of transcendental equations in engineering problems. Numerical solutions of ordinary differential equations. System of linear equations: Gaussian elimination with pivoting strategy. Gauss-Seidel iterative method and Jacobi iterative method.Evaluation of determinant and inverse matrix. Eigensystem analysis: system stability, stability of Gauss-Seidel and Jacobi solutions.
    (Remarks: Trapezoidal, Simpson are the fundamental techniques to be taught under numerical integration. Runge Kutta method is also under numerical solutions of ordinary differential equations. Amplitude and time scaling for model studies have been proposed to be cancelled, since the computational problem due to numbers have been covered while introducing Gaussian elimination with pivoting strategy.)