Engineering Mathematics I
 
 
Subject Code: EEM1016
Aim of Subject: To provide various mathematical concepts and analysis methods in calculus, algebra, vectors, complex functions, probability, and Fourier analysis in the engineering context.
Learning Outcome of Subject: At the completion of the subject, students should be able to:
  • define functions and evaluate limits of a function
  • determine continuity and differentiability of a function
  • differentiate and integrate functions of one variable
  • apply calculus techniques in problem solving
  • determine the convergence of sequence and series
  • express a function in terms of power series
  • determine the partial derivatives of functions of several variables
  • apply partial differentiation in some application problems
  • define and use the notions of polar form of complex numbers, Euler’s identity, de Moivre theorem and nth-roots of a complex number.
  • define some elementary complex functions and identify their properties
  • perform dot and cross products and their applications of finding equations of line and plane.
  • define the concepts of linear combination and linear independence.
  • obtain the Fourier series representation of function of arbitrary period, even function and odd function.
  • Use the method of full-range and half-range to obtain Fourier series of non-periodic functions
  • define and apply the basic concepts of probability
  • identify discrete and continuous random variables and their distributions, and apply their basic properties in some engineering problems
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: None
References:
  • Lecture Notes Series: Engineering Mathematics Volume 1 - 2nd edition, Pearson Prentice Hall, 2006. (Textbook)
  • Calculus - 6th edition, James Stewart, BrooksCole, 2008.
  • Thomas'Calculus - 11th edition, Ross L. Finney, Maurice D. Weir and Frank R. Giordano, Addison-Wesley, 2008.
  • Calculus - A New Horizon - 6th edition, Howard Anton, John Wiley & Sons Inc, 1999.
  • Modern Engineering Mathematics 4th edition, Glyn James, Prentice Hall, 2008
  • Advanced Engineering Mathematics - 9th edition, Erwin Kreysig, John Wiley & Sons Inc, 2006.
  • Probability and Statistics for Engineers and Scientists - 8th edition, R.E.Walpole, R.H.Myers, S.L.Myres and K. Ye, Prentice Hall, 2007.
  • Schaum's 3000 Solved Problems in Calculus, E. Mendelson, McGraw Hill, 1990.

Subject Contents

    Calculus

  • Calculus of One Variable (8)
    Elementary functions: polynomials, rational functions, exponential and logarithm functions, trigonometric and their inverses, principal values, hyperbolic and their inverses,graphs.Limits, continuity and differentiability. Mean-value theorem.Applications of differentiation: maximum and minimum values. Techniques of Integration. Applications of integration (eg. work done by variable force, centre of mass).
     
  • Sequences and Series (5)

    • Sequences of real numbers, convergence.Series of real numbers. Tests of convergence (eg. ratio test),Power series, radius of convergence. Taylor’s series expansion. Term-by-term integration and differentiation. Applications of Taylor polynomial in approximation problems.
     
  • Partial Differentiation (6)
    Functions of several variables, continuity and partial derivatives. Total differentials, approximations using differentials and their applications to engineering problems. Chain rule. Implicit differentiation. Extremum problems, without and with constraints, Lagrange multipliers, global extremum. Applications of extremum problems in engineering.
     

    • Complex Functions and Vector Algebra

    DeMoivre’s theorem, powers and nth-roots of complex numbers. Euler formula. Elementary functions of a complex variable, polynomials, rational, exponential, trigonometric, hyperbolic, logarithmic, inverse trigonometric and inverse hyperbolic functions.Dot product and its use in defining physical quantities (eg. work, etc). Cross product and its use in defining angular velocity, motion of charged particles in electromagnetic field, triple products and their geometrical applications. Vectors in Rn space, addition and scalar multiplication, linear combination of vectors, basic ideas of linear dependence and independence.
     

    Fourier Series

    Periodic functions,Fourier series of 2L-periodic functions, convergence and sum of Fourier series, even and odd functions. Half range sine and cosine series expansions. Complex form of Fourier series. Applications of Fourier series in engineering problems.
     

    Probability

    Probability space. Probability theory. Conditional probability and independence, concept of a random variable, discrete and continuous distributions, mean and variance. Bernoulli, Binomial, Poisson, hypergeometric, exponential, normal distributions and their characteristics. Examples involving experimental measurement and reliability.