Topics & Recommended Readings of Servicing Courses

Topics covered

1. Polynomials; Mathematical induction; Binomial theorem
2. Coordinate geometry and conic sections; Basic trigonometry
3. Functions and inverses; Limits, Continuity and differentiability
4. Techniques of differentiation, implicit, logarithmic and parametric differentiation; Successive differentiation
5. Applications of differentiation: rate of change, local extrema, optimization problems, Taylor's series, De l’Hôpital rule

Recommended readings
Text(s):

1. a. Frank Ayres, Jr. and Elliott Mendelson, Calculus (Schaum's Outlines), 6th ed., McGraw Hill, 2013
   b. Fred Safier, Precalculus (Schaum's Outlines), 3rd ed., McGraw Hill, 2013
2. Basic Calculus and Linear Algebra (Compiled by Department of Mathematics, City University of Hong Kong), Pearson Custom Publishing, 2007 (Available at CityU Bookstore)
   Note: This book is compiled from:
   1. Thomas' Calculus (11th Ed.) by George B. Thomas, Maurice D. Weir, Joel D. Hass, & Frank R. Giordano. (Chapters 1–8, 10–12, selected appendices)
   2. Linear Algebra: A First Course in Pure and Applied Math by Edgar G. Goodaire (Chapters 1–3)
3. Ron Larson and Bruce Edwards, Calculus I with Precalculus: A One-Year Course, 3rd ed., Brooks/Cole, 2012
4. C. Henry Edwards and David E. Penney, Calculus: Early Transcendentals, 7th ed., Pearson Prentice Hall, 2008
5. Robert A. Adams, Calculus: A Complete Course, 6th ed., Pearson Addison Wesley, 2006
6. Glyn James, Modern Engineering Mathematics, 4th ed., Pearson Prentice Hall, 2008

Topics covered

1. Definite and indefinite integrals; Techniques of integration, integration of rational functions, integration by substitution, integration by parts
2. Physical and geometric applications of integration
3. Vectors in R² and R³; Scalar products, vector products, triple scalar products; Linear (in)dependence
4. Matrices; Determinants, cofactor expansion; Systems of linear equations, Gaussian elimination, Cramer’s rule; Matrix inverses, Gauss-Jordan elimination method
5. Arithmetic of complex numbers; Polar and Euler forms; De Moivre’s theorem and its applications

Recommended readings
Text(s):

1. a. Frank Ayres, Jr. and Elliott Mendelson, Calculus (Schaum's Outlines), 6th ed., McGraw Hill, 2013
   b. Fred Safier, Precalculus (Schaum's Outlines), 3rd ed., McGraw Hill, 2013
2. David C Lay, Linear Algebra and its applications, 3rd Pearson International Edition, 2006
3. Basic Calculus and Linear Algebra (Compiled by Department of Mathematics, City University of Hong Kong), Pearson Custom Publishing, 2007
   Note: This book is compiled from:
   1. Thomas’ Calculus (11th Ed.) by George B. Thomas, Maurice D. Weir, Joel D. Hass, & Frank R. Giordano. (Chapters 1–8, 10–12, selected appendices)
   2. Linear Algebra: A First Course in Pure and Applied Math by Edgar G. Goodaire (Chapters 1-3)
4. C. Henry Edwards and David E. Penney, Calculus: Early Transcendentals, 7th ed., Pearson Prentice Hall, 2008
5. Robert A. Adams, Calculus: A Complete Course, 6th ed., Pearson Addison Wesley, 2006
6. Glyn James, Modern Engineering Mathematics, 4th ed., Pearson Prentice Hall, 2008

Topics covered

1. Polynomials; Mathematical induction
2. Coordinate geometry and conic sections; Basic trigonometry
3. Functions and inverses
4. Limits of sequences and infinite series
5. Limits, continuity and differentiability of functions
6. Techniques of differentiation, implicit, logarithmic and parametric differentiation; Successive differentiation

Recommended Textbook

Single Variable Calculus (6th edition) by J. Stewart, Thomson Brooks/Cole, 2009

Topics covered

1. Basic theorems of differentiation
2. Applications of differentiation: rate of change, local extrema, optimization problems, power and Taylor series, l’Hôpital rule
3. Definite and indefinite integrals; Techniques of integration, integration by substitution, integration by parts; Improper integrals
4. Physical and geometric applications of integration
5. Vectors in R² and R³; Scalar products, cross products, triple scalar products; Linearly (in)dependence; Applications to equations of lines and planes
6. Matrices; Determinants, cofactor expansion; Systems of linear equations, Gaussian elimination, Cramer’s rule; Matrix inverses, Gauss-Jordan elimination method
7. Arithmetic of complex numbers; Polar and Euler forms; De Moivre’s theorem and its applications

Recommended Textbook (tentative)

• Single Variable Calculus (7th edition) by J. Stewart, Pacific Grove, CA: Brooks/Cole, 2011.
• Modern Engineering Mathematics (4th edition) by G. James, Upper Saddle River, NJ: Prentice Hall, 2010.

Keyword syllabus

• Linear Algebra: Orthogonality; Eigenvalues and eigenvectors; Eigenvalue decompositions.
• Multi-variable Calculus: Functions of several variables; Partial differentiation; Multi-variable Taylor series; Multiple integration; Gradient, divergence and curl; Line and surface integrals; Theorems of Gauss, Stokes and Green.

Textbook

Mathematics for Engineering and Science, Department of Mathematics, City University of Hong Kong, Prentice Hall, Pearson Education South Asia, 2008
Note: This book is compiled from:

1. Calculus – Early Transcendentals (7th ed.) by C. Henry Edwards & David E. Penny (Chapters 11, 12, 13 and 14)
2. Linear Algebra – A Pure and Applied First Course (1st ed.) by Edgar G. Goodaire (Chapters 1, 2, 3, 7, Solutions to True/False Questions)
3. Differential Equations and Boundary Value Problems (4th ed.) by C. Henry Edwards & David E. Penny (Chapters 1, 3, 9 and Answers to Selected Problems)

Reference

1. Thomas’, Calculus (13th ed.) by George B. Thomas, Maurice D. Weir, Joel D. Hass and Frank R. Giordano, Upper Saddle River, N.J.: Pearson Addison Wesley, 2014
2. Linear Algebra and Its Applications (4th ed.) by David C. Lay, Pearson, 2012
3. Advanced Engineering Mathematics (7th ed.) by Peter V. O'Neil, Cengage Learning, 2012
4. Advanced Engineering Mathematics (10th ed.) by Erwin Kreyszig, Wiley 2011

Keyword syllabus (as shown in Form 2B)

Introduction to Computer Systems, Numerical tools, Statistical packages and Mathematical packages.

Topics covered

The Use of EXCEL

• Introduction to EXCEL workplace; Probability Distributions, histogram, random number generation; Statistical Inference; Regression and Correlation; ANOVA

The Use of MATLAB

• Introduction to MATLAB workplace; Solving of system of linear equations; basic commands in MATLAB, random number generation, elementary MATLAB programming techniques

Reference books

1. Elementary Statistics (11th Ed.) by R. Johnson & P. Kuby, Duxbury, Brooks/Cole Cengage Learning, 2012
2. Elementary Statistics Using Excel (4th Ed.) by M F Triola, Addison Wesley, 2010

Keyword syllabus (as shown in Form 2B)

Mathematical logic. Methods of mathematical proof. Predicate calculus. Sets and relations. Castesian product. Functions. Permutations and combinations. Inclusion-exclusion principle. Recurrence relations. Complexity analysis of algorithms.

Topics covered

Mathematical logic

• Basic concepts, propositions and logical operators, algebra of propositions, predicates and quantifiers, logical equivalence, logical inference, valid arguments, inference rules

Methods of proving theorems

• Direct proof, indirect proof, prove by contradiction, conditional proof, proof by cases, mathematical induction

Basic set theory

• Elements, equality of sets, subsets, Venn diagrams, power set, algebra of sets, set operators, set identities, Cartesian product

Relations

• Binary relations, n-ary relations, representations of relations, reflexive relations, symmetric relations, antisymmetric relations, transitive relations, composite relation, inverse relation

Functions

• Basic definitions, composite function, injective surjective and bijective functions, inverse function

Combinatorics

• Pigeonhole principle, product rule, sum rule, selections with and without replacement, permutations, combinations, binomial coefficients, inclusion-exclusion principle

Recurrence relations

• Linear homogeneous recurrence relations with constant coefficients, characteristic polynomial, Solutions of special linear nonhomogeneous recurrence relations

Analysis of algorithms

• Computation of maximum number of operations of some simple algorithms

Suggested book

Discrete Mathematics And Its Applications (6th Ed.) by Kenneth H. Rosen, McGraw-Hill Edition

Keyword syllabus (as shown in Form 2B)

Eigenvalues and eigenvectors. Applications in elasticity. First-and second-order ordinary differential equations and applications. Vector calculus. Partial differentiation. Multiple integration. Gradient, divergence and curl. Theorems of Gauss, Stokes and Green. Applications in energy methods, stress and strain transformations, etc. Fourier series.

Topics covered

Eigenvalues and Eigenvectors

• Determination of Eigenvalues and Eigenvectors, diagonalization

Ordinary Differential Equations

• Methods for solving first-order ODEs, homogeneous second-order ODEs, solving second-order ODEs by method of undetermined coefficients and method of variation of parameters, higher-order ODEs

Multivariable Calculus

• Limit, continuity and partial derivative, partial differentiation, chain rule, implicit function theorem, Taylor’s theorem, Maxima and minima, directional derivative and gradient

Vectors Differential Calculus

• Vector fields, divergence of a vector field, curl of a vector field, vector identities

Multiple Integrals

• Double integrals in Cartesian coordinates, change of variable in double integrals; triple integrals in Cartesian coordinates, change of variable in triple integrals

Vector Integral Calculus

• Line integrals, conservative fields; Surface integrals; Divergence theorem; Theorems of Stokes and Green, applications

Fourier Series

• Fourier series for periodic, even and odd functions, half range series

Textbook

Mathematics for Engineering and Science, Department of Mathematics, City University of Hong Kong, Prentice Hall, Pearson Education South Asia, 2008
Note: This book is compiled from:

1. Calculus – Early Transcendentals (7th Ed.) by C. Henry Edwards & David E. Penny (Chapters 11, 12, 13 and 14)
2. Linear Algebra – A Pure and Applied First Course (1st Ed.) by Edgar G. Goodaire (Chapters 1, 2, 3, 7, Solutions to True/False Questions)
3. Differential Equations and Boundary Value Problems (4th Ed.) by C. Henry Edwards & David E. Penny (Chapters 1, 3, 9 and Answers to Selected Problems)

References

1. Differential Equations with Boundary-Value Problems (6th ed.) by Zill, Dennis G. and Cullen, Michael R., Belmont, CA: Thomson Brooks/Cole, 2005

2. Thomas’ Calculus (11th Ed.) by Thomas, George B., Weir, Maurice D., Hass, Joel D. and Giordano, Frank R., Upper Saddle River, N.J.: Pearson Addison Wesley, 2004

Keyword syllabus (as shown in Form 2B)

Complex numbers. Vectors, matrices and determinants. Linear dependence, orthogonality. Systems of linear equations.  Eigenvalues and eigenvectors. Functions of several variables. Partial differentiation. Taylor series. Double integrals

Textbook

Mathematics for Engineering and Science, Department of Mathematics, City University of Hong Kong, Prentice Hall, Pearson Education South Asia, 2008
Note: This book is compiled from:

1. Calculus – Early Transcendentals (7th ed.) by C. Henry Edwards & David E. Penny (Chapters 11, 12, 13 and 14)
2. Linear Algebra – A Pure and Applied First Course (1st ed.) by Edgar G. Goodaire (Chapters 1, 2, 3, 7, Solutions to True/False Questions)
3. Differential Equations and Boundary Value Problems (4th ed.) by C. Henry Edwards & David E. Penny (Chapters 1, 3, 9 and Answers to Selected Problems)

Reference

1. Thomas’, Calculus (11th ed.) by George B. Thomas, Maurice D. Weir, Joel D. Hass and Frank R. Giordano, Upper Saddle River, N.J.: Pearson Addison Wesley, 2004
2. Linear Algebra and Its Applications (3rd ed.) by David C. Lay, Pearson, 2006
3. Multivariable Calculus with Matrices (6th ed.) by C. Henry Edwards and David E. Penney, Prentice Hall, 2002
4. Elements of Advanced Engineering Mathematics by Peter V. O'Neil, Cengage Learning, 2010

Keyword syllabus (as shown in Form 2B)

Random variables. Distribution. Data and sample description. Estimation of parameters. Tests of hypothesis. Regression. ANOVA.

Topics covered

Probability

• Basic concepts, rules of probability, conditional probability, Bayes’ Rule

Probability Distribution (discrete variables)

• Discrete probability distributions, binomial random variable

Normal Probability Distributions

• Normal probability distributions and its applications

Sample Variability

• Central limit theorem

Introduction to Statistical Inferences

• Basic concepts of estimation and hypothesis testing, classical and probability approach

Inferences Involving One Population

• Confidence intervals, inference about population mean, inference about the binomial probability of success, t test, inference about the population standard deviation, chi-square distribution

Inferences Involving Two Populations

• Independent and dependent samples; Comparison on two populations using the mean of paired differences, the difference between two means and the ratio of two variances, F test

Linear Correlation and Regression Analysis

• Linear correlation and regression analysis, line of best fit, confidence interval estimations

ANOVA* (*to be covered if time permits)

Suggest books

1. Elementary Statistics (11th Ed.) by R. Johnson & P. Kuby, Duxbury, Thomson Learning, 2012
2. Elementary Statistics Using Excel (6th Ed.) by M F Triola, Addison Wesley, 2018

Keyword syllabus (as shown in Form 2B)

Ordinary differential equations. Fourier series. Laplace transforms. Random variables. Probability. Distributions. Data and sample description. Estimation of parameters. Test of hypothesis. Simple linear regression.

Topics covered

Ordinary Differential Equations

• Methods for solving first-order ODEs, homogeneous second-order ODEs, solving second-order ODEs by method of undetermined coefficients and method of variation of parameters

Fourier Series

• Fourier series for periodic, even and odd functions, half range series

Laplace Transforms

• Definition and the Inverse Laplace Transform, properties of Laplace Transform

Probability Distribution

• Random variables, discrete probability distributions, normal probability distributions

Sample Variability

• Central limit theorem

Statistical Inferences

• Basic concepts of estimation and hypothesis testing, confidence intervals, statistical inferences

Simple Linear Regression

Suggested books

1. Mathematics for Engineering and Science, Department of Mathematics, City University of Hong Kong, Prentice Hall, Pearson Education South Asia, 2008
2. Advanced Engineering Mathematics (9th Ed.) by Erwin Kreyszig, Wiley, 2006
3. Elementary Statistics (11th Ed.) by R. Johnson & P. Kuby, Duxbury, Thomson Learning, 2012
4. Elementary Statistics Using Excel (4th Ed.) by M F Triola, Addison Wesley, 2010

Keyword syllabus (as shown in Form 2B)

Eigenvalues and eigenvectors. First- and higher order ordinary differential equations. Partial differentiation. Laplace transforms. Fourier series.

Topics covered

Eigenvalues and Eigenvectors

• Determination of Eigenvalues and Eigenvectors, diagonalization

Ordinary Differential Equations

• Methods for solving first-order ODEs, homogeneous second-order ODEs, solving second-order ODEs by method of undetermined coefficients and method of variation of parameters, higher-order ODEs

Multivariable Calculus

• Limit, continuity and partial derivative, partial differentiation, chain rule, Taylor’s theorem, Maxima and minima

Laplace Transforms

• Definition and the Inverse Laplace Transform, properties of Laplace Transform

Fourier Series

• Fourier series for periodic, even and odd functions, half range series

Textbooks

Mathematics for Engineering and Science, Department of Mathematics, City University of Hong Kong, Prentice Hall, Pearson Education South Asia, 2008
Note: This book is compiled from:

1. Calculus – Early Transcendentals (7th Ed.) by C. Henry Edwards & David E. Penny (Chapters 11, 12, 13 and 14)
2. Linear Algebra – A Pure and Applied First Course (1st Ed.) by Edgar G. Goodaire (Chapters 1, 2, 3, 7, Solutions to True/False Questions)
3. Differential Equations and Boundary Value Problems (4th Ed.) by C. Henry Edwards & David E. Penny (Chapters 1, 3, 9 and Answers to Selected Problems)

References

Advanced Engineering Mathematics (9th Ed.) by Erwin Kreyszig, Wiley 2006

Topics Covered

Mathematical Logic, Set Theory, Counting and Probability, Number Theory, and Graph

Textbook

1. K. Bogart, C. Stein, R. L. Drysdale, Discrete Mathematics for Computer Science, Key College Publishing, 2006

2. K. H. Rosen, Discrete Mathematics and Its Applications, 7th Edition, McGraw Hill, 2012

Keyword syllabus (as shown in Form 2B)

• Ordinary differential equations: First order differential equations, Second and higher order linear differential equations; Laplace transform; System of linear differential equations.
• Partial differential equations: Diffusion, wave and Laplace equations; Initial value problems; Fourier series; Boundary value problems.

Recommended readings
Text(s):

• "Differential Equations & Linear Algebra", 2nd ed.
  By Farlow, Hall, McDill and West
  Publisher: Pearson Education
• "Differential Equations with Boundary-value Problems", 7th ed.
  By D. G. Zill, M. R. Cullen
  Publisher: Brooks/Cole – CENGAGE Learning

Keyword syllabus (as shown in Form 2B)

Probability. Random variables. Distributions. Stochastic processes. Queuing theory.

Topics covered

Elementary Probability

• Probability: Probability, independence, conditional probability, expectation
• Random variables: a random variable, random vector, conditional random variable, conditional expectation
• Distributions: Bernoulli trials, binomial distribution, Poisson distribution, uniform distribution, exponential distribution

Stochastic Processes

• Poisson processes
• Markov chains

Queuing Theory

• Little’s formula, balance equations, m/m/1 queue, m/m/k queue

References

• Introduction to probability models / Sheldon M. Ross / 9th ed. / Academic Press, c2007