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Course Code            : PMAT 31303

Title                          : Complex Variables

Pre-requisites          : PMAT 22293

Learning Outcomes:

At the end of this course, the student should be able to

• Perform basic mathematical operations with complex numbers in cartesian and polar form
• demonstrate knowledge of complex numbers and complex valued functions
• analyze and discuss limit, continuity and differentiability of complex valued functions
• analyze and interpret results of complex numbers and complex valued functions in applications
• evaluate real integrals using complex integrals.

Course Contents:

Complex Numbers: Basic Algebraic Properties, Exponential Form, De Movers’ Theorem, nth root of unity, Roots of Complex Numbers, Argand Diagram, Regions in the Complex Plane.

Analytic Functions: Complex Valued Functions, Limits, Theorems on Limits, Continuity and Uniform continuity, Derivatives, Differentiation Formulas, Cauchy–Riemann Equations, Sufficient Conditions for Differentiability, Analytic Functions, Harmonic Functions.

Elementary functions: Polynomial functions, rational functions, exponentials, trigonometric functions, hyperbolic functions, logarithmic functions, power series.

Integrals: Definite Integrals of Complex-Valued Function of a Complex Variable, Contour Integrals, Properties of integrals, Cauchy Theorem, Cauchy Integral Formula.

Series: Taylor Series, Laurent Series, Classification of Singular Points.

Residues and Poles: Residues, Residues at Poles, Cauchy’s Residue Theorem, Residue at Infinity.

Applications of Residues: Evaluation of Real Integrals.

Method of Teaching and Learning: A combination of lectures and tutorial discussions.

Assessment: Based on tutorials, tests and end of course examination.

Recommended Reading:

1. Brown, J.W. & Churchill, R.V. (9th Ed., 2014). Complex variables and applications, McGraw-Hill
2. Spiegel, M., Lipschutz, S. & Schiller, J. (2nd Ed., 2009). Schaum’s Outline of Complex Variables, McGraw-Hill
3. Hann, L. & Epstein, B. (1st Ed., 1996). Classical Complex Analysis, Jones and Bartlett Publishers
4. Ponnusamy, S. (2nd Ed., 2005). Foundation of Complex Analysis, Alpha Science

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Course Code            : PMAT 22293

Title                          : Functions of Several Variables

Pre-requisites          : PMAT 21263

 Learning Outcomes:

At the end of this course, the student should be able to:

1. appraise the geometrical aspects of functions of several variables in different coordinate systems
2. parametrize curves and surfaces
3. give an account of the concepts of limit, continuity, partial derivative, gradient and differentiability for
functions of several variables
4. prove and apply Young’s Theorem, Schwarz’s Theorem
5. classify local and global extremes of functions of two variables
6. determine local extremes under constrains using Lagrange Multiplier Method
7. outline the definition of the multiple integral, compute multiple integrals and use multiple integrals to
compute volumes in different coordinate system
8. make use of change of variables to evaluate double integrals.

 Course Contents:

Geometrical Aspects: Domain and range of functions of several variables, Level curves, Parametric surfaces;
Some special surfaces; planes, spheres, cylinders and cones; Surface area.
Analytical Aspects: Domain of a function of two variables; Neighborhoods in the Plane, Limits and Continuity;
Partial derivatives; Clairaut’s Theorem (Young’s Theorem), Schwarz Theorem, Differentials, Differentiability;
Tangent planes and linear approximations; Chain rules; Gradient of a Function, Directional Derivative, Tangent
Planes and Normal lines, Maxima and Minima; Critical Points and Second Partial Test, Lagrange multipliers.
Coordinate Systems: Cartesian, Polar, Spherical and Cylindrical Coordinate Systems
Multiple Integrals: Double integrals and Volume; Iterated integrals; Fubini’s Theorem, Double integrals over
general regions, Double integrals in Polar coordinates, Change of variables in double integrals, Triple integrals in
Cartesian, Cylindrical and Spherical coordinates

 Method of Teaching and Learning: A combination of lectures and tutorial discussions.

 Assessment: Based on tutorials, tests and end of course examination.

 Recommended Reading:

1. Stewart, J. (2020). Calculus Early Transcendental, Cengage Learning, Inc.
2. Larson, R. & Edwards, B.H. (2018). Calculus, Brooks/Cole, Cengage Learning
3. Ross, K.S. (2015). Elementary Analysis: The Theory of Calculus, Springer.
4. Malik, S.C. & Arora, S. (2017). Mathematical Analysis, New Age International.

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Course Code            : PMAT 22213

Title                          : Mathematical Methods for Computing

 Learning Outcomes:

At the end of this course, the student should be able

1. demonstrate knowledge in basic mathematical concepts of calculus
2. use basic mathematical concepts of calculus for further studies
3. classify ordinary differential equations
4. solve linear ordinary differential equations
5. derive the ordinary differential equations for certain applications
6. compute inner products and determine orthogonality on vector spaces
7. apply orthogonal decomposition of inner product spaces.

Course Contents:

Limits and Derivatives: Limit of a Function, Calculating Limits Using the Limit Laws, Continuity, Limits at
Infinity-Horizontal Asymptotes, Derivatives and Rates of Change, Derivative as a Function, Differential rules,
Applications of Differentiation
Integrals: Definite Integral, Fundamental Theorem of Calculus, Integration by parts, Partial fractions, Substitution
Rule.
Applications of Integration: Areas between Curves, Volumes, Volumes by Cylindrical Shells, Arc Length, Area
of a Surface of Revolution, Techniques of Integration
Ordinary Differential Equations (ODEs): Introduction to Differential Equations, ODEs, Order, Degree,
classification of linear and non-linear ODEs, solution of a differential equation, Family of curves, first order ODEs:
Separable ODEs, Exact ODEs, Integrating Factors, Linear ODEs, Higher Order Linear ODEs: Homogeneous
Linear ODEs, Homogeneous Linear ODEs with Constant Coefficients, Differential Operators, Special types of
ODEs: Bernoulli Equations, Euler–Cauchy Equations, Applications of ODEs.
Analytic Geometry: Norms, Inner Products, Lengths and Distances, Angles and Orthogonality, Orthonormal Basis
Orthogonal Complement, Inner Product of Functions, Orthogonal Projections, Rotations, Applications in computer
Science

 Method of Teaching and Learning: A combination of lectures and tutorial discussions.

 Assessment: Based on tutorials, tests and end of course examination.

 Recommended Reading:

1. Kreyszig, E. (10th Ed., 2011). Advanced Engineering Mathematics, Wiley.
2. Ross, K.S. (2nd Ed., 2015). Elementary Analysis: The Theory of Calculus, Springer.
3. Stewart, J. (8th Ed., 2015). Calculus Early Transcendentals, Thomson Learning, Inc.
4. Deisenroth, M.P. (2020). Mathematics for Machine Learning Cambridge University Press.

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Course Code            : PMAT 22282

Title                          : Ordinary Differential Equations

Pre-requisites          : PMAT 12253

 Learning Outcomes:

At the end of this course, the student should be able to

1. classify the differential equations with respect to their order and linearity
2. find a particular solution of a differential equation using initial conditions
3. solve first-order and higher-order linear ordinary differential equations
4. examine the existence and uniqueness of a solution of an initial value problem
5. solve linear differential equations using Laplace transform method
6. solve differential equations involving real-life applications.

 Course Contents:

Introduction: Differential Equations, Ordinary Differential Equations (ODE), Order, Degree, classification of
linear and non-linear ODEs, solution of a differential equation, Family of curves
First-Order ODEs: Separable ODEs., Homogeneous equations, Exact ODEs., Integrating Factors, Linear
ODEs, Bernoulli Equation. Orthogonal Trajectories, Existence and Uniqueness of Solutions for Initial Value
Problems, applications of first order ODEs.
Second/Higher Order Linear ODEs: Homogeneous Linear ODEs with Constant Coefficients, Homogeneous
Linear ODEs of Second Order, method of order reduction, Existence and Uniqueness of Solutions, Wronskian,
Differential Operators, Euler–Cauchy Equations, Nonhomogeneous ODEs, Solution by Variation of Parameters,
Method of undetermined coefficients, applications of higher order ODEs.
The Laplace Transform: Definition of Laplace transforms, Basic properties, Inverse Laplace transform,
Convolution theorem, Solve Linear Differential Equations with constant coefficients using Laplace transform

Teaching/Learning methods: A combination of lectures and tutorial discussions.

 Assessment: Based on tutorials, tests and end of course examination.

 Recommended Reading:

1. Kreyszig, E. (2018). Advanced Engineering Mathematics, Wiley.
2. Shepley, L.R, (1989). Introduction to Ordinary Differential Equations, John Wiley and Sons.
3. Nagle R.K., Saff, E.B. & Snider A.D. (2011) Fundamentals of Differential Equations, Pearson.
4. Krantz, S.G. (2014). Differential equations: theory, technique and practice (Vol. 17). CRC Press.
5. Tenenbaum, M. & Pollard, H. (1985). Ordinary Differential Equations, Dover Publications.
6. Murray, R., Murray, R., & Spiegal. (1974). Theory and problems of Laplace transforms. Shaum's
Outline Series.