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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 21272

Title                         : Infinite Series

Pre-requisites         : PMAT 12253

 Learning Outcomes:

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

1. define the meaning of convergence of a real sequence of real numbers
2. use definitions to discuss the behavior of a given sequence
3. describe the nature of the convergence of infinite series and conditions under what differentiation and
integration can be performed
4. demonstrate knowledge on power series representation of a series.
5. use applications of Taylor polynomial.

 Course Contents:

Sequences: Limits and limit theorems for sequences, Monotone sequences and Cauchy sequences, Bounded
sequences, Monotone sequence theorem, Subsequences, Bolzano-Weierstrass theorem

Series: Convergence of Infinite Series, Geometric series, Harmonic Series, the Integral Test and Estimates of Sums,
The Comparison Tests and Estimates of Sums, Alternating Series and estimates of Sums, Absolute and conditional
Convergence, Ratio Test and Root Test

Power Series: Representation of Functions as Power Series, Differentiation and Integration of Power Series, Taylor
and Maclaurin Series, Binomial Series, Applications of Taylor Polynomials

 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 Transcendentals, Cengage Learning.
2. Knopp, K. (1956). Infinite sequences and series. Courier Corporation.
3. Knopp, K. (1990). Theory and application of infinite series. Courier Corporation.
4. Hirschman, I.I. (2014). Infinite series. Courier Corporation.
5. Bonar, D.D. & Khoury Jr., M.J. (2018). Real infinite series (Vol. 56). American Mathematical Soc.
6. Bromwich, T.J.I.A. (2005). An introduction to the theory of infinite series (Vol. 335). American
Mathematical Soc.

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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.

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Course code  :  PMAT 21263

Title               :  Linear Algebra

Pre-requisites :  PMAT 11232

 Learning outcomes:

Upon successful completion of the course students should be able to

1. demonstrate understanding of the concepts of vector space, subspace linear independence, span and basis
2. determine eigenvalues and eigenvectors and solve eigenvalue problems
3. describe algebraic and geometric multiplicities of eigenvalues and linearly independent eigenvectors
4. apply principles of matrix algebra to linear transformations
5. demonstrate an understanding of inner products and associated norms

 Course Contents:

Vector Spaces: Vector Spaces, Subspaces, Spanning Sets and Linear Independence, Basis and Dimension,
Extension Theorem, Coordinates, Change of Basis and Transition Matrix, Similarity, Dimensional Theorem.

Linear transformations: Linear Transformation, Kernel and Range of Linear Transformation, Rank and Nullity
Theorem, Isomorphisms, Matrix Representation of Linear Transformation, Applications of Linear Transformation.

Eigenvalues and Eigenvectors: Characteristic Polynomial, Eigenvalues and Eigenvectors, Eigen Spaces,
Diagonalization, Inner Product Spaces, Gram-Schmidt Orthogonalization Process, Orthogonal Complement,
Orthogonal Projections, Cayley-Hamilton Theorem, Minimum Polynomial of Matrices of Order Three

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

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

 Recommended readings:

1. Larson, R. & Falvo, D.C. (2016). Elementary Linear Algebra, Brooks Cole.
2. Andrilli, S. & Hecker, D. (2016). Elementary Linear Algebra, Elsevier Science.
3. DeFranza, J. & Gagliardi, D. (2015). Introduction to Linear Algebra with Applications, Waveland Press.
4. Lay, D.C., Lay, S.R. & McDonald, J.J. (5th Ed., 2015). Linear Algebra and Its Applications, Pearson