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Course Code    : PMAT 42373

Title                   : Advanced Mathematical Methods

Pre-requisites   : PMAT 22293, PMAT 22282

 Learning Outcomes:

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

1. discuss the properties of special functions as solutions of differential equations

2. classify partial differential equations using various techniques learned

3. solve hyperbolic, parabolic and elliptic equations using fundamental principles

4. apply a range of techniques to find solutions of standard PDEs

5. demonstrate accurate and efficient use of Fourier analysis techniques and their applications in the theory of PDEs

6. solve real world problems by identifying them appropriately from the perspective of partial derivative equations

 Course Content:

Special Functions: Legendre Polynomials, Bessel Functions

Partial Differential Equations: Classification of PDE, First order partial differential equations: Lagrange’s method and Charpit’s method, Second order partial differential equations: Linear Partial Differential Equations with Constant Coefficients, Partial Differential Equations of Order two with Variables Coefficients, Classification of Partial Differential Equations Reduction to Canonical or Normal Form: Parabolic, elliptic and hyperbolic partial differential equations.

Fourier Series: Fourier Series, Even and Odd Functions, Half-Range Expansions, Fourier Integral, Fourier Cosine and Sine Transforms

Solutions of PDEs: Separations of variables, D’Alembert’s Solution and Characteristic solutions of the Wave Equation,

Integral Transforms: Fourier Transforms, Laplace Transforms, Hankel Transforms

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

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

 Recommended Reading:

1. Kreyszig, E. (10th Ed., 2011). Advanced Engineering Mathematics, Wiley.
2. Pinsky, M.A. (2011). Partial Differential Equations and Boundary Value Problems with Application, American Mathematical Soc.
3. Raisinghania, M.D. (1995). Advanced Differential Equations, S.Chands, India.

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Course Code    : PMAT 41353

Title                   : Differential Geometry

Pre-requisites   : PMAT 22293

Learning Outcomes:

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

• demonstrate the fundamental knowledge of curves and surfaces in space
• identify the importance of the two factors curvature and torsion, and their intrinsic properties
• determine the local shape of the surface using first and second fundamental forms
• examine the theory of abstract manifolds
• use the theory, methods and techniques of the course to solve mathematical problems.

 Course Contents:

Theory of Curves: Concept of a curve, Parametrized curves, Regular curves: Arc length, Tangent vectors, Normal and binormal vector, Curvature and torsion, Frenet-serret formulae (Frenet formulas), Frenet frame, Isoperimetric inequality for a plane curve, The four-vertex theorem, General helix, intrinsic equations, Fundamental existence and uniqueness theorems for space curves, Canonical representation of a curve. Involutes and Evolutes, Bertrand curves, Theory of contact.

Theory of Surfaces: Concept of a surface, Tangent Plane, Topological properties of a surface, Surface of revolution, Ruled surfaces, Length of arc on a surface, Vector element of an area, First and second fundamental forms, Curves on a surface, Direction coefficients, Direction ratios, Family of curves on a surface, Double family of curves. Umbilical point, Intrinsic properties of a surface, Geodesics. Principal curvatures and directions, Gaussian and Mean curvatures, Lines of curvature, Rodrigues formula,

Introduction to Riemannian geometry: Riemannian Manifolds, Smooth Manifolds

 Method of Teaching and Learning: A combination of lectures, tutorials and presentations.

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

 Recommended Reading:

1. Lipschutz, L. (1969). Differential Geometry, McGraw-Hill.
2. Willmore, T.J. (2013). An Introduction to Differential Geometry, Oxford University Press.
3. Do Carmo, M.P. (2016). Differential Geometry of Curves and Surfaces, Prentice-Hall, New Jersey.

User Rating: 5 / 5

Course Code    : PMAT 42793

Title                   : Theory of Riemann Integration

Pre-requisites   : PMAT 12253

 Learning Outcomes:

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

1. identify the Darboux integrability considering the convergence of upper and lower Darboux sums.

2. make use of first and second Cauchy’s to decide the integrability of a function

3. compare Riemann and Darboux integrals of a function and discuss the properties of Riemann Integrable functions

4. prove and apply Intermediate Value Theorem for Integrals, Dominated Convergence Theorem and Monotone Convergence Theorem, and distinguish the Fundamental theorem of calculus and Second Mean Value Theorem for Integral

5. categorize various types of improper integrals based on the locally integrability and make use of properties of improper integrals

6. appraise improper integrals of nonnegative functions using given tests and determine absolute and conditional convergence of improper integrals

7. identify appropriate change of variables to simplify improper integrals.

 Course Content:

Darboux Integration: Upper and Lower Darboux sum, Darboux Integrability, Properties of darboux Integrability, First and Second Cauchy criterion for Integrability.

Riemann Integration: Riemann sum and the Riemann integral, Relationship between Darboux Integrability and Riemann Integrability, Properties of the Riemann integrability, Intermediate Value Theorem for Integrals, Dominated Convergence Theorem, Monotone Convergence Theorem, Fundamental theorem of calculus, Second Mean Value Theorem for Integral, Change of Variables.

Improper Integrals: Improper Integrals of first and second kind based on locally integrability, Improper integrals of unbounded functions (at left end, right end, both end and an interior point) with a finite domain of integral, Comparison test, Limit comparison test, Cauchy test, absolute and conditional convergence, convergence of beta function. Improper integrals of bounded functions with infinite domain of integrals, convergence at infinity, Comparison test, Limit comparison test, Absolute and conditional convergence, convergence of Gamma function, Abel’s test, Dirichlet’s test.

 Method of Teaching and Learning: A combination of lectures, tutorials and presentations

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

 Recommended Reading:

1.  Ross, K.A. (2nd Ed., 2015). Elementary Analysis. The Theory of Calculus, Springer.
2. Trench, W.F. (Hyperlinked Edition 2.04 December 2013), Introduction to Real Analysis, Library of Congress Cataloging-in-Publication Data.
3. Widder, D.V. (2nd Ed., 2012). Advanced Calculus. Courier Corporation.

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Course Code    : PMAT 41343

Title                   : Complex Analysis

Pre-requisites   : PMAT 22293

Learning Outcomes:

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

• identify regions in the complex plane
• explain the concepts of analytic functions and harmonic functions and then the important of Cauchy-Riemann equations
• explain the convergence power series and develop analytical capabilities in Taylor or Laurent series in a given domain
• describe the basic properties of singularities and zeros of analytic functions and calculate residues and use these to evaluate real integrals
• describe conformal mappings between various plane regions

 Course Contents:

Complex Numbers, Complex Valued Functions, Limits, Continuity, Differentiability, Analytic functions, Cauchy Riemann Equations, Elementary Functions, Line Integrals, Cauchy-Goursat Theorem, Morera’s Theorem, Cauchy’s Integral Formula, Cauchy’s Inequality, Lioville’s Theorem, Fundamental Theorem of Algebra, Maximum Modulus Principle, Minimum modulus Principle, Taylor and Laurent Series, Singularities, Argument Principle, Rouche's theorem, Residue Theorem, Maximum modulus theorem, Conformal mappings, Schwarz Christoffel Transformation.

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

 Assessment: Based on tutorials, tests, presentations 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.