Department of Mathematics

: Mathematics; Subjects; 03 December 2023; Hits: 2979

AMAT 11223: Vector analysis

User Rating: 4 / 5

Course Code : AMAT 11223

Title : Vector analysis

Pre-requisites : A/L Combined Mathematics

Learning Outcomes :

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

1. demonstrate knowledge of coplanarity of three vectors, orthogonal vectors, vector product.
2. calculate unit vectors, scalar and vector products, triple scalar & vector products, arc lengths, unit
tangent and normal vectors.
3. define Gradient, Divergence, Curl Operators.
4. calculate derivatives of vector fields and normal and directional derivatives of surfaces.
5. identify conservative, solenoidal, rotational and irrotational vector fields.
6. apply Fundamental Theorem of Line Integrals to compute work.
7. establish the Divergence and Stoke’s theorems.

Course Content:

Vector Algebra: Introduction to vectors, Condition for coplanarity of three vectors and collinearity of two
vectors, Orthogonal triads of unit vectors, Scalar and vector products, Triple scalar and vector products, Solution
of vector equations.

Vector Analysis: Scalar and vector fields, Differentiation of vector functions, sketch curves defined by position
vectors, arc length, unit tangent and unit normal vectors of curves, Gradient, Divergence and Curl operators and
identities involving them, Surfaces and normal, Directional Derivative, Conservative, solenoidal, rotational &
irrotational vector Fields, Sketching 3D objects,
Line Integrals, Fundamental Theorem of Line Integrals, Area Integrals, Surface Integrals, Volume Integrals,
Divergence and Stokes’ theorems.

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

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

Recommended Reading :

Spiegel, M. & S. Lipschutz. Vector Analysis, (2e) McGraw-Hill Education, 2009.
Chatterjee, D. Vector Analysis, PHI Learning private limited, India, 2009.
Davis, H.F. & Snider, A.D. An Introduction to Vector Analysis, C. Brown, New York, 1992.

: Mathematics; Subjects; 03 December 2023; Hits: 13

AMAT 42453: Computational Fluid Dynamics Using Appropriate Software

Course Code : AMAT 42453

Title : Computational Fluid Dynamics Using Appropriate Software.

Pre-requisites : AMAT 42393

Learning Outcomes :

After the completion of this course unit, the student will be able to:
1. identify the suitable governing equations of fluid flow problem
2. identify appropriate initial and boundary conditions
3. design the flow geometry and mesh
4. apply the appropriate software to solve the real-life fluid flow problem
5. simulate the flows fields
6. describe, interpret and communicate the simulation results
7. asses the quality of the numerical results and the efficiency of the solver.

Course Content:

Fluid Flow Simulations: Introduction to computational fluid dynamics, Review on governing equations of
different fluid flow problems, defining flow variables and parameters, designing geometry & generating mesh,
Implementation of Boundary Conditions,
Newtonian fluid flow simulation: Laminar Flow: viscous and inviscid flows, Boundary layers and separation,
Lid driven flow, Heat transfer in fluids, turbulent flows
Non-Newtonian fluid flow simulation

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

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

Recommended Reading :

Sharma, A. (2016). Introduction to Computational Fluid Dynamics, Wiley-Hall.
Tu, J., Yeoh, G.H. & Liu, C. (2018). Computational Fluid Dynamics, Butterworth Heinemann.
Niyogi, P., Chakrabartty, S.K. & Laha, M.K. (2002). Introduction to Computational Fluid Dynamics,
Pearson.
Pereira, A.C., Inacio, M. & Pereira, H.C. (2020). A Brief Introduction to CFD and Electromagnetism.
Matsson, J. (2020). An Introduction to ANSYS Fluent 2019. SDC.

: Mathematics; Subjects; 27 February 2020; Hits: 2036

PMAT 42423: Measure Theory

User Rating: 5 / 5

Course Code : PMAT 42423

Title : Measure Theory

Pre-requisites : PMAT 42363

Learning Outcomes:

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

1. Develop familiarity with measures, Integration theory and convergence as presented in the course

2. Understand the construction of the integral and know its key properties

3. Describe and apply the notion of measurable functions and sets and use Lebesgue monotone and dominated convergence theorems and Fatou’s Lemma

4. Apply theory of integral and understand the definition and basic properties of integral with respect to a measure

5. Analyze and understand how the major theorems presented in the course depend on preliminary results

6. Demonstrate knowledge of the concepts and theorems of abstract Measure Theory and to apply them in Lebesgue integral.

Content:

Sets and classes: Set inclusion, Number sets, Operations on sets, Unions, Intersections, Complement, Differences, Symmetric differences, Monotone sequence of sets, Limits, Inferior and Superior limits, Set of extended real numbers, Countable sets

Abstract Measure Theory: Algebra, 𝜎-Algebra, Measure, Additivity properties of a set function, Borel sets, Lebesgue measure, Outer measure, Extension of measures ( Measure on a larger class of subsets), Measurable subsets, Measurable functions, Integration with respect to a measure, Properties of integral, Properties that hold almost everywhere, Integrable functions, Additivity theorem, Monotone convergence theorem, Fatou’s lemma, Lebesgue Dominated convergence theorem, Relation between Lebesgue and Riemann integrals, Modes of convergence

Further results in Measure and Integration: Jordan-Hahn Decomposition theorem, Absolute continuity, Radon Nikodym theorem

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:

Cohn, D.L., (2^nd edition, 2015) Measure Theory, Springer New York.
Barra, G., (2^nd edition, 2003) Measure Theory and Integration, Elsevier.

: Mathematics; Subjects; 27 February 2020; Hits: 2184

PMAT 42433: Group Theory

User Rating: 5 / 5

Course Code : PMAT 42433

Title : Group Theory

Pre-requisites : PMAT 21263

Learning Outcomes:

After the completion of this course unit, the student will be able to:

1. analyze and use basic definitions in Abstract algebra including binary operations, groups, subgroups, homomorphism, group actions and Sylow theorems

2. examine the fundamental principles including the laws and theorems arising from the concepts covered in this course

3. develop and apply the fundamental properties of abstract algebraic structures, the substructures, their quotient structures and their mappings

4. build experience and confidence in proving theorems about the structure size and nature of groups, subgroups, group Isomorphism, group actions, Sylow theorems and the associated mappings

5. apply course materials along with techniques and procedures covered in this course to solve problems

6. develop specific skills, competencies and thought processes sufficient to support further studies or work in this or related fields

7. demonstrate knowledge of the structure of Groups and to apply the knowledge in solving problems in different areas in Algebra.

Course Contents:

Groups: Definition and Examples, Basic properties, Subgroups, symmetric Groups, Dihedral Groups, Cyclic Groups, Abelian Groups, Finite & Infinite Groups, order of a group, order of an element, Structure theory of finite Abelian Groups

Normal subgroups: Definition and examples, Quotient groups, Cosets, Lagrange theorem, Internal direct product

Group isomorphism: Definition of Group Homomorphism, Kernel of a homomorphism, Image of a homomorphism, Group Isomorphism, Isomorphism Theorems (First, Second and Third Isomorphism Theorems)

Group action: Orbits and Stabilizers, Orbit Stabilizer theorem, Burnside's theorem, Fixed point convergence theorems, Cauchy’s theorem, Fermat’s Little theorem

Sylow Theorems: Sylow’s p- subgroups, First, Second and Third Sylow’s Theorems, Applications of the Sylow’s Theorem, Groups of small order

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. Khanna, V.K. & Bhambri, S.K. (2016). A Course in Abstract Algebra, Vikas Publishing House.

2. Frakeigh, J.B. (2003). A first course in Abstract Algebra, Pearson Education India.

3. Baumslag, B. & Chandler, B. (1968). Group theory, McGraw-Hill, New York.

4. Narayan, S. & Pal, S. (1992). A Textbook of Modern Abstract Algebra, S. Chands, India.

5. Rotman, J.J. (4th Ed., 2014). An Introduction to the Theory of Groups, Springer-Verlag.

6. Gilbert, L. (8th Ed., 2014). Elements of Modern Algebra, Cengage Learning.

: Mathematics; Subjects; 27 February 2020; Hits: 1787

PMAT 43976: Research Project

User Rating: 5 / 5

Course Code : PMAT 43976

Title : Research Project

Learning Outcomes:

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

1. plan a research project in the field of Mathematics

2. investigate a research problem

3. develop a theoretical/practical/conceptual framework towards achieving the research objectives

4. document a research project proposal, and progress in detail

5. solve a research problem using theoretical principles and practical knowledge

6. analyze research results critically

7. communicate research findings, information, and solutions to a specialized audience in a form of dissertation and oral presentations

8. practice research ethics, technical skills.

Course Contents:

An undergraduate research project is an inquiry, investigation, or creation produced by a final year honours degree undergraduate that contributes to the discipline and reaches beyond the traditional curriculum. An undergraduate research project is designed to provide students with the opportunity to develop and practice advanced discipline-specific projects in collaboration with senior academics in the department.

Method of Teaching and Learning: self-studies, discussions and student presentations, seminars and colloquiums

Assessment: A combination of self-study, seminars, presentations, reports and dissertation

Recommended Reading:

1. Robson, C. (2nd Ed., 2016). How to do a research project - A guide for undergraduate students, Wiley

2. Reading list and material relevant for each selected topic to be provided at the beginning of the academic year by the supervisor

Page 7 of 13