Course Code    : PMAT 32342

Title                   : Number Theory

Pre-requisites   :  PMAT 12242

Learning Outcomes:

After the completion of this course unit, the student will be able to:
1. demonstrate a foundational understanding of number theory, including the definitions, conjectures, and
theorems that permit exploration of topics in the field.
2. work with numbers and polynomials modulo a prime, linear congruences, and systems of linear
congruences, including their solution via the Chinese remainder theorem.
3. Apply mathematical ideas and concepts within the context of number theory.
4. Solve a range of problems in number theory.
5. demonstrate an understanding of the mathematical underpinning of cryptography.

 Course Contents:

Divisibility: Division and Linear Diophantine Equations, Greatest common divisor, Euclidean Algorithm, Prime
and Composite Numbers, Fibonacci and Lucas Numbers
Congruences: Linear Congruences, Chinese remainder theorem, systems of linear congruences,
Applications of Congruences: Divisibility tests, Round-robin tournament, Special Congruences: Wilson’s
theorem, Fermat’s little theorem, Euler’s theorem
Multiplicative Functions: Euler’s phi-functions
Primitive Roots: order of an integer and primitive roots, Primitive roots for primes, Primality testing using
primitive roots, Perfect Numbers
Cryptology: Affine Ciphers, Hill Ciphers, Exponentiation Ciphers.
Quadratic Residues and Reciprocity: Quadratic Residues, The Legendre Symbols, Quadratic Reciprocity

 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. K. Rosen, Elementary Number Theory and its Applications (6th Edition), Pearson (2010).
2. T. Koshy, Elementary Number Theory with Applications (2 nd Edition),‎ Academic Press (2007)
3. G. Andrews, Number Theory, Dover Publications (1994)
4. O. Ore, Number Theory and Its History, Dover Publications (1988)
5. W. Trappe, L. Washington, Introduction to Cryptography with Coding Theory (2 nd Edition), Pearson
International Press (2006)

Course Code    : AMAT 42443

Title                   : Advanced topics in Geometry

Pre-requisites   :  PMAT 22293

Learning Outcomes:

After the completion of this course unit, the student will be able to:
1. develop an intuitive understanding of the nature of general equation of second degree
2. apply transformations and use symmetry to analyze mathematical situations
3. define conic sections and their properties
4. explain and use various techniques for calculating tangents, normal, pair of tangents, pole and polar of
conic sections
5. compute lines, plane, cone, sphere, ellipsoid and hyperboloid in three dimensions
6. analyze characteristics and properties of two dimensional and three-dimensional geometric shapes and
develop mathematical arguments about geometric relationships
7. demonstrate the knowledge of geometry and its applications in the real world.
8. develop knowledge and intuitive understanding on projective geometry.

 Course Contents:

Analytical Geometry in Two Dimensions: Pairs of straight lines, General equation of second degree,
Translations and rotations of axes, Invariants of transformations
Conic sections: Classifying general equation of second degree into Parabolas, Ellipses, Hyperbolas, eccentricity.
Equation of tangent, Pairs of tangents and chords of contact, Harmonic conjugates, Pole and Polar, Parametric
treatment, Degenerate conic, Properties of conic.
Analytical Geometry in Three Dimension: Coordinate system, Direction cosines and direction ratios of a line,
Angle between two lines, Parallel lines, Perpendicular lines, Plane and Straight line, Shortest distance between
two-non intersecting lines, Skewed lines, General Equation of the second degree, Sphere, Cone, Ellipsoid and
hyperboloid, Tangent plane, Normal, Pole and Polar.
Projective Geometry: Projective Spaces; Definitions and Properties, Hyperplane at Infinity, Projective lines;
Projective Transformation of Projective Line Space, The Cross Ratio, Introduction to projective plane

 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. Chatterjee, D., (2008) Analytical Geometry, Alpha Science International.
2. Thomas, G.B. and Finney, R.L., (2008) Calculus and Analytic Geometry.
3. Maxwell, E.A., (1962) Elementary Coordinate Geometry, Oxford University press.
4. Jain, P.K. & Ahmad, K. (1994) Analytical Geometry of Two Dimensions, Wiley.
5. Kishan, H. (2006) Coordinate Geometry of Two Dimensions. Atlantic Publishers & Dist.
6. Jain, P.K. (2005) A Textbook of Analytical Geometry Of Three Dimensions. New Age International.
7. McCrea, W.H. (2006). Analytical Geometry of Three Dimensions, Dover Publications, INC.
8. Richter-Gebert J. (2011) Perspectives on Projective Geometry: A Guided Tour Through Real and
Complex Geometry, Springer.

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Course Code    : AMAT 42853

Title                   : Tensor Analysis and General Relativity

Pre-requisites   :  PMAT 21553

Learning Outcomes:

At the end of the course the student should be able to demonstrate knowledge of concepts and theorems in tensor algebra, tensor analysis and the formalism of general relativity, and to solve Einstein’s field equations in simple cases, and to apply the solutions in Astrophysics.

 Course Contents:

Covariant and contravariant vectors, Metric tensor, Invariants, Inner products, Differential-forms, Tensor analysis, Covariant differentiation of tensors, Einstein’s field equations, Schwarzchild interior and exterior solutions, Rotating systems, Dragging of inertial frames, Gravitation red shift, Bending of light and gravitational lenses.

 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. Adler R., Bazin M. and Schiffer M., Introduction to General Relativity, McGraw Hill, New York. (1975).
2. Misner, C.W.,Thorne, K.S. & Wheeler, J.A., Gravitation, Princeton University Press, 2017.
3. Hawking S.W. & Ellis G.F.R., The Large Scale Structure of Space-time, Cambridge University Press. (1975).
4. James B. Hartle, Gravity: An Introduction to Einstein's General Relativity, Pearson (2013).

Course Code    : AMAT 42463

Title                   : Advanced Topics in Mechanics

Pre-requisites   :  AMAT 21272

Learning Outcomes:

After the completion of this course unit, the student will be able to:
1. apply Lagrange’s equations to solve motion involving impulse
2. solve dynamic problems using Hamilton’s equations
3. verify canonical variables, determine generating functions and use canonical transformations
4. use Poisson brackets and their properties for finding constants of motion, determining canonical
transformations
5. approximate problems for small oscillations and determine normal frequencies and normal modes
6. solve motion of a heavy symmetric top by Hamiltonian formulation
7. determine stability of fixed points of a nonlinear dynamical system

 Course Contents:

Impulsive motion: Equations of motion, body acted on by a given impulse, body acted on by an impulsive
couple, impact of inelastic bodies, elastic bodies, applications of principle of virtual work, Lagrange's equations
for impulsive motion
Hamiltonian dynamics: Generalized Momentum, Hamilton's equations of motion, Liouville's Theorem, Poisson
brackets, Canonical transformation, Hamilton-Jacobi equation
Small oscillations: Lagrange's method, Normal modes, Roots of the Lagrangian determinant, Oscillations under
constraint, Stationary property of the normal modes
The Motion of a Top: Euler's Equations, Free Tops, Euler's Angles, The Heavy Symmetric Top
Chaos theory: Fixed points and their linear stability, Elements of bifurcation theory, Limit cycles,
Synchronization and phase dynamics, Chaos and strange attractors, Cellular automata

 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. Rao, A.V. (2011). Dynamics of Particles and Rigid Bodies: A Systematic approach, Cambridge
University Press.
2. Chorlton, F. (2 nd Ed., 2019). Textbook of Dynamics, D. Van Nostrand.
3. Gignoux C & Silvestre-Brac, B. (2014). Solved Problems in Lagrangian and Hamiltonian Mechanics,
Springer Netherlands.
4. Kelley, J.D. & Leventhal, J.J. (2016). Problems in Classical and Quantum Mechanics: Extracting the
Underlying Concepts. Springer.
5. Strauch, D. (2009). Classical Mechanics, An Introduction, Springer.
6. Goldstein, H., Poole, C. P. & Safko, J. (3 rd Ed., 2011). Classical Mechanics, Pearson.
7. Mondal, C.R. (2005). Classical Mechanics, Prentice Hall of India.
8. Ramsey, A.S. (1975). Dynamics, Parts I & II, Cambridge University Press.