NATIONAL UNIVERSITY

Syllabus
Department of Mathematics

One Year Preliminary to Master’s Course

Effective from the Session: 2016-2017
 National University
Subject: Mathematics
Syllabus for One-Year Preliminary to Master’s Course
Effective from the Session: 2016-2017

Paper Code Paper Title Credits

413701 Real and Complex Analysis 4
413703 Abstract Algebra 4
413705 Theory of Numbers 4
413707 Partial Differential Equations 4
413709 Hydrodynamics 4
413710 Math Lab 4
(Optional Course) Any Two
413711 Linear Programming 4
413713 Tensor Analysis 4
413715 Mechanics 4
413717 Discrete Mathematics 4
Total = 32
 Detailed Syllabus
Course Code 413701 ------ Credits: 4 Class Hours: 120 hrs.
Course Title: Real and Complex Analysis

Group A
Real Analysis (50 Marks)

Completeness of real number system. Supremum principle. Infimum principle. Dedekind

principle. Their wquivalence and simple consequences.
Countable and uncomuntable sets of real numbers. Open and closed sets of real numbers. Limit
points of a set of real numbers. Bolzano-Weierstrass theorem. Compactness. Heine-Borel
Theorem.
Real sequences: Convergence. Algebraic combinations. Subsequential limits. Limit superior and
limit inferior. Monotone sequence. Cauchy sequence.
Infinite series of real numbers (tests of convergence, alternating series, absolute convergence).
Real functions: Limits and continuity. Boundedness theorem. Intermediate value theorm.
Uniform continuity.
Differentiation: Basic properties. Rolle’s theorem. Mean-value theorems. Taylor’s theorem.
L’Hospital rules.
Integration: Riemann integration (Riemann sum, integrability conditions, properties of
integrals, fundamental theorem of calculus)
Sequence and series of real functions: Point-wise convergence. Uniform convergence.
Weierstrass M-test. Intergation and differentiation of series.

Group B
Complex Analysis (50 Marks)

Open and closed sets of complex numbers. Limit points. Convergence of sequences of complex
numbers.
Functions of complex variable: Limits. Continuity and differentiability. Analytic function. C-R
equations. Harmonic conjugates of elementary complex functions. Complex integration.
Cauchy’s integral theorem. Cauchy-Goursat theorem. Cauchy’s integral formulae. Morera’s
theorem. Liouville’s theorem. Fundamental theorem of Algebra. Toylor;s and laurent’s series.
Singularities and their classifications. Residues, Cauchy’s residue theorem. Evaluation of real
integrals by contour integrations. Evaluation of Bilinear transformation.

Books Recommended:
1 Kenneth A. Ross : Elementary Analysis. The Theory of
Calculus.
2 Robert G. Bartle, Douald R. Sherbert : Introduction to Real analysis
3 Walter Rudin : Principle of Mathematical Analysis
4 Ruel V. Charchill : Complex Variable and Application
5 Schaum’s Outline Series : Complex Variables
 Course Code 413703 ------ Credits: 4 Class Hours: 120 hrs.
Course Title: Abstract Algebra

Part-I: Group Theory

1. Groups and subgroups. Groups of symmetry. Permutation groups. The symmetric group
on n letters. Cyclic groups.
2. Left and right congruence module a sub group. Cosecs. Lagrange’s theorem, Product of
cosecs. Forbenious’s counting formula.
3. Normal subgroups, quotient (factor) groups. Homomorphisms and automorphisms The
isomorphism theorems. Conjugacy: the class equation. Direct product. Groups of small
orders.

Part-II: Rings & Modules

1. Rings, ideals and quotient rings, prime and maximal ideals.
2. Intergral domain. Fiels of fractions.
3. Principal ideal domains. Euclideam domains. Unique factorization domains.
4. Polynomial rings. Primitive polynomials (Gauss’s theorem). Eisenstein’s irreducibility
criterion.
5. Characeristic of ring or integral domain. Prime Fields; structure of prime fields.

References
1. P.B Bhattarcharya, S.K Jain & S.R Nagpaul. Basic Abstract Algebra.
2. W.K Nicholson, Introduction to Abstract Algebra.
3. J.B Fraleigh, Introduction to Abstract Algebra
4. M.Artin, Algebra

Course Code 413705 ------ Credits: 4 Class Hours:120 hrs.

Course Title: Theory of Numbers

Arithmetic in Z. Euclidean Algorithm. Continued fractions.

The ring Z and its group of units. Chinese remainder theorem. Linear diphantine quuations
Arthmetical functions. Diriehlet convolution. Multiplicative function.
Representation by sum of two and four squares.
Arithmetic of quridratic fields. Euelidean quadratic firlds.

Book Recommended:
1. Nivan. H.S Zuckerman: An Introduction to the Theory of Numbers.
2. G.H Hardy & E.M Wright. An Introduction to Theory of Numbers.
3. I.S Niven and H.S Zuckerman, An introduction to Theory of Numbers.
4. W.J. LeVeque, Fundamentals of Number theory
5. Fazlur Rahman- Theory of Numbers.
 Course Code 413707 ------ Credits: 4 Class Hours: 120 hrs.
Course Title: Partial Differential Equations

First order equations: Complete integral. General solution. Cauchy problems. Method of
characteristics for linear and quasilinear equations. Charpit's method for finding complete
integrals. Methods for finding general solutions.
Second order equations: Classifications. Reduction to canonical forms. Characteristic curves.
Boundary value problems: Boundary value problems related to linear equations. Applictions of
Fourier methods (Coordinate systems and separability, Homogeneous equations,
Nonhomogeneous boundary conditions, Inhomogeneous equations.)
Problems involving symmetry: Problems involving cylindrical and spherical symmetry,
Boundary value problems involving special functions.
Transform methods for boundary value problems: applications of the Laplace transforms;
applications of Fourier sine and cosine transforms. Inhomogeneous equations.

Books Recommended :

1. J. N. Sneddon- Elements of Partial Differential Equations.

2. J. M. Kar – Partial Differential Equations.
3. B. Epsteim : Partial Differential Equations
4. Awal, Hafiz, Maidul – Partial Differentil Equation..

Course Code 413709 ------ Credits: 4 Class Hours: 120 hrs.

Course Title: Hydrodynamics

Velocity and acceleration of fluid particles. Relation between local and individual rates. Steady
and unsteady flows. Uniform and nonuniform flows. Stream lines. Path lines. Vortex lines.
Velocity potential. Bernoulli’s equations and its application.

Rotational and irrotational flows. Equation of continuity. Equation of continuity in spherical and
cylindrical polar coordinates. Boundary surfaces.

Euler’s equation of motion. Conservative field of force. Lamb’s hydrodynamical equations of

Motion. Motion under conservative body force. Vorticity equation(Helmholtz’s vorticity
equation)

Motion in two-dimensions. Stream function. Physical meaning of stream function. Velocity in

polar- Coordinates. Relation between stream function and velocity.

Circulation and vorticity. Relation between circulation and vorticity. Kelvin’s circulation
theorem, Kelvin’s minimum energy theorem. Generalized Joukowki’s transformation. Elliptic
coordinates and its application.
The circle’s theorem. Motion of a circular cylinder. Pressure at points on a circular cylinder.
Application of circle theorem. Blasius theorem.

Sources, sinks and doublets. Complex potential and complex velocity. Stagnation points.
Complex potential due to a source and a doublet, Image in two and three dimensions. Stoke’s
stream function.

Vortex motion. Complex potential due to vortex motion.

Books Recommended :

1. L. M. Milne, Thomson (CBE) - Theoretical Hydrodynamics.

2. F. Chorlton - A Text Book of Fluid dynamics.
3. P. P. Gupta – Hydrodynamics

Course Code 413710 ------ Credits: 4 -----

Course Title: Math Lab (Practical)

Problem solving in concurrent courses (Calculus, Complex Analysis, Numerical Analysis)

Lab Assignments: There shall be at least 15 lab assignments.

Evaluation: Internal Assessment (Laboratory works): 30 marks

Final Exam (Lab 3 hours): 50 marks.

Viva-Voce: 20 Marks

(Optional Course) Any Two

Course Code 413711 ------ Credits: 4 Class Hours: 120 hrs.

Course Title: Linear Programming
1. Convex sets and related theorems.
2. Introduction to linear programming. Feasibility and optimality.
3. Formulation of linear programming problems.
4. Graphical solutions.
5. Simplex method. Two phase and Big-M simplex methods.
6. Duality of linear programming and related theorems. Dual simplex method.
7. Sensitivity analysis in linear programming.
8. Transportation and assignment problems.
Book Recommended:

1. F. S. Hiller and G. T. Lieberman : Linear Programming.

2. P. R. Thie : Introduction to Linear Programming and Game theory.
3. N. S. Kambu : Mathematical Programming Techniques.
4. Hamdy A. Taha : Operations Research.

Course Code 413713 ------ Credits: 4 Class Hours: 120 hrs.

Course Title: Tensor Analysis

n-dimensional spaces. Summation convention. Transformation of coordinates. Transformation

by invariance. Transformation by covariance and contravariance.
The tensor concept. Contravariant and covariant tensors. Mixed tensors. Algebra of tensor
concept. Quotient laws Symmetric and skew-symmetric tensors. Relative tensors. Metric tensors.
rundamental and associated tensors. Christoffel’s symbols and their transformations. Covariant
differentiation of tensors. Formulas of covariant differentiation.
Ricci’s theorem. Riemann-Christofeel tensor. Ricci tensor. Bianchi identities. Einstein tensor.

Book Recommended:
1. I.S. Sokolnikoff : Tensor Analysis
2. Schaum’s Outline Series : Vector and Tensor Analysis.
3. B. Spain : Tensor Analysis
4. C. E. Weatherburn : An Introduction to Rimannian
Geometry and the Tensor Calculus.

Course Code 413715 ------ Credits: 4 Class Hours: 120 hrs.

Course Title: Mechanics

Group A. Statistics:
General conditions of equilibrium. Principles of virtual works. Center of gravity.
Stable and unstable equilibrium. Theory of strings and catenary.

Group B. Dynamics:
Motion in a straight line. Simple harmonic motion.
Motion in plane. Motion under central forces. Motion in space.
Motion of rigid bodies. Moments and products of inertia.
De Alembert’s Principle. Motion about fixed axes.
Motion of rigid bodies in two dimensions.
Book Recommended:
1. S.L. Loney : An elementary treatise on Statics.
2. S.L. Loney : An elementary treatise on the Dynamic of
a particle & of a Rigid Body.

Course Code 413717 ------ Credits: 4 Class Hours: 120 hrs.

Course Title: Discrete Mathematics

Mathematical reasoning: Inference and fallacies. Methods of proof. Recursive definitions.

Program verification.
Combinatorics: Counting- principles. Inclusion-exclusion principle. Pigeonhole principle.
Generating functions. Recurrence relations. Applications to computer operations.
Algorithms and their efficiency : Searching algorithms.' Sorting algorithms. Bin packing
algorithms. Algorithms on integer operations. Recursive algorithms.
Graphs: Structure and symmetry of graphs. Trees and connectivity. Eulerian and
Hamiltonian graphs Diagraphs. Directed graphs. Planar graphs.
Algorithms on graphs : Introduction to graphs, paths and trees. Shortest path problems
(Dijkstra's algorithm, Floyd-Warshall algorithm and their comparisons). Spanning tree problems.
(Kruskal's greedy algorithm, Prim's greedy algorithm and their comparisons).

Network flows: Flows and cuts. Flow augmentation algorithms. Application of max-flow min-
cut theorem.

Books Recommended :
1. Schaum's Outline Series – Discrete Mathematics.
2. Kenth H.Rosen – Discrete Mathematics and it's Application.
3. Fazlur Rahman - Discrete Mathematics.