Department of Mathematical Sciences

Bodoland University, Kokrajhar

(CBCS Syllabus)
2019-2020

Syllabus for M.Sc./M.A. in Mathematics

 COURSE STRUCTURE

Department: Mathematical Sciences, BU.

Programme: MA/MSc in Mathematics

L: Lectures, T: Tutorials, P: Practical , CH:Contact Hours (all per week) CR: Credit
C: Core Paper DSE: Discipline Specific Elective OE: Open Elective

SEMESTER I

Sl Paper Paper Paper Title L T P CH CR Full Marks: Remark

No. Code Type (Theory +
Practical+ Internal
Assessments)
01 MAT 101 C Algebra 3 1 0 4 4 100(80+ 0+20)
02 MAT 102 C Differential Equations 3 1 0 4 4 100(80+0+20)
03 MAT 103 C Mechanics 3 1 0 4 4 100(80+0+20)
04 MAT 104 C Real Analysis 3 1 0 4 4 100(80+0+20)
05 MAT 105 C Tensor Analysis 3 1 0 4 4 100(80+0+20)
06 MAT 106 OE Open Elective I 1 1 0 2 2 50(40+0+10)
Total Credit 22

Choices for Open Elective I (MAT 106)

1. Fundamentals of Mathematics. 2. History of Mathematics

SEMESTER II

Sl Paper Code Paper Paper Title L T P CH CR Full Marks: Remark

No. Type (Theory +
Practical+ Internal
Assessments)
01 MAT 201 C Complex Analysis 3 1 4 0 4 100(80+ 0+20)
02 MAT 202 C Continuum Mechanics 3 1 4 0 4 100(80+0+20)
03 MAT 203 C Functional Analysis 3 1 4 0 4 100(80+0+20)
04 MAT 204 C General Topology 3 1 4 0 4 100(80+0+20)
05 MAT 205 C Mathematical Methods 3 1 4 0 4 100(80+0+20)
06 MAT 206 OE Open Elective II 1 1 0 2 2 50(40+0+10)
Total Credit 22

Choices for Open Elective II (MAT 206)

1. Applications of Mathematics in Real Life. 2. Mathematics Education

 SEMESTER III

Sl Paper Paper Paper Title L T P CH CR Full Marks: Remark

No. Code Type (Theory +
Practical+ Internal
Assessments)
01 MAT 301 C Fuzzy Set Theory 3 1 4 0 4 100(80+ 0+20)
02 MAT 302 C Graph Theory 3 1 4 0 4 100(80+0+20)
03 MAT 303 C Number Theory 3 1 4 0 4 100(80+0+20)
04 MAT304 C Numerical Analysis 3 1 4 0 4 100(80+0+20)
05 MAT 305 C Special Theory of Relativity 3 1 0 4 4 100(80+0+20)
06 MAT 306 C Dissertation 6 6 100(80+0+20)
Total Credit 26

SEMESTER IV

Sl Paper Code Paper Paper Title L T P CH CR Full Marks: Remark

No. Type (Theory +
Practical+ Internal
Assessments)
01 MAT 401 DSE Elective I 3 1 0 4 4 100(80+ 0+20)
02 MAT 402 DSE Elective II 3 1 0 4 4 100(80+0+20)
03 MAT 403 DSE Elective III 3 1 0 4 4 100(80+0+20)
04 MAT404 DSE Elective IV 3 1 0 4 4 100(80+0+20)
05 MAT 405 DSE Dissertation 6 6 100(80+0+20)
Total Credit 22 100(80+0+20)
Total 94 credits
A. Choices for Elective I (MAT 401)(choose anyone)

1. Advanced Topology 2. Fluid Dynamics 3. Operator Theory

B. Choices for Elective II (MAT 402) (choose anyone)

1. Advanced Functional Analysis 2. Dynamical System 3. Category Theory

C. Choices for Elective III (MAT403) (choose anyone)

1. Fuzzy Logic and Fuzzy Control System 2. Operations Research 3. Relativity and Cosmology

D. Choices for Elective IV (MAT404)(choose anyone)

1. Advanced Graph Theory 2. Advanced Numerical Analysis 3. Advanced Number Theory

 Detail Syllabus

SEMESTER- I

Paper Code: MAT 101

Paper Title: Algebra

Theory Marks: 80
Internal Marks: 20

Unit-I: Homeomorphism and Automorphism of Groups, Permutation Groups, Cyclic Groups, Sylow’s Theorem
with applications.
Marks: 20
Unit-II: Ring Homorphisms, Ideals, maximal and Prime ideals, Ring with zero divisors and Ring without Zero
divisor, Definitions and Examples of Integral Domains and Fields. Ring Homomorphisms.
Marks: 20

Unit-III: Polynomial Rings, Factorization of Polynomials, Divisibility in Integral Domains. Prime and irreducible
elements , Principal Ideal Domains, Euclidean Domains.
Marks: 20
Unit-IV: Construction of fields, Prime field, Extension of fields. Marks: 20

Reference Books:
1. Modern Algebra by Surjeet Singh and Qasi Zameeruddin, Vikas Publishing House (Second Edition), New Delhi,
1975.
2. A First Course in Abstract Algebra by John B. Fraleigh, Published by Pearson Education(Singapore) Pte. Ltd.
3.Topics in Algebra, Second edition (Wiley Eastern Ltd.) by I.N. Herstein .

4.Joseph A. Gallian, Contemporary Abstract Algebra (Fourth Ed.), Narosa,1999.

5. P. B. Bhattacharya, S. K. Jain and S. R. Nagpaul, Basic Abstract Algebra (Second Ed.), Cambridge Univ. Press
(Indian Ed.1995).
 Paper Code: MAT 102

Paper Title: Differential Equations

Theory Marks: 80
Internal Marks: 20

Unit-I: Solution of 2nd order differential equations with variable coefficients, General theory of homogeneous and
non-homogeneous linear ODEs, Picard’s method of successive approximation, problems of Existence and
Uniqueness. Marks-20
nd
Unit-II: Method of series solution of 2 order differential equations with reference to Legendre, Bessel and Gauss,
Orthogonal set of functions. Marks: 20
Unit-III : Lagrange and Charpit methods for solving first order PDEs, Cauchy problem for first order PDEs.
Classification of second order PDEs, Partial differential equation of second order, linear equations with constant
coefficient. Marks: 20
Unit-IV: Laplace’s equation: Elementary solution of Laplace’s equation, Solutions in various coordinates systems.
Greens function for Laplace’s equation. The wave equation: the first Cauchy problem, General solutions of the
wave equation. Green’s function for the wave equation. Marks: 20
Reference Books:
1.Advanched differential Equations : M. D. Raisinghania
2.Ordinary differential Equations: M. D. Raisinghania
3.Elements of partial differential equations: Ian N Snedden, McGraw Hill.
4.Theory and problems of differential equations: Frank AyresJr. Schaum’s Outline Series. McGraw Hill.
 Paper Code: MAT 103

Paper Title: Mechanics

Theory Marks: 80
Internal Marks: 20

Unit-I: Variational principle and Lagranges Equations : Hamilton’s principle, Derivation of Lagrange’s Equations
from Hamilton’s Principle.
Marks: 20
Unit-II: Generalised coordinates, Halonomic & Non-holonimic systems, Scleronomic and Rheonomic systems,
Generalized potential, Lagranges Equations of first kind and second kind. Marks: 20

Unit-III: Canonical transformations: Introduction, Lagendre transformations, generating function and canonical
transformations. Conditions for a transformation to be canonical. Poisson’s brackets & Lagrange brackets their
properties.
Marks: 20

Unit-IV: Hamilton-Jacobi Equation, Hamilton-Jacobi Equation for Hamilton’s characteristic function function,
Separation of variables in the Hamilton-Jacobi equation.
Marks: 20
.
Reference Books:
1. Introduction to Classical Mecanics : R..Takwale & P.S. Puranik Tata Mc Graw Hill, 1983
2. Classical Mechanics : By H.Goldstein, Second edition, Narosa Publishing House, New Delhi.
3. Classical Mechanics : By N.C.Rana & P.S.Joag, Tata Mc Graw Hill,1991.
4. A.S.Ramsey Dynamics Part-II, the English Language Book Society and Cambridge University Press.
 Paper Code: MAT104

Paper Title: Real Analysis

Theory Marks: 80
Internal Marks: 20

Unit-I: Elements of set theory, finite and infinite sets, cardinal numbers, countable and uncountable sets, Axiom of
choice, Real number system.

Unit-II: Sequences and series of functions, point wise and uniform convergence, Cauchy criterion for uniform
convergence, relation of uniform convergence with continuity, differentiation and integration, Weierstrass
approximation theorem. Fourier Series, Dirichlet criterion for convergence of Fourier series and its application into
even and odd functions, function in any interval [a, b].

Unit-III: Functions of several variables, differentiation, implicit function theorem, inverse function theorem,
maxima and minima.

Unit-IV: Functions of Bounded Variation and their properties, Differentiation of a function of bounded variation,
Absolutely Continuous Function, Representation of an absolutely continuous function by an integral. Riemann-
Stieltjes integrals, properties, mean value theorems, the fundamental theorem of Calculus.

Unit-V: Metric spaces, convergence, continuity, compactness, connectedness, completeness, Heine-Borel theorem,
Intermediate value theorem, Baire Category theorem.

Reference Books:
1. Apostol, T. M., Mathematical Analysis, Narosa Publishing House, 1985.
2. Rudin, W., Principles of Mathematical Analysis, McGraw Hill, 1982.
3. Goldberg, R. R, Methods of real analysis, Oxford & IBH, 1970.
4. Simmons, G. F., Introduction to Topology and Modern Analysis, Tata McGraw Hill Book
Co. Ltd., 1963.
5. Mallik, S. C. and Arora S. Mathematical Analysis, New Age International (P) Limited, New Delhi, 1992
 Paper Code: MAT 105

Paper Title: Tensor Analysis

Theory Marks: 80
Internal Marks: 20

Unit 1: Curvilinear coordinates; Transformation of coordinates; Summation Convention; Dummy Suffix; Real
Suffix; Covariant and Contravariant vectors; Tensors of Second Order; Mixed Tensors; Kronecker Delta; Algebra of
Tensors; Symmetric and Skew-Symmetric tensors; Outer multiplication, Contraction and Inner Multiplication,
Quotient Law of Tensors, Reciprocal Symmetric Tensor. 20 marks

Unit 2: Christoffel’s symbols; Transformation of Christoffel’s symbols; Differential equation of a Geodesic;

Covariant differentiation of vectors; Covariant differentiation of tensors; Intrinsic derivative of a tensor; Laws of
covariant differentiation of tensors. 20 marks

Unit 3: The metric tensor; Riemannian metric; Riemannian space; Geodesic coordinates; Natural coordinates;
Riemannian Christoffel’s tensor; Curvature of a curve; First curvature; Covariant curvature tensor; Properties of
covariant curvature tensor. 20 marks

Unit 4: Parallelism of vector of constant magnitude; Parallelism for vector of variable magnitude along a curve;
Tensor differentiation. 20 marks

Reference Books :

1. Tensor Calculus and Riemannian Geometry: D. C. Agarwal.

2. An Introduction to Riemannian Geometry and Tensor Calculus: Cambridge University Press: C. E. Weatherburn
(1950).
3. Tensor Analysis : De Gruyter : Heinz Schade, Klaus Neemann, Andrea Dziubek, Edmond Rusjan (2018).
 Paper Code: MAT 106 (Open Elective)
Theory Marks: 40
Internal Marks: 10

Options:

1. Fundamental of Mathematics:
Unit I: Number System, Properties of Real Numbers, Sequences and Series, Infinite sequences and series.

Unit II: Sets and Their Properties, Cardinality, countable, and uncountable sets, more about relation and functions,
injective

Unit III: Data collection, types of data and its classification of data by type, organize data into tables, and
summarize data graphically, Normal Approximation for Data, central tendency, Mean, Median, Mode,

Unit IV: Sampling, Sample Surveys, Chance Errors in Sampling, Accuracy of Percentages, Accuracy of Averages.

Reference Books:
1. D. Freedman, R. Pisani, R. Purves, Statistics, 4th edition. W. W. Norton & Company (2007). ISBN: 978-
0393-92972-0.
2. Mallik, S. C. and Arora S. Mathematical Analysis, New Age International (P) Limited, New Delhi, 1992

2. History of Mathematics:
Unit I: Egypt and Babylon - Greek mathematics (Euclid, Archimedes, Apollonius, Ptolemy).

Unit II: Mathematics in the Islamic World - The transmission of the mathematics of antiquity to medieval Europe -
Algebra, trigonometry and arithmetic in the Renaissance - Analytic geometry in the seventeenth century (Descartes,
Fermat)

Unit III: The beginnings of calculus - The calculus of Newton and Leibniz - Newton’s Principia - Euler - Gauss

Unit IV: Geometry in the oulvasatras, the origins of zero (which can be traced to ideas of lopa in Paoini's
grammar), a cross-cultural view of the development of negative numbers (from Brahmagupta (c. 628 CE) to John
Wallis (1685 CE).

Recommended Texts

The course reader is Fauvel & Gray, History of Mathematics, a Reader (McMillan, C. £25), which is essential. Short
surveys like D J Struik’s Concise History (4th edn only) indicate a framework. General histories like those of Boyer
& Merzbach, and V. Katz, are useful adjuncts to the course. S. Hollingdale’s Makers of Mathematics is a readable
and informative introduction.

Books:

1. A short course in the History of Mathematics : W.W.Rouse Ball.

2. A History of Mathematics : Carl B.Boyer.
3. A History of Mathematics : Florian Cajori.
4. The History of Mathematics: A Very Short Introduction : Jacqueline Stedall
5. Studies in the History of Indian Mathematics (Culture and History of Indian Mathematics): C. S. Seshadri.