DISCRETE MATHEMATICS BHARATHIDASAN UNIVERSITY, TIRUCHIRAPPALLI M.Sc. MATHEMATICS SYLLABUS (ANNUAL PATTERN)

BHARATHIDASAN UNIVERSITY, TIRUCHIRAPPALLI – 620 024
M.Sc. MATHEMATICS SYLLABUS (ANNUAL PATTERN)
(FOR DISTANCE EDUCATION CANDIDATES ONLY)
(For the candidates admitted from the academic year 2006-2007 batch onwards)
PAPER VII – DISCRETE MATHEMATICS

Unit I
Mathematical Logic: Basic Notation, Connectives, Normal forms.

Unit II
Inference Theory: The inference theory for the statement calculus, Predicate Calculus, Inference ‘Theory of the Predicate Calculus.

Unit III
Algebraic Structures: Algebraic Systems, Semi groups and Monoids, Grammars and languages, Groups.

Unit IV
Lattices and Boolean Algebra: Lattices as partially ordered sets, Boolean Algeabra, Boolean Functions.

Unit V
Basic concepts of Graph Theory, Storage representation and Manipulation of Graphics, Simple Procedence Grammars, Fault Detection in Combinational Switching Circuits.

Text Book:
1. J.P. Tremblay and R. Manohar, “Discrete Mathematical Structures with Applications to Computer Science”, Tata McGraw Hill, Edition 1997, New Delhi.

References

1. Kenneth H.Rosen, “Discrete Mathematics and its Applications”, McGraw Hill Book Company, 1999, New Delhi.
2. Kolman, Busby and Ross, “Discrete Mathematical Structures”, Prenctice Hall of India, Fourth Edition 2002, New Delhi.

Unit I : Sections 1 – 1 to 1 – 3.5
Unit II : Sections 1 – 4 to 1 – 6.5
Unit III : Sections 3 - 1to 3 – 3.3 and 3 – 5 to 3 – 5.5
Unit IV : Sections 4 – 1 to 4 – 3.2
Unit V : Sections 5 – 1 to 5 – 4.4

DISCRETE MATHEMATICS BHARATHIDASAN UNIVERSITY, TIRUCHIRAPPALLI M.Sc. MATHEMATICS SYLLABUS (ANNUAL PATTERN)

BHARATHIDASAN UNIVERSITY, TIRUCHIRAPPALLI – 620 024
M.Sc. MATHEMATICS SYLLABUS (ANNUAL PATTERN)
(FOR DISTANCE EDUCATION CANDIDATES ONLY)
(For the candidates admitted from the academic year 2006-2007 batch onwards)
PAPER VII – DISCRETE MATHEMATICS

Unit I
Mathematical Logic: Basic Notation, Connectives, Normal forms.

Unit II
Inference Theory: The inference theory for the statement calculus, Predicate Calculus, Inference ‘Theory of the Predicate Calculus.

Unit III
Algebraic Structures: Algebraic Systems, Semi groups and Monoids, Grammars and languages, Groups.

Unit IV
Lattices and Boolean Algebra: Lattices as partially ordered sets, Boolean Algeabra, Boolean Functions.

Unit V
Basic concepts of Graph Theory, Storage representation and Manipulation of Graphics, Simple Procedence Grammars, Fault Detection in Combinational Switching Circuits.

Text Book:
1. J.P. Tremblay and R. Manohar, “Discrete Mathematical Structures with Applications to Computer Science”, Tata McGraw Hill, Edition 1997, New Delhi.

References

1. Kenneth H.Rosen, “Discrete Mathematics and its Applications”, McGraw Hill Book Company, 1999, New Delhi.
2. Kolman, Busby and Ross, “Discrete Mathematical Structures”, Prenctice Hall of India, Fourth Edition 2002, New Delhi.

Unit I : Sections 1 – 1 to 1 – 3.5
Unit II : Sections 1 – 4 to 1 – 6.5
Unit III : Sections 3 - 1to 3 – 3.3 and 3 – 5 to 3 – 5.5
Unit IV : Sections 4 – 1 to 4 – 3.2
Unit V : Sections 5 – 1 to 5 – 4.4

TOPOLOGY AND FUNCTIONAL ANALYSIS BHARATHIDASAN UNIVERSITY, TIRUCHIRAPPALLI – 620 024 M.Sc. MATHEMATICS SYLLABUS (ANNUAL PATTERN)

BHARATHIDASAN UNIVERSITY, TIRUCHIRAPPALLI – 620 024
M.Sc. MATHEMATICS SYLLABUS (ANNUAL PATTERN)
(FOR DISTANCE EDUCATION CANDIDATES ONLY)
(For the candidates admitted from the academic year 2006-2007 batch onwards)
PAPER VI – TOPOLOGY AND FUNCTIONAL ANALYSIS

Unit I
Topological spaces and continuous functions – Product topology – metric topology – The metric topology (continued) – connectedness.

Unit II
Compactness – Count ability axioms and Separation axioms.

Unit III
The Tychonoff theorem – Completer metric spaces and Function spaces

Unit IV
Banach spaces

Unit V
Hilbert spaces – finite dimensional spectral theory.

TEXT BOOK:

1. James R. Munkres, “Topology – A First Course”, PHI (Second edition)
Unit 1: Chapter 2 (Sec 2.1 to 2.10)
Chapter 3 (Sec 3.1 to 3.4)
Unit II: Chapter 3 (Sec 3.5 to 3.7)
Chapter 4
Unit III: Chapter 5
Chapter 7 (Sec 7.1 to 7.3)
Functional Analysis

Text Book
1. G.F. Simmons, “Introduction to Topology and Modern Analysis”, Mc-Graw Hill. 1963
Unit IV: Chapter 9.
Unit V : Chapter 10 and 11.

References:
1. J. Dugundji, “Topology”, PHI, New Delhi, 1975
2. Goffman and Pedrick, “First course in Functional Analysis”, PHI
3. B.V. Limays, “Functional Analysis”, Wiley Eastern.

TOPOLOGY AND FUNCTIONAL ANALYSIS BHARATHIDASAN UNIVERSITY, TIRUCHIRAPPALLI – 620 024 M.Sc. MATHEMATICS SYLLABUS (ANNUAL PATTERN)

BHARATHIDASAN UNIVERSITY, TIRUCHIRAPPALLI – 620 024
M.Sc. MATHEMATICS SYLLABUS (ANNUAL PATTERN)
(FOR DISTANCE EDUCATION CANDIDATES ONLY)
(For the candidates admitted from the academic year 2006-2007 batch onwards)
PAPER VI – TOPOLOGY AND FUNCTIONAL ANALYSIS

Unit I
Topological spaces and continuous functions – Product topology – metric topology – The metric topology (continued) – connectedness.

Unit II
Compactness – Count ability axioms and Separation axioms.

Unit III
The Tychonoff theorem – Completer metric spaces and Function spaces

Unit IV
Banach spaces

Unit V
Hilbert spaces – finite dimensional spectral theory.

TEXT BOOK:

1. James R. Munkres, “Topology – A First Course”, PHI (Second edition)
Unit 1: Chapter 2 (Sec 2.1 to 2.10)
Chapter 3 (Sec 3.1 to 3.4)
Unit II: Chapter 3 (Sec 3.5 to 3.7)
Chapter 4
Unit III: Chapter 5
Chapter 7 (Sec 7.1 to 7.3)
Functional Analysis

Text Book
1. G.F. Simmons, “Introduction to Topology and Modern Analysis”, Mc-Graw Hill. 1963
Unit IV: Chapter 9.
Unit V : Chapter 10 and 11.

References:
1. J. Dugundji, “Topology”, PHI, New Delhi, 1975
2. Goffman and Pedrick, “First course in Functional Analysis”, PHI
3. B.V. Limays, “Functional Analysis”, Wiley Eastern.

PAPER IV - CLASSICAL AND FLUID MECHANICS BHARATHIDASAN UNIVERSITY, TIRUCHIRAPPALLI – 620 024 M.Sc. MATHEMATICS SYLLABUS (ANNUAL PATTERN) SYLLABUS

BHARATHIDASAN UNIVERSITY, TIRUCHIRAPPALLI – 620 024
M.Sc. MATHEMATICS SYLLABUS (ANNUAL PATTERN)
(FOR DISTANCE EDUCATION CANDIDATES ONLY)
(For the candidates admitted from the academic year 2006-2007 batch onwards)
PAPER IV - CLASSICAL AND FLUID MECHANICS

Unit I
Introductory concepts – the mechanical systems – Generalized Coordinates – Constraints – Virtual work – Energy and momentum – Lagrange’s equation – Integrals of the motion – Small oscillation.

Unit II
Special applications of Lagrange’s equation – Rayleigh’s dissipation function – impulsive motion – Gyroscopic systems – Velocity dependent potentials – Hamilton’s equations – Hamilton’s principles – Other variational principles – Phase space.

Unit III
Real fluids and ideal fluids – Velocity of a fluid at a point – Stream lines and path lines – Steady and unsteady flows – the velocity potential – The vorticity vector – Local and particle rate of change – The equation of continuity – worked examples – Acceleration of a fluid – Pressure at a point in a fluid at rest – Pressures at a point in a moving fluid – Conditions at a boundary of the inviscid invisible fluids – Euler’s equation of motions – Bernoulli’s equation – worked examples.

Unit IV
Some flows involving Aerial symmetry – Some special two dimensional flows – Impulsive motion – some three dimensional flows – Sources, sinks and doublets – Images in a rigid infinite plane – Axi symmetric flows – Stokes streams functions.

Unit V
Some two-dimensional flows – Meaning of a two dimensional flow – Use of cylindrical polar coordinates – the stream function – The complex potential for two dimensional irrotational, incompressible flow – complex velocity potentials for standard two dimensional flows – Some worked examples – the Milne’s Thomson circle theorem and applications – the theorem of Blasius.

Text Books:
1. Donald T. Greenwood, “Classical Dynamics” PHI Pvt. Ltd., New Delhi, 1985
Unit I: Chapter 1 (1.1 to 1.5) Chapter 2 (2.1 to 2.4)
Unit II: Chapter 3 (3.1 to 3.4) Chapter 4 (4.1 to 4.4)
2. F. Chorton,”Text Book of Fluid Dynamics”, CBS Publications, New Delhi, 1985
Unit III: Chapter 2 (2.1 to 2.9) Chapter 3 (3.1 to 3.6)
Unit IV: Chapter 3 (3.9 to 3.11) Chapter 4 (4.1, 4.2, 4.3, 4.5)
Unit V: Chapter 5 (5.1 to 5.6, 5.8, 5.9)

PAPER IV - CLASSICAL AND FLUID MECHANICS BHARATHIDASAN UNIVERSITY, TIRUCHIRAPPALLI – 620 024 M.Sc. MATHEMATICS SYLLABUS (ANNUAL PATTERN) SYLLABUS

BHARATHIDASAN UNIVERSITY, TIRUCHIRAPPALLI – 620 024
M.Sc. MATHEMATICS SYLLABUS (ANNUAL PATTERN)
(FOR DISTANCE EDUCATION CANDIDATES ONLY)
(For the candidates admitted from the academic year 2006-2007 batch onwards)
PAPER IV - CLASSICAL AND FLUID MECHANICS

Unit I
Introductory concepts – the mechanical systems – Generalized Coordinates – Constraints – Virtual work – Energy and momentum – Lagrange’s equation – Integrals of the motion – Small oscillation.

Unit II
Special applications of Lagrange’s equation – Rayleigh’s dissipation function – impulsive motion – Gyroscopic systems – Velocity dependent potentials – Hamilton’s equations – Hamilton’s principles – Other variational principles – Phase space.

Unit III
Real fluids and ideal fluids – Velocity of a fluid at a point – Stream lines and path lines – Steady and unsteady flows – the velocity potential – The vorticity vector – Local and particle rate of change – The equation of continuity – worked examples – Acceleration of a fluid – Pressure at a point in a fluid at rest – Pressures at a point in a moving fluid – Conditions at a boundary of the inviscid invisible fluids – Euler’s equation of motions – Bernoulli’s equation – worked examples.

Unit IV
Some flows involving Aerial symmetry – Some special two dimensional flows – Impulsive motion – some three dimensional flows – Sources, sinks and doublets – Images in a rigid infinite plane – Axi symmetric flows – Stokes streams functions.

Unit V
Some two-dimensional flows – Meaning of a two dimensional flow – Use of cylindrical polar coordinates – the stream function – The complex potential for two dimensional irrotational, incompressible flow – complex velocity potentials for standard two dimensional flows – Some worked examples – the Milne’s Thomson circle theorem and applications – the theorem of Blasius.

Text Books:
1. Donald T. Greenwood, “Classical Dynamics” PHI Pvt. Ltd., New Delhi, 1985
Unit I: Chapter 1 (1.1 to 1.5) Chapter 2 (2.1 to 2.4)
Unit II: Chapter 3 (3.1 to 3.4) Chapter 4 (4.1 to 4.4)
2. F. Chorton,”Text Book of Fluid Dynamics”, CBS Publications, New Delhi, 1985
Unit III: Chapter 2 (2.1 to 2.9) Chapter 3 (3.1 to 3.6)
Unit IV: Chapter 3 (3.9 to 3.11) Chapter 4 (4.1, 4.2, 4.3, 4.5)
Unit V: Chapter 5 (5.1 to 5.6, 5.8, 5.9)

COMPLEX ANALYSIS BHARATHIDASAN UNIVERSITY, TIRUCHIRAPPALLI – 620 024 M.Sc. MATHEMATICS SYLLABUS (ANNUAL PATTERN) SYLLABUS

BHARATHIDASAN UNIVERSITY, TIRUCHIRAPPALLI – 620 024
M.Sc. MATHEMATICS SYLLABUS (ANNUAL PATTERN)
(FOR DISTANCE EDUCATION CANDIDATES ONLY)
(For the candidates admitted from the academic year 2006-2007 batch onwards)
PAPER III - COMPLEX ANALYSIS

Unit I
Analytic functions as mappings: Elementary point set topology – Conformality – Linear transformations – Elementary conformal mappings.

Unit II
Complex Integration: Fundamental theorems – Cauchy’s integral formula – Local properties of analytic functions – General form of Cauchy’s theorem.

Unit III
Complex Integration: Calculus of residues – Harmonic functions

Unit IV
Series and Product Development: Power series expansion – Partial fractions and factorization – Entire functions – Riemann zeta function – Normal; families.

Unit V
Riemann Mapping theorem – Elliptic functions: Simply periodic functions – Doubly periodic functions – the Weierstrass theory

Text Books:
1. L.V. Alfors, “Treatment as in Complex Analysis “, III Edition, I.S.E., McGraw Hill.

Unit I : Chapter 3
Unit II : Chapter 4, Sections: 1,2,3 and 4
Unit III: Chapter 4, Sections 5 and 6
Unit IV: Chapter 5
Unit V : Chapter 6 – Section 1, Chapter 7

Books for Reference :
1. E.Hille, “ analytic Functions”, Vol.I and II.
2. J.B.Conway, “Functions of one Complex Variable”.
3. Nevalinna and Paatro, “Complex analysis”.

COMPLEX ANALYSIS BHARATHIDASAN UNIVERSITY, TIRUCHIRAPPALLI – 620 024 M.Sc. MATHEMATICS SYLLABUS (ANNUAL PATTERN) SYLLABUS

BHARATHIDASAN UNIVERSITY, TIRUCHIRAPPALLI – 620 024
M.Sc. MATHEMATICS SYLLABUS (ANNUAL PATTERN)
(FOR DISTANCE EDUCATION CANDIDATES ONLY)
(For the candidates admitted from the academic year 2006-2007 batch onwards)
PAPER III - COMPLEX ANALYSIS

Unit I
Analytic functions as mappings: Elementary point set topology – Conformality – Linear transformations – Elementary conformal mappings.

Unit II
Complex Integration: Fundamental theorems – Cauchy’s integral formula – Local properties of analytic functions – General form of Cauchy’s theorem.

Unit III
Complex Integration: Calculus of residues – Harmonic functions

Unit IV
Series and Product Development: Power series expansion – Partial fractions and factorization – Entire functions – Riemann zeta function – Normal; families.

Unit V
Riemann Mapping theorem – Elliptic functions: Simply periodic functions – Doubly periodic functions – the Weierstrass theory

Text Books:
1. L.V. Alfors, “Treatment as in Complex Analysis “, III Edition, I.S.E., McGraw Hill.

Unit I : Chapter 3
Unit II : Chapter 4, Sections: 1,2,3 and 4
Unit III: Chapter 4, Sections 5 and 6
Unit IV: Chapter 5
Unit V : Chapter 6 – Section 1, Chapter 7

Books for Reference :
1. E.Hille, “ analytic Functions”, Vol.I and II.
2. J.B.Conway, “Functions of one Complex Variable”.
3. Nevalinna and Paatro, “Complex analysis”.

PAPER – I – ALGEBRA BHARATHIDASAN UNIVERSITY M.Sc. MATHEMATICS SYLLABUS (ANNUAL PATTERN) SYLLABUS

BHARATHIDASAN UNIVERSITY, TIRUCHIRAPPALLI – 620 024
M.Sc. MATHEMATICS SYLLABUS (ANNUAL PATTERN)
(FOR DISTANCE EDUCATION CANDIDATES ONLY)
(For the candidates admitted from the academic year 2006-2007 batch
onwards)

FIRST YEAR -- PAPER – I – ALGEBRA

Unit I
Another counting principle – Sylow’s theorems – Direct products – Finite abelian groups, Polynomial rings – Polynomials over the rational field – Polynomial rings over commutative rings.

Unit II
Extension fields – roots of polynomials – More about roots – The element of Galois theory – Finite fields – Wedderburn’s theorem on finite division rings – A theorem of Frobenius.

Unit III
The algebra of linear transformations – Isomorphism of vector spaces – Representations of linear transformations by matrices – Linear functionals- the double dual – the transpose of a linear transformation.

Unit IV
The algebra of polynomials – Lagrange Interpolation – Polynomial ideals – the prime factorization of a polynomial – Commutative rings – Determinant functions- Permutations and the uniqueness of determinant – classical adjoint of a matrix – Inverse of an invertible matrix using determinats.

Unit V
Characteristic values – Annihilating polynomial – Invariant subspaces –
Simultaneous triangulation – Simultaneous diagonalization – Direct sum decompositions.

Text books:

1. I.N. Herstein, “Topics in Algebra” Second Edition, Vikas Publishing House Pvt. Ltd., New Delhi.
Unit I: Chapter 2 ( 2.11,2.12,2.13,2.14) Chapter 3 (3.9,3.10,3.11)
Unit II: Chapter 5 (5.1,5.3,5.5,5.6) Chapter 7 (7.1,7.2,7.3)
2. K.Hoffman and R. Kunze, “ Linear Algebra”, Second Edition, Prentice – Hall of India Pvt. Ltd.
Unit III: Chapter 3 ( all sections )
Unit IV Chapter 4 (all sections ) Chapter 5 (5.1 to 5.4)
Unit V: Chapter 6 (6.1 to 6.6)
Books for References:
1. P.B. Bhattacharya, S.K. Jain, S.R. Nagpaul “Basic Abstract Algebra”, Cambridge University Press, Second Edition, 1995.
2. J.b. Fraleigh, “A First Course in Abstract Algebra”, Narosa Publishing House, New Delhi
3. N. Jacohson, Basic Algebra, Volume I and II, Freeman 1980.

PAPER – I – ALGEBRA BHARATHIDASAN UNIVERSITY M.Sc. MATHEMATICS SYLLABUS (ANNUAL PATTERN) SYLLABUS

BHARATHIDASAN UNIVERSITY, TIRUCHIRAPPALLI – 620 024
M.Sc. MATHEMATICS SYLLABUS (ANNUAL PATTERN)
(FOR DISTANCE EDUCATION CANDIDATES ONLY)
(For the candidates admitted from the academic year 2006-2007 batch
onwards)

FIRST YEAR -- PAPER – I – ALGEBRA

Unit I
Another counting principle – Sylow’s theorems – Direct products – Finite abelian groups, Polynomial rings – Polynomials over the rational field – Polynomial rings over commutative rings.

Unit II
Extension fields – roots of polynomials – More about roots – The element of Galois theory – Finite fields – Wedderburn’s theorem on finite division rings – A theorem of Frobenius.

Unit III
The algebra of linear transformations – Isomorphism of vector spaces – Representations of linear transformations by matrices – Linear functionals- the double dual – the transpose of a linear transformation.

Unit IV
The algebra of polynomials – Lagrange Interpolation – Polynomial ideals – the prime factorization of a polynomial – Commutative rings – Determinant functions- Permutations and the uniqueness of determinant – classical adjoint of a matrix – Inverse of an invertible matrix using determinats.

Unit V
Characteristic values – Annihilating polynomial – Invariant subspaces –
Simultaneous triangulation – Simultaneous diagonalization – Direct sum decompositions.

Text books:

1. I.N. Herstein, “Topics in Algebra” Second Edition, Vikas Publishing House Pvt. Ltd., New Delhi.
Unit I: Chapter 2 ( 2.11,2.12,2.13,2.14) Chapter 3 (3.9,3.10,3.11)
Unit II: Chapter 5 (5.1,5.3,5.5,5.6) Chapter 7 (7.1,7.2,7.3)
2. K.Hoffman and R. Kunze, “ Linear Algebra”, Second Edition, Prentice – Hall of India Pvt. Ltd.
Unit III: Chapter 3 ( all sections )
Unit IV Chapter 4 (all sections ) Chapter 5 (5.1 to 5.4)
Unit V: Chapter 6 (6.1 to 6.6)
Books for References:
1. P.B. Bhattacharya, S.K. Jain, S.R. Nagpaul “Basic Abstract Algebra”, Cambridge University Press, Second Edition, 1995.
2. J.b. Fraleigh, “A First Course in Abstract Algebra”, Narosa Publishing House, New Delhi
3. N. Jacohson, Basic Algebra, Volume I and II, Freeman 1980.

REAL ANALYSIS BHARATHIDASAN UNIVERSITY

BHARATHIDASAN UNIVERSITY, TIRUCHIRAPPALLI – 620 024
M.Sc. MATHEMATICS SYLLABUS (ANNUAL PATTERN)
(FOR DISTANCE EDUCATION CANDIDATES ONLY)
(For the candidates admitted from the academic year 2006-2007 batch onwards)
PAPER II - REAL ANALYSIS

Unit I
Basic topology: Finite, Countable sets – Metric spaces - Compact sets – Perfect sets – Connected sets.
Numerical sequences and series, sequences – convergence – subsequences – Cauchy sequences – Upper and Lower limits – some special sequences – Tests of Convergence – Power series – Absolute convergence – Addition and Multiplication
series.

Unit II
Continuity: Limits of functions – Continuous Functions - Continuity and Compactness – Continuity and connectedness – discontinuities – Monotonic functions- Infinite limits and Limit at infinity.
Differentiation: Derivative of real functions – mean value theorems – intermediate value theorems for derivatives – L’ hospital rule – Taylor’s Theorem – differentiation of vector – valued functions.

Unit III
Riemann – Stieltjes integrals: definition and existence – properties – integration and differentiations – Integration of vector valued functions.

Unit IV
Sequence and Series of functions – Discussions of main problem – uniform convergence – Uniform convergence and continuity – Uniform convergence and integration – Uniform convergence and different ion – equi continuous – family of
functions – Stone – Weierstrass theorem.

Unit V
The Lebesgue theory – set functions. Construction of Lebesgue measure – measure spaces – measurable functions – simple functions – integration – comparisons with the Riemann integral – integration of Complex function – function of class L2.

Text Books:
1. Walter Rudin, “Principles of Mathematical Analysis”, Third edition, Mc-Graw Hill,
1976
Unit I: Chapter 2 and 3
Unit II: Chapter 4 and 5
Unit III: Chapter 6
Unit IV: Chapter 7
Unit V: Chapter 11
Books for Reference:
1. T.M. Apostol, “Mathematical Analysis”, Second edition, Addison Wesley publication,
Tokyo 1981.
2. V.Ganapathy Iyer, “Introduction to Real Analysis”, PHI.

REAL ANALYSIS BHARATHIDASAN UNIVERSITY

BHARATHIDASAN UNIVERSITY, TIRUCHIRAPPALLI – 620 024
M.Sc. MATHEMATICS SYLLABUS (ANNUAL PATTERN)
(FOR DISTANCE EDUCATION CANDIDATES ONLY)
(For the candidates admitted from the academic year 2006-2007 batch onwards)
PAPER II - REAL ANALYSIS

Unit I
Basic topology: Finite, Countable sets – Metric spaces - Compact sets – Perfect sets – Connected sets.
Numerical sequences and series, sequences – convergence – subsequences – Cauchy sequences – Upper and Lower limits – some special sequences – Tests of Convergence – Power series – Absolute convergence – Addition and Multiplication
series.

Unit II
Continuity: Limits of functions – Continuous Functions - Continuity and Compactness – Continuity and connectedness – discontinuities – Monotonic functions- Infinite limits and Limit at infinity.
Differentiation: Derivative of real functions – mean value theorems – intermediate value theorems for derivatives – L’ hospital rule – Taylor’s Theorem – differentiation of vector – valued functions.

Unit III
Riemann – Stieltjes integrals: definition and existence – properties – integration and differentiations – Integration of vector valued functions.

Unit IV
Sequence and Series of functions – Discussions of main problem – uniform convergence – Uniform convergence and continuity – Uniform convergence and integration – Uniform convergence and different ion – equi continuous – family of
functions – Stone – Weierstrass theorem.

Unit V
The Lebesgue theory – set functions. Construction of Lebesgue measure – measure spaces – measurable functions – simple functions – integration – comparisons with the Riemann integral – integration of Complex function – function of class L2.

Text Books:
1. Walter Rudin, “Principles of Mathematical Analysis”, Third edition, Mc-Graw Hill,
1976
Unit I: Chapter 2 and 3
Unit II: Chapter 4 and 5
Unit III: Chapter 6
Unit IV: Chapter 7
Unit V: Chapter 11
Books for Reference:
1. T.M. Apostol, “Mathematical Analysis”, Second edition, Addison Wesley publication,
Tokyo 1981.
2. V.Ganapathy Iyer, “Introduction to Real Analysis”, PHI.

ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS BHARATHIDASAN UNIVERSITY

BHARATHIDASAN UNIVERSITY, TIRUCHIRAPPALLI – 620 024
M.Sc. MATHEMATICS SYLLABUS (ANNUAL PATTERN)
(FOR DISTANCE EDUCATION CANDIDATES ONLY)
(For the candidates admitted from the academic year 2006-2007 batch onwards)
PAPER V – ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS

Unit I
The general solution of the homogeneous equation – The use of one known solution to find another – The method of variation of parameter – Power series solutions– series solutions of first order equations – Second order linear equations – ordinary points – Regular singular points – Gauss hyper geometric equations – the point oat infinity.

Unit II
Legendre Polynomials - Properties of Legendre polynomials – Bessel functions – The gamma function – Properties of Bessel function – linear systems – Homogeneous Linear system with constant coefficients.

Unit III: The existence and uniqueness of solutions
The method of Successive approximation – Picard’s theorem – Types of critical points – Critical points and stability for linear systems – Stability by Liapunov’s direct method.

Unit IV
First order partial differential equations – Linear equations of the first order – Pfafian differential equations – Compatible systems – Charpit’s method – Jacobi’s method – Integral surface through a given circle.

Unit V
Genesis of second order PDE – Classifications of second order PDE – one dimensional wave equation – Vibration of an infinite string, Vibrations of semi-infinite string, Vibrations of a string of finite length (Method of separation of Variables) - Heat conduction problem – Heat conduction – Infinite rod case and heat conduction – finite rod case.

Text Book:
1. G.F. Simmons – Differential Equations with Applications and Historical Notes, TMH, New Delhi
Unit I - Chapter 3: Sections – 15, 16, 19, Chapter 5: Sections - 26 to 31
Unit II: Chapter 6: Sections – 32 to 36, Chapter 7 : Sections – 37,38.
Unit III: Chapter 8: Sections – 41 to 43, Chapter 7: Sections – 56, 57.
2. T.Amarnath, “An Elementary Course in Partial Differential Equations”, Narosa, New Delhi, 1997.
Unit IV – Chapter 1: Sections – 1.4 to 1.9
Unit V - Chapter 2: Sections – 2.1, 2.2, 2.3.1, 2.3.2, 2.3.3, 2.3.5, 2.5.1, 2.5.2
References
1. W.T.Reid, Ordinary Differential Equations, John Wiley, New York, 1971.
2. E.A.Coddington and E.Levinson, Theory of ODE, Mc Graw Hill Publishing Company,
New york, 1955
3. J.N. Sneddon, Elements of Partial Differential Equations, Mc Graw Hill Publishing
Company, New york, 1957.

ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS BHARATHIDASAN UNIVERSITY

BHARATHIDASAN UNIVERSITY, TIRUCHIRAPPALLI – 620 024
M.Sc. MATHEMATICS SYLLABUS (ANNUAL PATTERN)
(FOR DISTANCE EDUCATION CANDIDATES ONLY)
(For the candidates admitted from the academic year 2006-2007 batch onwards)
PAPER V – ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS

Unit I
The general solution of the homogeneous equation – The use of one known solution to find another – The method of variation of parameter – Power series solutions– series solutions of first order equations – Second order linear equations – ordinary points – Regular singular points – Gauss hyper geometric equations – the point oat infinity.

Unit II
Legendre Polynomials - Properties of Legendre polynomials – Bessel functions – The gamma function – Properties of Bessel function – linear systems – Homogeneous Linear system with constant coefficients.

Unit III: The existence and uniqueness of solutions
The method of Successive approximation – Picard’s theorem – Types of critical points – Critical points and stability for linear systems – Stability by Liapunov’s direct method.

Unit IV
First order partial differential equations – Linear equations of the first order – Pfafian differential equations – Compatible systems – Charpit’s method – Jacobi’s method – Integral surface through a given circle.

Unit V
Genesis of second order PDE – Classifications of second order PDE – one dimensional wave equation – Vibration of an infinite string, Vibrations of semi-infinite string, Vibrations of a string of finite length (Method of separation of Variables) - Heat conduction problem – Heat conduction – Infinite rod case and heat conduction – finite rod case.

Text Book:
1. G.F. Simmons – Differential Equations with Applications and Historical Notes, TMH, New Delhi
Unit I - Chapter 3: Sections – 15, 16, 19, Chapter 5: Sections - 26 to 31
Unit II: Chapter 6: Sections – 32 to 36, Chapter 7 : Sections – 37,38.
Unit III: Chapter 8: Sections – 41 to 43, Chapter 7: Sections – 56, 57.
2. T.Amarnath, “An Elementary Course in Partial Differential Equations”, Narosa, New Delhi, 1997.
Unit IV – Chapter 1: Sections – 1.4 to 1.9
Unit V - Chapter 2: Sections – 2.1, 2.2, 2.3.1, 2.3.2, 2.3.3, 2.3.5, 2.5.1, 2.5.2
References
1. W.T.Reid, Ordinary Differential Equations, John Wiley, New York, 1971.
2. E.A.Coddington and E.Levinson, Theory of ODE, Mc Graw Hill Publishing Company,
New york, 1955
3. J.N. Sneddon, Elements of Partial Differential Equations, Mc Graw Hill Publishing
Company, New york, 1957.