Thiagarajar college (autonomous), madurai – 9




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Variational Principle and Lagrange’s Equation : Hamilton’s Principle , Some Techniques of the Calculus of Variation – Derivation of Lagrange’s Equation from Hamilton’s Principle- Extension of Hamilton’s Principle through non Halonomic Systems, Conservation Theorem and Symmetric Properties.




Unit – III (18 Hours)

Two Body Central Force Problem : Reduction to one body Problem-Equations of motion and first integral- The Equivalent One body Problem and Classification of Orbits- Virial theorem- The differential equation of the Orbit and integrable power law potentials- The kepler problem and inverse square law of force- The motion in time in the kepler problem – Laplace- Runge- Lenz vector


Unit – IV (18 Hours) The Hamilton’s equations of motion : Legendre’s transformation and the Hamilton’s equation of motion- Cyclic Co-ordinates and Conservation Theorem- Derivation of Hamilton’s equation from a variational principle- The principle of least action.


Unit – V (18 Hours) Canonical transformations: Equation of canonical transformations- examples of Canonical Transformation- simple approach to Canonical transformation- Poisson brackets and other Canonical invariants.

Text Book: Classical Mechanics

- Herbert Goldstein-2nd Edition, Narosa Publishing House, 1986.


Unit

Chapter/Section

I

Chapter 1

II

Chapter 2 (except 2.5)

III

Chapter 3 (except 3.10, 3.11)

IV

Chapter 8 : 8.1, 8.2, 8.5, 8.6 only

V

Chapter 9 : 9.1 , 9.2 , 9.4 only



Reference Book: Classical Mechanics - V.B.Bhatia

Narosa Publishing House – 1998




THIAGARAJAR COLLEGE (AUTONOMOUS), MADURAI – 625 009

(Re- Accredited with 'A' Grade by NAAC)

Department of Mathematics

(From 2011 – 2013 batch onwards)

Course : M.Sc. Code No. : 3PM2

Semester : III No. of hours allotted : 6

Paper : Core 12 No. of credits : 4

Title of the Paper : Functional Analysis


Course objective: To develop the skills in theorem through geometry and to get knowledge in using classes of functions rather than individual functions.

Unit – I (18 Hours)

Fundamentals of Normed Linear Spaces: Normed Linear Spaces – Continuity of Linear Maps – Hahn-Banach Theorems – Completeness of a Norm.


Unit – II (18 Hours)

Bounded Linear Maps on Banach Spaces: Uniform Boundedness Principle – Closed Graph Theorem – Open Mapping Theorem – Spectrum of a Bounded Operator.


Unit – III (18 Hours)

Spaces of Bounded Linear Functionals: Duals and Transposes – Duals of Lp([a,b]) and C([a,b]) – Weak and Weak* Convergence.

Unit – IV (18 Hours)

Geometry of Hilbert Spaces: Inner Product Spaces – Orthonormal Sets – Projection and Riesz Representation Theorems.


Unit – V (18 Hours)

Bounded Operators on Hilbert Spaces: Bounded Operators and Adjoints – Normal, Unitary and Self-Adjoint Operators – Spectrum and Numerical range – Compact self-adjoint operators.


Text Book : Functional Analysis - Balmohan Vishnu Limaye – Wiley eastern Limited, 1989.


Unit

Chapter/Section

I

Chapter II (Except Section 7.14)

II

Chapter III (Except Section 9.5, 9.6, 9.8, 11.4, 11.5, 11.6)

III

Chapter IV (Except Section 14.9, 15.9 & 16)

IV

Chapter VI (Except Section 23)

V

Chapter VII



Reference Books: 1) Introductory Functional Analysis with Applications - Erwin Kreyszig

John Wiley & Sons, Third Print – 2007.

2) Functional Analysis – George Bachman Lawrence Narici

Academic Press International, New York, 1996


THIAGARAJAR COLLEGE (AUTONOMOUS), MADURAI – 625 009

(Re- Accredited with 'A' Grade by NAAC)

Department of Mathematics

(From 2011 – 2013 batch onwards)

Course : M.Sc. Code No. : 3PM3

Semester : III No. of hours allotted : 4

Paper : Core 13 No. of credits : 3

Title of the Paper : Number Theory


Course objective: To study some importance tools in number theory and study of distributions of primes

Unit - I (12 Hours)

Congruences: Congruences – Solutions – Congruence of degree 1 – The function (n)


Unit - II (12 Hours)

Quadratic reciprocity: Quadratic residues – Quadratic reciprocity – The Jacobian symbol


Unit - III (12 Hours)

Some functions of Number Theory: Greatest integer function – Arithmetic function – The Mobius formula.


Unit - IV (12 Hours)

Diophantine equations: Diophantine equations – The equation ax + by = c – Positive solutions – Other linear equations – The equation x2 + y2 = z2 – The equation x4 + y4 = z4 – Sums of four squares and five squares.


Unit - V (12 Hours)

Distribution of primes: The function (x) – The sequence of primes – Betrand’s postulate


Text Book: An introduction to Number Theory

– Ivan Niven and Zuckerman, Wiley Eastern, Reprint 1989


Unit

Chapter/Section

I

2.1 – 2.4

II

3(Full)

III

4.1 – 4.3

IV

5.1 – 5.7

V

8(Full)



Reference book: Introduction to Analytic Number Theory

– Martin Erickson and Anthony Vazzana,

Chapman and Hall /CRC publications,2009.


THIAGARAJAR COLLEGE (AUTONOMOUS), MADURAI – 625 009

(Re- Accredited with 'A' Grade by NAAC)

Department of Mathematics

(From 2011 – 2013 batch onwards)

Course : M.Sc. Code No. : 4PM1

Semester : IV No. of hours allotted : 6

Paper : Core 14 No. of credits : 4

Title of the Paper : Statistics


Course objective: To present the importance of theoretical approach in statistical methods.

Unit – I (18 Hours)

Theoretical Distributions: Binomial, Trinomial distributions, - Poisson distributions, Gamma and Chi-square distribution – Normal distribution


Unit – II (18 Hours)

Distributions Functions of Random Variables: Sampling theory-Transformation of variables of discrete type - Transformation of variables of continuous Type – The t and F distributions- The moment generating function technique- The Distributions of X and n S2/s2


Unit – III (18 Hours)
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