MT- 605 : Partial Differential Equations

1. First Order P.D.E. : Introduction, Charpit’s Method, Jacobi’s Method, Quasi-Linear

Equations, Non-Linear First Order P.D.E.

2. Second Order P.D.E.: Introduction, One Dimensional Wave Equation, Laplace

Equation, Boundary Value Problems, the Cauchy Problem, Dirichlet and Neumann

Problem for different regions, Harnack’s Theorem, Heat Conduction Problem,

Duhamel’s Principle, Classification of P.D.E. in the case of n-variables, Families of

Equipotential Surfaces, Kelvin’s Inversion Theorem.

Text Book

T. Amarnath : An Elementary Course in Partial Differential Equations (2nd edition)

(Narosa Publishing House) [Chapters 1 and 2].

Reference Books :

1. K. Sankara Rao: Introduction to partial differential equation, third edition.

2. W. E. Williams: Partial Differential equations (Clarendon press-oxford)

3. E. T. Copson : Partial differential equations (Cambridge university press)

4. I.N. Sneddon: Elements of partial differential equations (Mc-Graw Hill book company).

- Teacher: IshratJahan Ansari
- Teacher: Naushad Shaikh

- Teacher: IshratJahan Ansari
- Teacher: Naushad Shaikh

- Teacher: IshratJahan Ansari

MT-604: Linear Algebra

1. Vector spaces: Definition and Example, Subspace, Basis and Dimension (revision)2. Linear mapping and matrices: Linear Mappings, Quotient Spaces, Vector Space of

Linear Mapping, Linear Mapping and Matrices, Change of Basis, Rank of a Linear

Mapping, Decomposition of a Vector Space

3. Reduction of matrices to canonical forms: eigenvalues and eigenvectors,

triangularization of a matrix, jordan canonical form

4. Metric vector space: Bilinear form, Symmetric Bilinear Forms, Quadratic Forms,

Hermitian Forms, Euclidean Vector Space, Canonical Representation of Unitary

Operator, Euclidean Space, Classification of Quadrics in Three- Dimensional Euclidean

Space

Text Book: First Course in Linear Algebra -

Jain [Chap 4: Revision, Chapters 5, 6, 7].

P.B. Bhattacharya, S.R. Nagpaul, S.K.

Reference Books:

i) K. Hoffman and Ray Kunje : Linear Algebra (Prentice -Hall of India private Ltd.)

ii) M. Artin : Algebra (Prentice -Hall of India private Ltd.)

iii) A.G. Hamilton : Linear Algebra (Cambridge University Press),1989.

iv) N.S. Gopala Krishnan : University algebra (Wiley Eastern Ltd.).

v) J.S. Golan : Foundations of linear algebra (Kluwer Academic publisher),1995.

vi) Henry Helson : Linear Algebra, (Hindustan Book Agency), 1994.

vii) I.N. Herstein : Topics in Algebra, Second edition, (Wiley Eastern Ltd.)

- Teacher: IshratJahan Ansari
- Teacher: Yashwant Madke

1. Preliminaries: Rings- Definition, Examples, types of the rings: matrix, polynomial,

power series, Laurent series, Boolean rings, opposite rings

2. Ideals, maximal ideal, quotient rings, local rings

3. Homomorphism of rings, fundamental theorems, endomorphism rings, field of

fractions, prime fields

4. Fractions in domain: Euclidean Domains, P.I.D.’s, U.F.D.’s.

3. Polynomial Rings: Definition, properties, Polynomial Rings over Fields, Polynomial

Rings that are U.F.D.’s, Irreducibility Criteria.

4. Basic Definitions and Examples of Modules, Quotient Modules and Module

Homomorphisms

Text Book: C. Musili, Rings and Modules, 2 nd Revised Edition, Narosa Publishing

House, [Chapters 1, 2, 3, 4, 5].

Reference Books:

1. Dummit and Foote, Abstract Algebra, second edition (Wiley India).

2. Luther and Passi, Algebra II, Narosa Publishing House.

3. Jain and Bhattacharya, Basic Abstract Algebra, Second Edition, Cambridge

University Press.

- Teacher: IshratJahan Ansari
- Teacher: Yashwant Madke

1. Countable and uncountable sets : Infinite sets, the Axiom of Choice, Continuum

Hypothesis, Well-ordered sets, The maximum principle.

2. Topological spaces and continuous functions : Basis for topology, Order topology,

Continuous functions, Product topology, Metric topology, Quotient topology.

3. Connectedness and compactness : Connected spaces, Components and localconnectedness, Compact spaces, Limit point compactness, Local compactness. One

point compactification.

4. Countability and Separation Axioms : The Countability Axioms, Separation Axioms,

Normal spaces, The Urysohn Lemma, The Urysohn Metrization Theorem (statement

only), The Tietze extension theorem (statement only).

5. Tychonoff theorem, Completely regular spaces.

Text Book : J.R. Munkres : Topology, a first course (Prentice Hall of India).

Sections : 1.7, 1.9, 1.10, 1.11, 2.1 to 2.11, 3.1 to 3.8, 4.1 to 4.4, 5.1 and 5.2.

Reference Books :

1. J. Dugundji : Topology (Allyn and Bacon, Boston, 1966.)

2. K. D. Joshi : Introduction to General Topology (Wiley Eastern Limited).

3. J. L. Kelley : General Topology (Springer Verlag, New York 1991.)

4. L. A. Steen and J. A. Seebach Jr. : Counterexamples in Topology (Holt Rinehart and

Winston, Inc. New York 1970.)

5. S. Willard : General Topology (Addison-Wesley Publishing company, Inc., Reading,

Mass., 1970)

- Teacher: IshratJahan Ansari
- Teacher: Yashwant Madke

MT- 601: Complex Analysis

1. Preliminaries : Functions on Complex Plane, Integration along curves

2. Cauchy’s theorem and its applications: Goursat’s theorem, Local existence of

primitives and Cauchy’s theorem in a disc, Evaluation of some integrals, Cauchy

integral formulas, Further applications - Morera’s theorem, sequences of

holomorphic functions, holomorphic functions defined in terms of integrals,

Schwarz reflection principle, Runge’s approximation theorem.

3. Meromorphic functions and the Logarithm, Zeros and poles, the residue

formulae, singularities and meromorphic functions, the argument principle and

applications, homotopies and simply connected domains, the complex logarithm,

Fourier series and harmonic functions.

Text Book: Complex Analysis, E. Stein and Shakarchi, Overseas Press (India) Ltd.,

Princeton Lectures in Analysis. Chapters 1, 2 and 3.

Reference Books:

1.John B. Conway : Functions of one complex variable (Narosa Publishing house)

2. Lars V. Ahlfors : Complex Analysis (McGraw Hill)

3. Ruel V. Churchill / James Ward Brown : Complex Variables and Applications

(McGraw Hill)

- Teacher: IshratJahan Ansari