Review : General remarks on solutions of differential equations, Families of curves,
1. Second order linear equations : The general solution of the homogeneous equations,
1. G. Birkhoff and G.C. Rota : Ordinary differential equations. (John Wiley and Sons)
MT-503 : Group Theory
1. Revision of definition and examples of groups, subgroups.
2. Cyclic Groups, Classification of subgroups of cyclic groups.
3. Permutation Groups - Revision
4. Isomorphism, Cayley’s theorem, properties of isomorphisms, automorphism,
5. Revision of Cosets and Lagrange’s theorem. Orbit-stabilizer theorem, the rotation group of a cube and a soccer ball
6. External Direct Products
7. Normal subgroups and factor groups, Internal direct products
8. Group Homomorphism
9. Fundamental theorem of finite abelian groups
10. Sylow theorems
11. Finite simple groups
Text Books: Joseph Gallian – Contemporary Abstract Algebra (Narosa Publishing House). Chapter 2 to 11, 24, 25.
1. I.S. Luthar and I.B.S. Passi : Algebra (Volume 1) Groups (Narosa Publishing House )
2. I.N. Herstein : Topics in Algebra (Wiley -Eastern Ltd)
3. M. Artin : Algebra (Prentice Hall)
4. N.S. Gopala Krishnan : University Algebra (Wiley-Eastern Ltd)
5. Fraleigh : A First Course in Abstract Algebra
6. Dummit and Foote : Abstract Algebra ( Wiley-Eastern Ltd)
MT - 502: Advanced Calculus
1. Derivative of a scalar field with respect to a vector, Directional derivative, Gradient of a scalar field, Derivative of a vector field, Matrix form of the chain rule, Inverse function theorem and Implicit function theorem.
2. Path and line integrals, The concept of work as a line integral, Independence of path, The first and the second fundamental theorems of calculus for line integral, Necessary condition for a vector field to be a gradient.
3. Double integrals, Applications to area and volume, Green's Theorem in the plane, Change of variables in a double integral, Transformation formula, Change of variables in an n-fold integral.
4. The fundamental vector product, Area of a parametric surface, Surface integrals, The theorem of Stokes, The curl and divergence of a vector field, Gauss divergence theorem, Applications of the divergence theorem.
Text Book: T. M. Apostol: Calculus, Vol. II (2nd edition) (John Wiley and Sons, Inc.) Chapter 1: Sections 81 to 8.22 Chapter 2: Sections 10.1 to 10.11 and 10.14 to 10.16 Chapter 3: Sections 11.1 to 11.5 and 11.19 to 11.22 and 11.26 to 11.34 Chapter 4: Sections 12.1 to 12.15, 12.18 to 12.21 (For Inverse function theorem and Implicit function theorem refer the book “Mathematical Analysis” by T. M. Apostol.)
Reference Books :
1. T. M. Apostol: Mathematical Analysis (Narosa publishing house)
2. W. Rudin: Principles of Mathematical Analysis (Mc-Graw Hill)
3. A. Devinatz: Advanced Calculus, (Holt, Rinehart and Winston), 1968
MT - 501: Real Analysis
1. Measure Theory: Preliminaries, Exterior Measure, Measurable Sets and Lebesgue Measure, Measurable Functions.
2. Integration Theory: The Lebesgue Integral, basic properties and convergence theorems. The space L^1 of integrable functions, Fubini’s theorem.
3. Differentiation and Integration: Differentiation of the integral, Good kernels and approximation to the identity, differentiation of functions.
Text Book: Real Analysis, E. Stein and R. Shakharchi, New Age International Publishers, Princeton Lecture Notes III. Chapter 1 - Sections 1 to 4, Chapter 2 - Sections 1 to 3, Chapter 3 - Sections 1 to 3.
1. Karen Saxe : Beginning Functional Analysis (Springer International Edition)
2. N. L. Carothers: Real Analysis (Cambridge University Press)
3. W. Rudin : Principles of Mathematical Analysis (Mc-Graw Hill)
4. H. Royden, Real Analysis, McMillan Publishing Company