M.Sc. I Semester Mathematics (CBCS) 2015-16 Teaching Plan Paper : M1 MAT 01-CT01 (ALGEBRA-I) Day Topic Reference Book Chapter UNIT-I 1-4 External and Internal direct product of Algebra (Agrawal, R.S.) Chapter 6 two and finite number of subgroups; 5-6 Commutator subgroup - do - Chapter 6 7-9 Cauchy’s theorem for finite abelian and - do - Chapter 6 non abelian groups. 10-11 Tutorials UNIT-II 12-15 Sylow’s three theorem and their easy - do - Chapter 6 applications 16-18 Subnormal and Composition series, - do - Chapter 6 19-21 Zassenhaus lemma and Jordan Holder - do - Chapter 6 theorem. 22-23 Tutorials - do - UNIT-III 24-26 Solvable groups and their properties Modern Algebra (Surjeet Chapter 5 Singh and Quazi Zameeruddin) 27-29 Nilpotent groups - do - Chapter 5 30-32 Fundamental theorem for finite abelian - do - Chapter 4 groups. 33-34 Tutorials UNIT-IV 35-38 Annihilators of subspace and its Algebra (Agrawal, R.S.) Chapter 12 dimension in finite dimensional vector space 39-41 Invariant, - do - Chapter 12 42-45 Projection, - do - Chapter 12 46-47 adjoins. - do - Chapter 12 48-49 Tutorials UNIT-V 50-54 Singular and nonsingular linear Algebra (Agrawal, R.S.) Chapter 12 transformation 55-58 quadratic forms and Diagonalization. - do - Chapter12 59-60 Tutorials - do - 61-65 Students Interaction & Difficulty solving - - M.Sc. I Semester Mathematics (CBCS) 2015-16 Teaching Plan Paper : M1 MAT 02-CT02 (REAL ANALYSIS) Day Topic Reference Book Chapter UNIT-I 1-2 Length of an interval, outer measure of a Theory and Problems of Chapter 2 subset of R, Labesgue outer measure of a Real Variables: Murray subset of R R. Spiegel – 3-4 Existence, non-negativity and - do - Chapter 2 monotonicity of Lebesgue outer measure 5-7 Translation Invariance of outer measure, - do - Chapter 2 Countable sub-additivity of Lebesgue outer measure 8 The relation between outer measure and - do - Chapter 2 length of an interval. 9 Tutorial UNIT-II 10-11 σ- algebra, Definition of measures with Measure and Integration: Chapter 3 equations, signed measure, measurable P.K. Jain & V.P. Gupta space – 12-14 measurable sets (Lebesgure), - do - Chapter 3 Complement, Lebesgue measure, union intersection and difference of measurable sets 15-17 Denumerable union and intersection of - do - Chapter 3 measurable sets, countable additivity of measure, 18-19 counting measurem Haar measure, Borel - do - Chapter 3 measure 20-21 Measures of infinite union of measure - do - Chapter 3 sets (intersection), convergence in measure, Fσ and Gσ sets 22-23 the measure of the intersection of a - do - Chapter 3 decreasing sequence of measure sets. 24 Tutorial UNIT-III 25-26 Measurable functions; Different Measure Theory by Pal Chapter 4 equivalent definition of a measurable R. Halmos – function; almost every where Scalar multiple constant function 27-29 modules f/g, inverse image of a mean - do - Chapter 4 function, sum, difference and product of measurable functions of measurable function 30-32 Measurability of a continuous function - do - Chapter 4 and measurability of a continuous image of measurable function. 33-35 Supremum, infimum, limit superior, - do - Chapter 4 limit inferior and limit of a sequence of measurable functions 36-38 Convergence pointwise and convergence - do - Chapter 4 in measures of a sequence of measurable functions. 40 Tutorial UNIT-IV 41-43 Convergence of sequence of Measurable Introduction to Measure Chapter 6 function: Convergence pointwise, and Integration : M.E. uniform Convergence, Convergence Munroe almost every where (a.e.), 44-45 Convergence in measure of a sequence - do - Chapter 6 of measurable functions 46-47 characteristic function of a set, simple - do - Chapter 6 function, step function, pointwise Convergence, theorem of measure functions. 48 Tutorial UNIT-V 4-50 Lebesgue Integral; Characteristic P.K. Jain, V.P. Gupta & Chapter 5 function of a set; Simple functions, Pankaj Jain Lebesgue integral of a simple function 51-52 Lebesgue integral of a bounded - do - Chapter 5 measurable function; Lebesgue integral and Riemann integral of a bounded function defined on a closed interval 53-54 Lebesgue integral of a non-negative - do - Chapter 5 function; Lebesgue integral of a measurable function; 55 Properties of Lebesgue integral - do - Chapter 5 56-57 Convergence Theorems and Lebesgue - do - Chapter 5 integral; the bounded convergence theorem; 58-59 Fatou's Lemma: Monotone convergence - do - Chapter 5 theorem; Lebesgue convergence theorem. 60 Tutorial 61-65 Students Interaction & Difficulty solving - - M.Sc. I Semester Mathematics (CBCS) 2015-16 Teaching Plan Paper : M1 MAT 03-CT03 (DIFFERENTIONAL EQUATIONS & CALCULUS OF VARIATION) Day Topic Reference Book Chapter UNIT-I 1-2 Introduction and pre requisites and revision of - Differential equations 3-7 Lipschitz condition and question based on it. Existence Bansal, Dhami: Chapter and Uniqueness theorem and Existence and uniqueness Differential 9 solutions of I.V.P. equations Vol. II. 8-11 Solution of second order partial differential equations - do - Chapter through Mange’s method. 12 12 Tutorials UNIT-II 13-18 Canonical forms and reduction of second order Semilinar - do - Chapter partial differential equations to canonical forms. 12 19-22 Classification of second order partial differential - do - Chapter equations having more than two independent variables. 13 Cauchy’s problem. 23-24 Tutorials UNIT-III 25-29 Boundary value problems of second order ordinary - do - Chapter differential equations. Orthogonality, Strum-Liouville 13 B.V.P. 30-34 Lagrange’s Identity. Relevent theorems and properties - do - Chapter based on eigen values and eigen functions. Expansion of 13 function in terms of eigen functions. Periodic Strum problem. 35-36 Tutorials UNIT-IV 37-42 Solution of second order P.D. equations by the method of - do - Chapter separation of variables. 13 42-48 Green’s function and its construction and solution of - do - Chapter second order Homogonous B.V.P’s through Green’s 13 function. Dirac delta function and its important properties. 49-50 Tutorials UNIT-V 51-55 Calculus of variations: Functionals, Euler- Largrauge - do - Chapter differential equation for externals and it’s alternative 14 forms. 55-58 Variational problems involving several dependent - do - Chapter variables, several in dependent variables and higher order 14 derivatives. Isoperimetric Problems. Solution of variational problems using Ritz method. 59-60 Tutorials 61-65 Students Interaction & Difficulty solving - - M.Sc. I Semester Mathematics (CBCS) 2015-16 Teaching Plan Paper : M1 MAT 04-CT04 (MECHANICS-I) Day Topic Reference Book Chapter UNIT-I 1-3 Fluid motion, material derivative, Equation of Hydrodynamics Chapter 1 continuity by Lagrange’s and Euler’s method (Gaur, Mathew and Goyal) 4-6 Equation of Continuity in different coordinate - do - Chapter 1 system and related problems 7-10 Path line, Stream line velocity potential, - do - Chapter 1 Boundary surface and related problems 11-12 Tutorials UNIT-II 13-15 Euler’s Dynamical equations of motion, - do - Chapter 2 Bernoulli’s equations in Cartesian coordinates Bernoulli’s Theorem 16-18 Problems and exercise based on above - do - Chapter 2 19 Equations of motion under impulsive forces - do - Chapter 2 20-21 Principle of permance of errotational motion - do - Chapter 2 with derivation 22 Principle of energy for incompressible fluids - do - Chapter 2 23-24 Tutorials UNIT-III 25-27 Different forms of equation of a central orbit - do - Chapter 2 and derivations 28-30 Questions and exercise based on above - do - Chapter 2 31-32 Keplar Laws and Planetary motion - do - Chapter 2 33-34 Question an exercise based on Planetary motion - do - Chapter 2 35-36 Tutorials UNIT-IV 37-38 Galilean Transformation and properties Theory of Relativity Chapter 1 (S.R. Roy & Raj Bali) 39-41 Michelson and Morley experiment and - do - Chapter 1 explanations of null results 42-44 Relative character of space and time, Lorentz - do - Chapter 1 transformation. 45-46 Realistic transformation formulas (velocity, - do - Chapter 1 acceleration and force) 47-48 Tutorials UNIT-V 49-51 Variation of mass with velocity and its - do - Chapter 1 derivations. 52-54 Equivalence of mass and energy and problems - do - Chapter 1 55-57 Relativistic Lagrangian, Hamiltonian, Four - do - Chapter 1 dimensional space time 58-59 Tutorials 60-64 Students Interaction & Difficulty solving - - M.Sc. I Semester Mathematics (CBCS) 2015-16 Teaching Plan Paper : M1 MAT 05-CT05 (DIFFERENTIAL GEOMETRY-I) Day Topic Reference Book Chapter UNIT-I 1-4 The Axes of Plane Sections: Circular sections, Coordinate Chapter Axes of central sections of a central conicoid. Geometry of the 8 three dimensions. (Robert, L., Bell, J.T.) 5-8 Axes of non central section of a central conicoid. - do - Chapter 8 UNIT-II 9-10 Axes of any section of a central conicoid, Circular - do - Chapter sections. 8 11-12 Generating Lines: Introduction, properties of - do - Chapter generating line 9 13-15 intersection of generators of hyperboloid of one - do - Chapter sheet, generators through any point of hyperboloid. 9 16-18 direction cosines of generating line, The section of - do - Chapter a surface by a tangent plane. 9 UNIT-III 19-23 Systems of generators of a central hyperboloid, - do - Chapter Locus of the points of intersection of perpendicular 9 24-28 generators, The projection of generators, - do - Chapter Generators' of the hyperbolic paraboloid. 9 UNIT-IV 29-30 The three confocals through a point,- Elliptic - do - Chapter coordinates. 10 31-32 confocal touching a given plane, confocal touching - do - Chapter a given line. 10 33-35 The parameter of the confocals through a point on a - do - Chapter central conicoid, The normals. 10 36-38 The self polar tetrahedron, The axes of an - do - Chapter enveloping cone. 10 UNIT-V 39-42 Conoids; Equation to a conoid, surface in general, - do - Chapter The degree of a surface. 10, 13 43-45 Tangents and tangent planes, The inflexional - do - Chapter tangents.