{"id":363,"date":"2020-12-13T17:33:24","date_gmt":"2020-12-13T12:03:24","guid":{"rendered":"https:\/\/mahabharti.in\/university\/?p=363"},"modified":"2020-12-13T21:15:25","modified_gmt":"2020-12-13T15:45:25","slug":"rtmnu-m-sc-sem-3-new-syllabus-2020","status":"publish","type":"post","link":"https:\/\/mahabharti.in\/university\/rtmnu-m-sc-sem-3-new-syllabus-2020\/","title":{"rendered":"RTMNU MSc 3 Sem Mathematics Syllabus 2020"},"content":{"rendered":"

RTMNU MSc Sem 3\u00a0New Syllabus 2020<\/strong> | RTMNU M.Sc Revised Sem 3 Syllabus 2020 | Nagpur University M.Sc Syllabus Part 2| RTMNU Second Year M.Sc Syllabus
\nRTM Nagpur University M.Sc Semester New Revised Syllabus is given below for Downloading. The students can Download the respective Syllabus from following given details. Just go through the given links & read the given syllabus carefully. Nagpur University Second Year New Semester Online Detail syllabus given below.<\/p>\n

M Sc Mathematics
\nSemester-III Syllabus
\nPaper \u2013 XI (Code\u00a0 3T1)<\/h3>\n

Complex Analysis<\/strong><\/h3>\n

Unit 1<\/strong>
\nImpossibility of ordering Complex numbers. Extended complex plane and stereographic projection. Elementary properties and examples of analytic Functions: Power series, analytic functions.<\/p>\n

Unit 2<\/strong>
\nAnalytic functions as mappings, Mobius transformations. Power series representation of analytic functions, zeros of an analytic function, index of a closed curve.<\/p>\n

Unit 3<\/strong>
\nCauchy\u2019s theorem and integral formula, the homotopic version of cauchy\u2019s theorem and simple connectivity, counting zeros; the open mapping theorem, Goursat\u2019s theorem, Classification of singularities, residues, the argument principle.<\/p>\n

Unit 4<\/strong>
\nThe maximum principle. Schwarz\u2019s lemma. convex functions and Hadamards three circles theorem. Phragmen-Lindelof theorem.<\/p>\n

Text Book<\/strong>
\nFunctions of one complex variable: John B. Conway, Second edition, Springer international Student Edition.<\/p>\n

Reference Book<\/strong>
\nComplex Analysis, L.V. Ahlfors. Mc-Graw Hill, 1966<\/p>\n

___________________________________________________________________________<\/p>\n

M. Sc Mathematics<\/strong>
\nSemester-III<\/strong>
\nPaper \u2013 XII (Code 3T2)<\/strong><\/h3>\n

Functional Analysis<\/strong><\/h3>\n

Unit 1<\/strong>
\nNormed spaces, Banach spaces, Further properties of normed spaces. Finite dimensional normed spaces and subspaces. Compactness and finite dimension. Bounded and continuous linear operators.<\/p>\n

Unit 2<\/strong>
\nLinear functionals. Normed spaces of operators. Dual spaces. Inner product space. Hilbert space. Further properties of inner product spaces. Orthogonal complements and direct sums. Orthonormal sets and sequences. Total orthonormal sets and sequences.<\/p>\n

Unit 3<\/strong>
\nRepresentation of functionals on Hilbert spaces. Hilbert adjoint operators, self adjoint, unitary and normal operators. Hahn-Banach Theorem, Hahn-Banach Theorem for complex vector spaces and normed spaces. Reflexive spaces.<\/p>\n

Unit 4<\/strong>
\nCategory theorem, Uniform boundedness theorem, strong and weak convergence, Convergence of sequences of operators and functionals. Open mapping theorem, Closed linear operators and closed graph theorem.<\/p>\n

Text Book<\/strong>
\nIntroductory Functional Analysis with Applications by E. Kreyszig, John Wiley and Sons.<\/p>\n

Reference Books<\/strong>
\n1. Introduction to Functional Analysis by A.E. Taylor and D.C. Lay, John Wiley and Sons.
\n2. Introduction to Topology and Modern Analysis: G.F. Simmons, Mc Graw Hill<\/p>\n

__________________________________________________________________________<\/p>\n

M. Sc Mathematics<\/strong>
\nSemester-III<\/strong>
\nPaper \u2013 XIII (Code 3T3)<\/strong><\/p>\n

Mathematical Methods<\/strong><\/p>\n

Unit 1<\/strong>
\nFourier integral theorem. Fourier transform. Fourier cosine and sine transform. The convolution integral. Multiple Fourier transform. Solution of partial differential equation by means of Fourier transform.<\/p>\n

Unit 2<\/strong>
\nCalculations of the Laplace transform of some elementary functions. Laplace transform of derivatives. The convolution of two functions. Inverse formula for the Laplace transform. Solutions of ordinary differential equations by Laplace transform.<\/p>\n

Unit 3<\/strong>
\nFinite Fourier transform. Finite Sturm-Liouville transforms. Generalized finite Fourier transform.<\/p>\n

Unit 4<\/strong>
\nFinite Hankel transform. Finite Legendre transform. Finite Mellin transform.<\/p>\n

Text Book<\/strong>
\nThe use of integral transforms: I N. Sneddon, Tata Mc Graw Hill Publishing Company Ltd.<\/p>\n

References Books<\/strong>
\nModern Mathematics For Engineers: Edwin F Beckenbach, Second series, Mc Graw Hill Book Company.<\/p>\n

__________________________________________________________________________<\/p>\n

M. Sc Mathematics<\/strong>
\nSemester-III<\/strong>
\nCore Elective*<\/strong>
\nPaper \u2013 XIV (Code 3T4)<\/strong><\/h3>\n

Fluid Dynamics-I<\/strong><\/h3>\n

Unit 1<\/strong>
\nReal fluids and ideal fluids. Velocity of a fluid at a point. Stream lines and path lines. Steady and unsteady flows. Velocity potential. Velocity vector. Local and particle rate of change. Equation of continuity. Acceleration of a fluid. Condition at a rigid boundary. General analysis of fluid motion. Euler\u2019s equation of motion. Bernoulli\u2019s equation. Worked examples. Discussion of the case of steady motion under conservative body forces. Some further aspects of vortex motion.<\/p>\n

Unit 2<\/strong>
\nSources, sinks and doublets. Images in a rigid infinite plane. Images in solid spheres. Axisymmetric flows. Stokes\u2019 stream function. The complex potential for twodimensional irrotational, incompressible flow. Complex velocity potential for standard two dimensional flow. Uniform stream. Line source and line sink. Line doublets. Line vortices. Two dimensional image systems. The Milne-Thomson circle theorem. Circle Theorem. Some applications of circle theorem. Extension of circle theorem. The theorem of Blasius.<\/p>\n

Unit 3<\/strong>
\nThe equations of state of a substance, the first law of thermodynamics, internal energy of a gas, functions of state, entropy, Maxwell\u2019s thermodynamic relation, Isothermal Adiabatic and Isentropic processes. Compressibility effects in real fluids, the elements of wave motion. One dimensional wave equation, wave equation in two and three dimensions, spherical waves, progressive and stationary waves.<\/p>\n

Unit 4<\/strong>
\nThe speed of sound in a gas, equation of motion of a gas. Sonic, subsonic, supersonic flows; isentropic gas flow. Reservoir discharge through a channel of varying section, investigation of maximum mass flow through a nozzle, shock waves, formation of shock waves, elementary analysis of normal shock waves.<\/p>\n

Text Book<\/strong>
\nF. Chorlton, Text book of Fluid Dynamics, CBS Publishers, Delhi 1985.<\/p>\n

Reference Books<\/strong>
\n1. G.K. Batchelor, An Introduction to fluid Mechanics, Foundation Books, New Delhi 1994.
\n2. M.D. Raisinghania, fluid Mechanics, S. Chand and Company, Delhi.<\/p>\n

_________________________________________________________________________<\/p>\n

M. Sc. Mathematics<\/strong>
\nSemester-III<\/strong>
\nCore Elective<\/strong>
\nPaper \u2013 XIV (Code 3T4)<\/strong><\/h3>\n

General Relativity<\/strong><\/h3>\n

Unit 1
\nTensor Algebra, Riemannian geometry, Curvature Tensor: Covariant Curvature tensor, Ricci tensor, Einstein Tensor, The Bianchi identity.<\/p>\n

Unit 2<\/strong>
\nThe principle of covariance, The principle of equivalence, Geodesic principle, Newton\u2019s equations of motion as an approximation of geodesic equations, Poisson\u2019s equations as an approximation to Einstein field equations.<\/p>\n

Unit 3<\/strong>
\nGravitational field equations in free space, Exterior Schwarzchild\u2019s solution and its isotropic form, Birkhoff\u2019s theorem, Schwarzchild singularity, planetary orbit, Advance of Perihelion of a planet, Bending of light rays in the gravitational filed, Gravitational Red shift in the spectral lines.<\/p>\n

Unit 4<\/strong>
\nGravitational field equations for non empty space, Linearization of the field equations, The Weyl\u2019s solution of linearized Field equations, Interior Schwarzchild\u2019s solution.<\/p>\n

Text Book<\/strong>
\nIntroduction to General Relativity: Ronald Adler, Maurice Bezin and Manamen Schiffer, McGraw-Hill Kogakusha Ltd.<\/p>\n

References Books<\/strong>
\n1. Introduction to theory of relativity, Rosser W.G.V., ELBS(1972).
\n2. Lecture on General Relativity, Sonu Nilu Publication (2004) by T M Karade, G S Khadekar and Maya S Bendre
\n3. Relativity Special, General and Cosmology, Rindler W., Pub. Oxford University Press (2003).
\n4. The Classical Theory of Fields By Landau I.D. and Lifshitz E.M., Pub. Pergamon Press (1978).<\/p>\n

________________________________________________________________________<\/p>\n

M. Sc. Mathematics<\/strong>
\nSemester-III<\/strong>
\nCore Elective<\/strong>
\nPaper \u2013 XIV (Code 3T4)<\/strong><\/h3>\n

Algebraic Topology- I<\/strong><\/h3>\n

Unit 1<\/strong>
\nThe Elements of Homotopy theory: Introduction. Homotopic mappings. Essential and inessential mappings. Homotopically equivalent spaces. Fundamental group. Knots and related embedding problems. Higher homotopy groups. Covering spaces.<\/p>\n

Unit 2<\/strong>
\nPolytopes and triangulated spaces: En as a vector space over E1 .Barycentric coordinates. Geometrical complexes and polytopes. Barycentric subdivision. Simplicial mappings and simplicial approximation theorem.<\/p>\n

Unit 3<\/strong>
\nAbstract simplicial complexes. Embedding theorem for polytopes. Simplicial homology theory: Introduction. Oriented complexes. Incidence numbers. Chains, cycles and groups.<\/p>\n

Unit 4<\/strong>
\nDecomposition theorem for abelian groups. Betti numbers and torsion coefficients. Zero dimensional homology groups. Universal coefficients. Euler Poincare formula. Universal coefficients.<\/p>\n

Text Book\u00a0<\/strong>
\nTopology : J.G. Hocking and G.S. Young : Addison Wesley, 1961<\/p>\n

Reference Books\u00a0<\/strong>
\n1. Topology : J.R.Munkres, Prentice Hall, Second Edition, 2000
\n2. Basic Concepts of Algebraic Topology : Fred H.Croom , Springer Verlag 1978.<\/p>\n

_________________________________________________________________________<\/p>\n

M. Sc. Mathematics<\/strong>
\nSemester-III<\/strong>
\nCore Elective<\/strong>
\nPaper \u2013 XIV (Code 3T4)<\/strong><\/h3>\n

Non-linear Programming-I<\/strong><\/h3>\n

Unit 1\u00a0<\/strong>
\nThe non-linear programming problem and its fundamental ingredients. Linear inequalities and the theorem of the alternative. The optimality criteria of linear programming. Tucker\u2019s lemma and existence theorems.<\/p>\n

Unit 2<\/strong>
\nTheorems of the alternative Convex sets \u2013 Separation theorems. Convex and concave functions – basic properties and some fundamental theorems for convex functions. Generalised Gordan theorem. Bohnenblust \u2013 Karlin \u2013 Shapley theorem. Saddle point optimality criteria without differentiability \u2013 The minimization and the local minimization problems and some basic results.<\/p>\n

Unit 3<\/strong>
\nSufficient optimality theorem. Fritz John Saddle point necessary optimality theorem. Slater\u2019s and Karlin\u2019s constraint qualifications and their equivalence. The strict constraint qualification. Kuhn \u2013 Tucker saddle point optimality theorems. Differentiable concave and convex functions – Some basic properties. Twice differentiable convex and concave functions. Theorems in cases of strict convexity and concavity of functions.<\/p>\n

Unit 4<\/strong>
\nOptimality criteria with differentiability- Optimality theorems, Fritz John stationary point necessary optimality theorem. The Arrow \u2013 Hurwicz \u2013 Uzawa constraint qualification. Kuhn \u2013 Tucker stationary \u2013 point necessary optimality theorem.<\/p>\n

Text Book\u00a0<\/strong>
\nO.L. Mangasarian, Non- linear programming. Mc Graw Hill, New York.<\/p>\n

Reference Book\u00a0<\/strong>
\nMokhtar S. Bazaraa and C.M.Shetty, Non- linear programming, Theory and
\nAlgorithms, Wiley, New York.<\/p>\n

__________________________________________________________________________<\/p>\n

M. Sc. Mathematics<\/strong>
\nSemester-III<\/strong>
\nCore Elective<\/strong>
\nPaper \u2013 XIV (Code 3T4)<\/strong><\/h3>\n

Operator Theory<\/strong><\/h3>\n

Unit 1<\/strong>
\nBasic concepts about spectrum. Spactral properties of bounded linear operators. Further properties of resolvent and spectrum. Use of complex analysis in spectral theory.<\/p>\n

Unit 2<\/strong>
\nBanach Algebras. Further properties of Banach Algebras. Compact linear operators on normed spaces. Further properties of Compact linear operators. Spectral properties of compact linear operators.<\/p>\n

Unit 3<\/strong>
\nFurther spectral properties of Compact linear operators. Operator equations involving compact linear operators. Further theorems of Fredholm type. Fredholm alternative.<\/p>\n

Unit 4<\/strong>
\nSpectral properties of bounded self adjoint linear operators. Further Spectral properties of bounded self adjoint linear operators. Positive operators. Square roots of a positive operator. Projection operator. Further properties of projections. Spectral family. Statement of spectral representation theorem.<\/p>\n

Text Book<\/strong>
\nIntroductory Functional Analysis with Applications by E. Kreyszig, John Wiley and Sons<\/p>\n

Reference Book\u00a0<\/strong>
\nIntroduction to Functional Analysis by A.E.Taylor and D.C.Lay, John Wiley and Sons<\/p>\n","protected":false},"excerpt":{"rendered":"

RTMNU MSc Sem 3\u00a0New Syllabus 2020 | RTMNU M.Sc Revised Sem 3 Syllabus 2020 | Nagpur University M.Sc Syllabus Part 2| RTMNU Second Year M.Sc Syllabus RTM Nagpur University M.Sc Semester New Revised Syllabus is given below for Downloading. The students can Download the respective Syllabus from following given details. Just go through the given […]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6],"tags":[],"class_list":["post-363","post","type-post","status-publish","format-standard","hentry","category-rtmnu-syllabus"],"_links":{"self":[{"href":"https:\/\/mahabharti.in\/university\/wp-json\/wp\/v2\/posts\/363","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mahabharti.in\/university\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mahabharti.in\/university\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mahabharti.in\/university\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/mahabharti.in\/university\/wp-json\/wp\/v2\/comments?post=363"}],"version-history":[{"count":0,"href":"https:\/\/mahabharti.in\/university\/wp-json\/wp\/v2\/posts\/363\/revisions"}],"wp:attachment":[{"href":"https:\/\/mahabharti.in\/university\/wp-json\/wp\/v2\/media?parent=363"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mahabharti.in\/university\/wp-json\/wp\/v2\/categories?post=363"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mahabharti.in\/university\/wp-json\/wp\/v2\/tags?post=363"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}