Department of Mathematics

 Head of Department Dr. Lalita Dhurve Ph.D.,M.Sc. Faculties 1. Shri.- Prajendra Raghuwanshi(Guest faculty) M.Sc. 2. Miss.- Pooja Bhute (JBS) M.Sc. Cources B.Sc./M.Sc.

Cources

• B.Sc. / M.Sc.

Eligibility Criteria for B.Sc. / M.Sc.

1.  Eligibility Criteria for B.Sc.12th  in Maths Group
2. Eligibility Criteria for M.Sc. Graduate with Maths one subject

## B.Sc. I Year (Mathematics)

 Course Name Course Outcomes Major –I Paper Algebra , ,  Vector Analysis & Geometry  110163 Knowledge gained: Concept of rank and nullity of matrix and some theorems about Normal forms Certain relation between Eigen values and Eigen vectors Concept of gradient, divergent and curl Basic concept of product of four vectors and reciprocal vectors Vector differentiation Vector integration Gauss’s, Stoke’s and green’s theorem Tracing of conics, confocal conics Polar equation of conic The cone, the cylinder, the central conicoids, the paraboloid   Skills Gained: Solving problems using the powerful concept of application of matrices. Ability to understand vector differentiation and vector integration Gain practical knowledge of tracing of conics Ability to understand problem solving and gain knowledge of grad,       divergent and curl Competency developed: Vector integration and its application developed practical knowledge of area of curve in daily life Ability to handle daily life problems with the help of cone and cylinder etc.

 Major-II/Minor/ Open Elective – Calculus  &                  Differential Equations 110164/110165/110166 Knowledge gained: Successive differentiation. Leibnitz theorem, Taylor series Asymptotes, curvature Concavity and convexity, points of inflexion Tracing of curves Integration of Transcendental functions Definite integral Quadrature and rectification Differential equations of first order and higher degree Linear differential equations Method of variation of parameters Skills gained: Generalization of Concept like successive differentiation Understand practical knowledge of curve tracing Find the area of curves to understand the practical knowledge of definite integrals

B.Sc. IInd Year (Mathematics)

 Major –I Paper-I Abstract Algebra & Linear Algebra 210332 Knowledge gained: Recognize the algebraic structures as a group and classify them as ableian, cyclic and permutation groups etc. Link the fundamental concepts of groups and symmetrical figures. Analyze the subgroups of cyclic      groups.  Explain the significance of the notion of cossets, normal subgroups and quotient groups. The fundamental concept of rings, fields, sub rings, integral domains and the corresponding orphisms. Analyze whether a fini9te set of vectors in a vector space is linearly independent. Understand the linear transformations, rank and nullity,matrics of a linear transformations algebra of transformations and change of basic. Compute the characteristic polynomial, Eigen values, Eigen vectors and Eigen spaces as well as the geometric and the algebraic multiplicities of an Eigen value and apply the basis diagonalization result. Skills gained : Linear algebra helps you understand geometric concepts such as planes, in higher dimensions, and perform mathematical operations on them. The idea of groups used in abstract algebra has helped students in finding out areas and volumes of complex structures. Competency developed: Determining whether a transformation from one vector space V into another vector space W is linear.  Determining the general rule for a linear transformation, given its behavior on a basis of the domain space. Finding the standard matrix of a linear transformation. Proving that the kernel and the range of a general linear transformation are subspaces of the corresponding.

 Major II / minor /open elective  Advanced Calculus & Partial Differential Equations  210332 Knowledge gained: Convergence of sequences, absolute and conditional convergence Continuity and differentiability Concept of limit and continuity of functions of two variables Partial differentiation Change of variables, Taylor’s theorem, Jacobians Maxima and minima of functions of two variables. Beta and gamma functions Taylor’s theorem for functions of two variables. Euler’s theorem on homogeneous functions. Skills gained: Efficiency in handling various type of series for convergent or divergent nature Ability to understand the continuity and differentiability Idea about partial differentiation   Competency developed: Ability to solve maxima and minima for function of two variables. Ability to deal with homogeneous and non-homogeneous partial differential equation constant coefficients. Lagrange’s solution and standard forms Ability to use heat flow problems and transmission of signals.

## B.Sc. III Year (Mathematics)

 Course Name Course Outcomes Paper-I Linear Algebra and Numerical Analysis 310332 Knowledge gained: Concept of vector spaces, basis, finite dimensional, vector spaces Linear transformations and their representation as matrices Eigen values and Eigenvectors of a linear transformation and Diagonalization Inner product, solution of equations Interpolation Numerical Quadrature Linear Equations, ordinary Differential Equation Skill gained: Solving problems using the powerful concept of vector spaces Solving problems of linear transformations and their representation as matrices

 Ability to understand interpolation in practical way Competency gained: Understanding of vector spaces and inner product Applying the concept of interpolation and Quadrature in daily life problems such  engineering Understanding linear equations and ordinary differential equations Paper-II Real and Complex Analysis 310333 Knowledge gained: Concept of Riemann integral and some theorems Partial derivatives, Differentiability of real valued function of two variables Concept of improper integrals and their convergence Generalization of concepts like continuity and differentiability Metric spaces Concept of continuous functions, compact sets, connectedness Concept of analytic function, Cauchy-Riemann equation Skill gained: Idea about Riemann integration and some theorem like mean value theorem

 Ability to understand Matrix spaces, limit points, open and closed sets, Cauchy  sequences Geometric representation of complex numbers, idea about analytic function Competency developed: Ability to understand real number system and the relation between number line one to one correspondence of a function Paper-III Discrete Mathematics (Optional) 310346 Knowledge gained: Binary relations, equivalence relation, partial order relation Graph, multi graph and weighted graph Path and circuits Shortest paths, Eulerian paths and circuits Trees and their properties. Cutest and its properties, Planar graphs kuratowskis two graphs  Skill gained: Efficiency in handling with discrete structures Efficiency in set theory and binary relations Efficiency in notations of matching, ordering, planarity    Competency gained: Application to real life problems such as network theory Developed efficiency with planar graph Ability to apply combinatorial intuitions in network theory Ability to use graphs as unifying theme for various combinational problems Ability to deal with notations

M.Sc. I Semester (Mathematics)

 Course Code Course Name Course Outcomes 212100 Real Analysis-I Knowledge Gained: Definition and existence of Riemamm stieltjes integral Integration and differentiation of vector valued function Point wise and uniform convergence of sequence and series, cauchy criterion, weirtars M-test. Uniqueness theorem for power series abels and taubers theorems Function of several variables Inverse function theorem Implicit function theorem Stoke’s theorem   Skills gained: • Viewing C [0, 1], i.e., the space of continuous functions on [0, 1] as a metric space. • The notion of convergence in c [0, 1] and related theorems. • Differentiability of functions in several variables and their relation to partial derivatives. • Realizing the differentials in terms of geometric properties.    Competency developed: • Ability to handle convergence of series and sequence of functions. • Ability to differentiate functions in Rn. • Apply Implicit and inverse function theorem, moving towards calculus on manifolds. 212101 Topology-I Knowledge Gained: Definition and examples of topological spaces Closed sets, closure , dense subsets neighborhoods Accumulation points and derived sets Kuratowski closure operator and Neighbourhood systems Connected space Path connectedness Linear continuum Topologist’s sine curve Axioms T1 ,T2,  T3, spaces Metric topology Skills Gained:` Solving problems using powerful concept of groups, Sylows gro commutator groups Ability to understand the Galois groups theory, extended field the Invariants etc. Competency developed: Applying the concept of groups and fields in real life problem thinking logically. Implies facility in working with matrices, a concept that find a la number of applicants in real life including graphs and networks. 212102 Complex analysis-I Knowledge Gained: Concept of analytic function and power series and some theorems Concept of Cauchy Riemann equation, singularities Cauchy’s theorem, Cauchy integral formula and its applications Cauchy residue theorem, mobius transformations Concept of complex integration Concept of conformal mapping and its theorems Generalization of concept of continuity and differentiability Skills Gained: Understanding of Conformal and Isogonal mappings Development of functions into power series, classifying power seri Applications to counting Zeros and Poles Competency developed: Ability to handle competitive exams like I.E.S. , I.A.S., I.R.S. etc. Ability to handle NET, GATE, SLET exams Ability to gain the concept of limit and continuity Ability   to   understand complex integration and        some theor 212099 Advanced Abstract algebra-I Knowledge Gained: Concept    of   nilpotent group, modules,            Cauchy theorem, sylows theorems Series of groups Solvable groups Finite field Skills Gained: Solving problems using powerful concept of groups, Slows group commutator groups Ability to understand the Galois groups theory, extended field the Invariants etc. Competency developed: Applying the concept of groups and fields in real life problem thinking logically.   Implies facility in working with matrices, a concept that find a la number of applicants in real life including graphs and networks. 212103 Advanced Discrete Mathematics-I Knowledge Gained: Binary relations, equivalence relation, partial order relation Graph, multi graph and weighted graph Path and circuits Shortest paths, Eulerian paths and circuits Trees and their properties. Cutest and its properties, Planar graphs kuratowski’s two graphs Skills Gained: Efficiency in handling with discrete structures Efficiency in set theory and binary relations Efficiency in notations of matching, ordering, planarity   Competency developed: Application to real life problems such as network theory Developed efficiency with planar graph Ability to apply combinatorial intuitions in network theory Ability to use graphs as unifying theme for various combinational problems Ability to deal with notations

 Limit and continuity of functions on metric spaces Compactness and connectedness of metric spaces Definition and existence of Riemann – stieltjes integral Integration and differentiation of vector valued function Point wise and uniform convergence of sequences and series, Cauchy  criterion, weirtars M- test Measurable sets and its properties borel sets Lebesgue integral of bounded function Integration of non-negative function Skills Gained: Calculation of different matrices spend examining the compactness Calculation of limit of a function on metric spaces and examining continuity Computation of R-S integral Application of different test of convergence for sequences and se Computation of measures of sets Competency developed: Concepts gained are helpful in the exams NET, JRF and UPSC Understanding the concept of metric as a distance function Enhancing   the   topological properties     of     metric           spaces completeness, compactness and connectedness Understanding the concept of R-S integral and measure theory.

## M.Sc. II Semester (Mathematics)

 Course Code Course Name Course Outcomes 220334 Lebesgue Measure Knowledge gained: Students acquired basic knowledge of measure and integration theory. Analyze measurable sets and Lebesgue measure. Describe the Borel sets and Measureable functions. The student will be able to describe the structure of measurable functions. 220335 Topology-II Knowledge Gained: Topological spaces with their bases and topological properties. Axioms T1 ,T2,  T3, spaces Continuous function and countability axioms Connectedness, path connectedness and components Compactness with local and sequential compactness Separation axioms and product topology All above properties with respect to product spaces The definitions of standard terms in topology. Some important concepts like continuity, compactness, connectedness, projection mapping etc Using new ideas in mathematics and also help them in communicating the subject with other subjects. 220336 Complex analysis-II Knowledge gained: Weierstrass factorization theorem. Riemanan zeta function Riemans functional equation. Runge’s theorem, mittage-lefflers theorem. Power series method of analytic continuation. Schwartz reflection principle. Borel’s theorem, Dirichlet problem Green’s functions. Students will learn more about everywhere differentiable function and they will learn how it helps them to decide analyticity. 220333 Advanced Abstract algebra-II Knowledge gained: The students will be able to define the concept of module over a ring and will be able to give examples of this kind of algebraic structures. The students will be able to define the concept of Linear transforms, algebra of linear transformations & linear operators, Nilpotent transformations, Jordan blocks and forms. The students will be able to give detail proof and work with the concepts of Schurz lemma. The students will be able to apply the basic concept of modules, including uniform and primary modules. The students will be able to define and work the concepts of homomorphism. The students will be able to apply the basic concept of field theory, including field extensions and finite fields. 220337 Advanced Discrete Mathematics-II Knowledge gained: Understand the basic principles of sets and operations in sets. Demonstrate different traversal methods for trees and graphs. Write model problems in mathematical science using trees and graphs. Finite state machines and their transition table and diagrams Equivalence of finite state machines, finite automatea. Moore and mealy machines Grammars and languages- phrase-Structure Grammars. The reverse polish notation.

M.Sc. III Semester (Mathematics)

 Course Cod Course Name Course Outcomes 320337 Functional Analysis-I Knowledge Gained: Properties of finite dimensional normed space with concept and dealing with examples. Concept of uniform boundedness principles and its applications. Inner product space and their elementary properties. Hahn Banach theorem for normed linear spaces. Study of orthonormal sets, Bessel’s inequality. Concept of  Cantor’s intersectionit theorem 320338 Advanced Functional Analysis-I Knowledge Gained:   Concept of normed space, Gateaux derivative, fretchet derivative. Concept of fixed point theorem and their applications, Banach contraction principle and its generalization. Application of Banach contraction principle. Definitions and examples of topological vector space , covex and their properties, Minkowski’s functional, subsace. Product space and quotient space of a topological vector space.

 320341 Algebric Topology-I Knowledge gained:     • Concept of structure of homology groups, connected simplex.    • Concept of chain map, simplicial map, examples.      • Concept of homology group of n-dimensional sphere.   • Concept of Geometrically independent set, k simplex. Face of a simplex, triangulation of sphere, Mobius band. Torusn Klein bottle. 320344 Machenics –I Knowledge gained:   Concept of holonomic and non-holonomic systems. Constraint and its classifications. Possible and virtual displacement. Larange’s equation of first kind. Concept of uniqueness of solution Hamilton’s variables, Hamilton’s canonical equations. Shortest distance, minimum surface of revolution. Isopreimetric problems. 320350 Integral Transform-I Knowledge gained:   Basic concept of Laplace transform, inverse Laplace transform Application of Laplace transform to solution of differential equation  and integral equations with convolution type kernel Application of L.T. to the solution of initial –boundary valuel problem, solution of heat equation, wave equation Concept of Fourier transform, inverse Fourier transform Application of F.T. to the solution of initial- boundary value problem, solution of heat equation, diffusion equation, w equations Skills for solving various types of differential equations, , initial values problems. Skills for solving various real life problems related to physical science mathematical sciences, engineering etc.

M.Sc. IV Semester (Mathematics)

 Course Cod Course Name Course Outcomes 420346 Applied Functional Analysis Knowledge Gained: Hilbet space obtained from Hilbert space and tensor product of Hilbert space , convex set and projections. Weak convergence, weak compactness properties. Minkowski functional support plane through a boundary point, support mapping. Linear operators and their adjoints, bounded and unbounded operators. Spectral theory of opertaors.

 420347 Advanced Functional Analysis-II Knowledge Gained: Finite dimensional topological vector space, locally convex topological vector space. Normable and metrizable topological vector spaces. Frechet space, uniform boundedness principle, open mapping theorem Banach- alaoglu theorem. Points and external sets krein miliman’s theorem.

 420350 Algebric Topology-II Knowledge Gained:   The fundamental group and its properties. Introduction homotopy definitions and examples. Simply connected space. Path lifting and homotopy lifting property. Covering pojections. Applications of homotopy lifting theorem. The monodromy theorem. 420353 Machenics-II Knowledge Gained: Fundamental lemma of calculus of variations. Euler’s equations for one dependent function and its generalization. Conditional extremum under geometry constraints. Hemilton – Jacobi equation, Jacobi theorem. Poisson brackets invariance of Lagrange brackets.

0

 420359 Integral Transform-II Knowledge Gained:   To apply the Fourier transform methods for solving functions. Applications of Laplace transform to boundary value problems. Electric circuits problems, related to application of Electric circuits. Applications to beams. Wave equations. Fourier cosine and sine transform.
Back to Department