MATHEMATICS B.A./B.Sc. Mathematics Course Structure B.A./B.Sc. Mathematics (First Year) Duration: One Year With effect from June 2008 Papers: Theory – Paper I Theory – Paper II Theory – Paper III
Periods Per Week 04
Algebra and Graph Theory Calculus and 04 Differential Equations Vector Analysis and 04 Geometry (Only for Science Students)
Marks 100 100 100
Note: Maximum Number of periods required for each paper is 120
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BA/B.Sc. (First Year) Mathematics Paper – I Algebra and Graph Theory Algebra: Unit1: Cartesian product of two sets, Functions or mappings, binary operations, Relations, Equivalence relations, Equivalence classes, partitions, Unit 2: Binary operation on a set Algebraic structure, Definition of group, finite and infinite groups, order of a finite group, General properties of groups, Definitions of a group based upon left axioms compositions tables for finite sets, Addition modulo m, Multiplication modulo p, Residue classes of the set of integers, An alternative set of postulates for a group, Unit 3: Permutations, group of permutations, cyclic permutations, Even and odd permutations. Integral powers of an element of a group order of an element of a group, Isomorphism of groups. Unit 4: Complexes and subgroups of a group, Cosete, Relation of congruence modulo Lagrange’s theorem Euler’ theorem, Format’s theorem, Cyclic groups. Graph Theory : Unit V : Definition of graph, applications of graphs, incidence and degree, isolated vertex, pendant vertex and null graph, isomorphism, subgraphs, walks, paths and circuits, connected graphs, components Euler graphs,
Hamiltonian paths and circuits, Incidence matrix and
adjacency matrix. (Only examples) Text Books : 1)
Modern Algebra (36th Edition, 1998) By A.R. Vasishtha ,Krishna Prakashan Median (p) Ltd. Meerut Scope : Chapter 1 : Art 18, to 40,Chapter 2 : Art 1, to 15 Chapter 2 : Art 16 to 19, art 22 to 29, art 32 to 33
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Graph Theory with applications to Engineering and computer Sciences By Narsingh Deo ,Prentice Hall of India,New Delhi 2005 Scope :Chapter 1: 1.1, 1.2, 1.3, 1.4, 1.5,Chapter 2: 2.1, 2.2, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9,Chapter 3: 3.1, 3.2,Chapter 7 : 7.1, 7.9
References: 1) I.N. Herstein, Topics in Algebra, Wiley Eastern Ltd. New Delhi 1975 2) C.L. Liu, Elements of Discrete Mathematics, McGraw-Hill (Second Edition) International Edition computer science series 1986. 3) P.B.Bhattachary, S.K.Jain, S.R.Nagpal, Basic Abstract Algebra (2nd edition). Cambridge university press, Indian Edition 1997.
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BA/B.Sc. (First Year) Mathematics Paper – II Calculus and Differential Equations Calculus: Unit1 :
Successive differentiation : Higher
order derivatives. Calculation of the nth
derivative. Determination of nth derivative of rational functions. The nth derivative of the products of the power of sines and cosines. Leibnitz’s Theorem. Expension of Functions : Maclaurin’s Theorem. Taylor’s Theorem. Mean Value Theorems : Rolle’s Theorem, Lagrange’s mean value theorem. Meaning of the sign of derivates. Cauchy’s mean value theorem. Higher derivative. Generalized mean value theorem. Unit 2: Functions of two variables. Neighbourhood of a point (a,b) continuity of a function of two variables . Limit of
a function of two variables. Partial derivatives. Homogenous
functions. Euler’s theorem on homogeneous functions. Theorem on total differentials. Composite functions. Curvature, curvature of a circle. Radius of curvature, length of arc as a functions (derivative of arc) Radius of curvature : Cartesian equations, parametric equations polar equations, pedal equations. Center of curvature. Differential Equations : Unit 3 : Equations of the first order and of the first degree. Ordinary differential equations (Def), order and degree of a differential equations (Def.), Equations homogeneous in x and y. Non –homogeneous equations conditions that an equations of the first order be exact. Rule for finding the solution of an exact differential equation. Integrating factors found by inspection. Rules for finding integrating factors. Rules I, II (Only). Linear equations. Equations reducible to the linear form. Unit 4: Linear differential equations with constant co-efficient The Complementary function. The particular integral. The complete integral. The linear equation with constant co-efficients and second member zero. Case of auxiliary equations having equal roots, imaginary roots, The symbol D. The linear equation with constant coefficient and second member of a functions of x. The symbolic function 1/F(D) Short methods of finding the particular integral in eax (where a is any constant), Xm (where m is+positive integer), Sin ax, Cos ax, eax v (where V is any function of x), X.V., Unit 5 : Linear equations with variable co-efficients The homogeneous linear equations Integral equations Corresponding to a term of form Equations reducible to the homogeneous linear form. in the second member. Exact differential -3-
equations. Criterion of an exact differential equations. Simultaneous differential equations which are linear. Simultaneous equations of the first order. Text Books (For unit 1 and unit 2) 1. Differential calculus, Shanti Narayan and P.K. Mittal S.Chand and company (7361), Ramnagar,New Delhi –110055 (Revised Edition 2005 Reprint 2007) Scope :- Chapter 5 : Articles 5.1, 5.2, 5.3, 5.4, 5.5(Complete),Chapter 6: Articles 6.1, 6.2 (Complete),Chapter 8: Articles 8.1 to 8.6, 8.6,Chapter 11 : Articles 11.1 to 11.6, 11.8, 11.8.1, 11.9.1, to 9.3.,Chapter 14: 14.1, 14.2, 14.3, 14.3.1, 14.3.2, 14.5 Text Books : (For Unit 3, 4 and 5), 1.
Introductory Course in Differential Equations By Daniel A.Murray,Orient Longman Co. (India) 2002,160, Anna Salai, Chennai600002 Scope : Chapter I : Articles 1,Chapter II : Articles 7, 9, 10, 11, 12,13, 16, 17, 20,21,Chapter VI : CompleteChapter, VII : Article 65, 70, 71,Chapter VIII : Article 73, 74,Chapter XI : Article 98, 74
Reference Books : (for Unit 1 and 2) 1. Text book on differential calculus by Gorakh Prasad, Pothisahala private limited. Allahabad 2. Theory and problems of Advanced calculus By Murray R. Spiegel Schawm Publishing (O.New york). 3. An introduction to Real Analysis By P.K. Jain and S.K. Kaushi K. S.Chand and Co.New-Delhi. Reference Books : [For unit 3, 4, and 5] 1. Differential Equations, G.F. Simmons,Tata McGrawhill, 1972 2. Differential Equations, N.P. Bali,Anmol publications Pvt.Ltd. New Delhi – 110002 (India) 3. Ordinary and Partial Differential Equations, M.D. Raisinghania. S.Chanel and company Ltd.Ramnagar, New Delhi – 110055 4. Lectures on (Calculus and Differential Equations) By T.M. Karade and Maya S.Bendre,Orion Printers Private Limited. Hyderabad – 500 004,[Sonu Nilu (Einstin Foundation International]
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B.A./B.Sc. (First Year) Mathematics Paper – III Vector Analysis and Geometry Unit 1: Vector Analysis Functions of scalar Variables: Vector Functions of single scalar variable, Limit of a vector function, continuity, Derivative, Derivability in relations to algebraic operations, Constant Vectors, Derivative of second & higher orders, Indefinite Integration, Vector functions of several scalar variables, Decomposition, Limit of a vector function, Continuity of partial Derivatives. Unit 2: Differential Operators: Point functions, Limits & Continuity, Directional derivatives, Cartesian Formulation. Directional Derivatives of Scalar & Vector point functions. Gradient of a scalar point function. Geometrical interpretation grad F. character of gradient as point function. The operators x & a.x. Divergence & Curl of sums & products. Second order differential operatiors, The laplacian operator x2. Unit 3: Geometry:Polar equation to a Conic, Tangent and normal. General Equation of second degree. Confocal Conics. Unit 4: Analytical Solid Geometry: The plane, Right line. Unit 5: Analytical Solid Geometry:The sphere, cones, Cylinders. For Unit 1 and 2: Text Book:A text book of Vector Calculus- Shanti Narayan and J.N. Kapoor, S.Chand and Co. Scope: Chapter 1 :Complete ,Chapter 2: 51.1, 51.2, 52.1, 52.2, 52.3, 53.3, 53.2, 54, 55.1, 55.2, 56, 56.1, 56.2, 57, 57.1, 57.2, 57.3, 58, 58.1, 58.2, 59, 60, 61, 62, 62.1, 62.2, 63, 63.1, 63.2, 64 to 67. For Unit 3: Text Book: The elements of co-ordinate Geometry by S.L. Loney S.Chand and Company Scope: Chapter XIV: Art. 335, 336, 339 to 342, 345Chapter XV:Art. 348 to 358 Chapter XVII: Art. 415 to 418 For Unit 4 and 5 : Text book: Analytical Solid Geometry by Shanti Narayan ,S.Chand and Company Scope: Chapter 2: Art. 2.1 to 2.3, 2.31, 2.32, 2.4, 2.41, 2.42, 2.5, 2.7 Chapter 3: Art. 3.1, 3.11 to 3.14, 3.2 to 3.4, 3.6, 3.7 Chapter 6: Art. 6.11 to 6.13, 6.2, 6.31 to 6.33, 6.4, 6.41, 6.5 to 6.71 Chapter7:Art.7.1,7.2,7.3,7.7,7.71,7.8,7.82 Reference Book: 1) Advanced Calculus by Murry R. Spiegel Schaum’s Outline Series. 2) Text Book on coordinate geometry by Gorakha Prasad and H.C. Gupta. Pothishala Ptv. Ltd. Allahabad. 3) Lectures on Vector Analysis and Geometry by T.M.Karde and Maya S. Bedre, -5-
Pattern of the question Paper B.A./B.Sc. (First Year) Mathematics Papers – 1st, 2nd and 3rd (Theory) Maximum Marks: 100 N.B. – 1) All Questions are Compulsory 2) Figures to the right indicate full marks.
Duration 3 Hours
Q.1 A) Attempt any one of the following a) Theory b) Theory B) Solve any two of the following a) Theory b) Problem c) Problem
Marks
Q.2 A) Attempt any one of the following a) Theory b) Theory B) Solve any two of the following a) Theory b) Problem c) Problem
Marks
10 10 05 05 05
10 10 05 05 05
Q.3 A) Attempt any one of the following a) Theory b) Theory B) Solve any two of the following a) Theory b) Problem c) Problem
Marks
Q.4 A) Attempt any one of the following a) Theory b) Theory B) Solve any two of the following a) Theory b) Problem c) Problem
Marks
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10 10 05 05 05
10 10 05 05 05
Q.5 A) Attempt any one of the following a) Theory b) Theory B) Solve any two of the following a) Theory b) Problem c) Problem
Note: 1) Questions to be set on each unit as below: Unit I : Question 1 only Unit II : Question 2 only Unit III : Question 3 only Unit IV : Question 4 only Unit V : Question 5 only
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Marks 10 10 05 05 05
