CORE COURSES

B.Sc. (Honours)-Mathematics

Semester-I

MATH.( Hons ) -CC-1: Calculus-I

(Total Marks: 100) Part-I (Marks: 75) Time – 3 hrs.

(Theory:Sem-60 Marks+Mid-Sem-15 Marks, Credit – 6 [4(Th.) + 2 (Pr.)]

Unit-I

Hyperbolic functions, higher order derivatives, Leibniz rule and its applications to problems of the type eax+bsinx, eax+bcosx, (ax+b)nsinx ,(ax+b)ncosx, concavity and inflection points, asymptotes. Unit-II

Curve tracing in Cartesian coordinates, tracing in polar coordinate sofstandard curves, LHospital”s rule, applications in business, economics and life sciences, Reduction formulae, Derivations and illustrations of reduction formulae of the type ∫ sinnxdx, ∫ cosnxdx,∫ tannxdx,∫ secnxdx,∫(logx)ndx,∫sinnxcosnxdx Unit-III Volumes by slicing, disks and washers methods, volumes by cylindrical shells, parametric equations, parameterizing a curve, arc length, arc length of parametric curves, area of surface of revolution. Unit-IV

Techniques of sketching conics ,reflection properties of conics, rotation of axe sand second degree equations, classification in to conics using the discriminant, polare quations of conics. Sphere, Cone, Cylinder, Central Conicoids.

Unit-V

Triple product, introduction to vector functions, operations with vector-valued functions, limits and Continuity of vector functions, differentiation and integration of vector functions, tangent and normal Components of acceleration.

Math. Part-II C – I (Practical,Marks:25)

List of Practicals (Using any software)

Practical/ Lab work to be performed on a Computer.

1. Plotting the graph soft the function seax+b,log(ax+b),1/(ax+b),sin(ax+b),cos(ax+b),|ax+b| and to illustrate the effect of a and both the graph. 2. Plotting the graphs of the polynomialofdegree 4 and 5, the derivative graph, the second derivative graph and comparing them. 3. Sketching parametric curves (Eg. Trochoid, cycloid, epicycloids, hypocycloid). 4. Tracing of conics in cartesian coordinates/ polar coordinates. 5. Sketching ellipsoid, hyper boloid of one and two sheets, ellipticcone, elliptic, paraboloid, hyperbolic paraboloid using Cartesian coordinates. 6. Matrix operation (addition, multiplication, inverse, transpose).

Books Recommended:

1. M. J. Strauss, G. L.BradleyandK.J.Smith,Calculus,3rd Ed., Dorling Kindersley (India) P. Ltd. (Pearson Education), Delhi, 2007.Chapters:4 (4.3,4.4,4.5 & 4.7), 9(9.4), 10(10.1-10.4).

2. H.Anton,I.BivensandS.Davis,Calculus,7thEd.,JohnWileyandSons(Asia)P.Ltd., Singapore, 2002.Chapters:6,(6.2-6.5),7(7.8),8(8.2-8.3,Pages:532-538),11(11.1),13(13.5)

3. Analytical Geometry of Quadratic Surfaces, B.P.Acharya and D. C. Sahu, Kalyani Publishers, New Delhi, Ludhiana.

4. Elements of vector calculus by Sarana & Prasad+878 Books for Reference:

1. G.B.ThomasandR.L.Finney, Calculus,9th Ed., Pearson Education, Delhi, 2005. 2. R. Courantand F. John, Introduction to Calculus and Analysis (Volumes I & II),

Springer-Verlag, New York, Inc., 1989. 3. Text Book of Calculus, Part-II- Shantinarayan, S.Chand &Co., 4. Text Book of Calculus, Part-III-Shantinarayan, S.Chand &Co., 5. Shanti Narayan and P.K.Mittal-Analytical Solid Geometry,

S.Chand&CompanyPvt.Ltd.,New Delhi.

CORE COURSES

MATHMATICS- SEMESTER- 1

Math ( Hons ) C-1I : Algebra-I Time – 3 hrs.

Total Marks: 100

Theory:Sem.-80 Marks + Mid- Sem: 20 Marks Credit - 6

5Lectures, 1 Tutorial (perweekperstudent)

Unit-I

Polar representation of complex numbers, n-throots of unity, DeMoivres the oremforration alindices and it sapplications.

Unit-II

Equivalence relations, Functions, Composition of functions, Invertible functions, One to one correspondence and cardinality of a set, Well- ordering property of positive integers, Division algorithm, Divisibility and Euclidean algorithm,.

Unit-III

Systems of linear equations, row reduction and echelon forms, vector equations, the matrix equation Ax=b, solution sets of linear systems, applications of linear systems, linear independence.

Unit-IV

Introduction to linear transformations, matrix of a linear transformation, inverse of a matrix, characterizations of invertible matrices

Sub spaces of Rn, dimension of sub spaces of Rn

Unit-IV

Congruence relation between integers, Principles of Mathematical Induction, statement of

Fundamental theorem of Arithmetic Rank of a matrix, Eigenvalues, EigenVectors and Characteristic Equation of a matrix.

BooksRecommended:

1. L.V.Ahlfors,ComplexAnalysis,McGraw-Hill(InternationalStudentEdn.)

2. TituAndreescuandDorinAndrica,ComplexNumbersfromAtoZ,Birkhauser,2006.Chapter:2

3. Edgar G. Goodaire and Michael M. Parmenter, Discrete Mathematics with Graph Theory, 3rdEd., Pearson Education (Singapore) P. Ltd., Indian Reprint, 2005. Chapters:2 (2.4), 3,4 (4.1-4.1.6,4.2-4.2.11,4.4(4.1-4.4.8),4.3-4.3.9,5(5.1-5.1.4).

4. David C.Lay, Linear Algebraand its Applications, 3rdEd., Pearson Education Asia, Indian Reprint

5. V. Krishanmurthy, V. P. Mainra & J. B. Arara – An Introduction of Linear Algebra Chapters: 1(1.1-1.9), 2(2.1-2.3,2.8,2.9), 5(5.1,5.2)

Semester-II

CORE COURSES

Math ( Hons) CC-III :Real Analysis (Analysis-I) Time – 3 hrs.

TotalMarks:100

Theory:Sem.-80Marks+Mid-Sem:20Marks Credit - 6

5 Lectures, 1Tutorial(per week per student)

Unit-I

Review of Algebraic and Order Properties of R, Neighborhood of a point in R,Idea of countable sets, uncountable sets and uncountability of R. Bounded above sets, Bounded below sets, Bounded Sets ,Unbounded sets, Suprema and Infima.

Unit-II

The Completeness Property of R, The Archimedean Property, Density of Rational (and Irrational ) numbers in R, Intervals . Limi points of a set, Isolated points, Illustrations of Bolzano –Weierstrass theorem for sets.

Unit-III

Sequences, Bounded sequence, Convergent sequence, Limit of a sequence. Limit Theorems, Monotone Sequences, Monotone Convergence Theorem. Subsequences, Divergence Criteria, Monotone Subsequence Theorem (statement only) ,Bolzano Weierstrass Theorem for Sequences. Cauchy sequence, Cauchys Convergence Criterion.

Unit-IV

Infinite series, convergence and divergence of infinite series, Cauchy Criterion, Tests for convergence: Comparison test, Limit Comparison test, Ratio Test, Cauchys n-th root test,

Integral test.

Unit-V

,Alternating series,Leibniz test, Absolute and Condition al convergence.

Books Recommended:

1.G.DasandS.Pattanayak,FundamentalsofMathematicsAnalysis,TMHPub-lishingCo.,Chapters:2(2.1to2.4,2.5to2.7),3(3.1-3.5),4(4.1to4.7,4.10,4.11,4.12,4.13).

Books for References:

1.R.G.BartleandD.R.Sherbert,IntroductiontoRealAnalysis,3rdEd.,JohnWileyandSons(Asia)Pvt.Ltd., Singapore, 2002.

2.GeraldG.Bilodeau , Paul R.Thie,G.E.Keough,AnIntroductiontoAnalysis,2nd Ed.,Jones&Bartlett,2010.

3.BrianS.Thomson,Andrew.M.BrucknerandJudithB.Bruckner,ElementaryRealAnalysis,PrenticeHall,2001.

4.S.K.Berberian,AFirstCourseinRealAnalysis,SpringerVerlag,NewYork,1994.

5.S.C.MallikandS.Arora-MathematicalAnalysis,NewAgeInternationalPublications.

6.D.SmasundaramandB.Choudhury-AFirstCourseinMathematicalAnalysis,NarosaPublishingHouse.

7.S.L.GuptaandNishaRani-RealAnalysis,VikasPublishingHousePvt.Ltd.,NewDelhi. 8. R.B. Dash & D.D. Dalai – A Course on Mathematical analysis, Kalyani Publisher

CORE COURSES

Semester-II

Math ( Hons ) – CC -IV : DifferentialEquations

Time – 3 hrs. Full Marks:100 [60 (Sem.) + 15 )In.) + 25 (Pr.)] Credit – 6 [4(Th.) + 2 (Pr.)]

Part-I (Marks:75)

Theory: 60 Marks + Mid-Sem:15 Marks

04 Lectures (per week per student)

Unit-I

Differential equations and mathematical models. First order and first degree ODE (variables separable, homogeneous, exact ,and linear). Equations of first order but of higher degree.

Applications of first order differential equations( Growth, Decay and Chemical Reactions, Heatflow, Oxygen debt, Economics).

Unit-II

Second order linear equations ( homogeneous and non-homogeneous ) with constant coefficients, second order equations with variable coefficients, variation of parameters, method of undetermined coefficients

Unit-III

Euler’s equations reducible to linear equations with constant coefficients, Euler’s equation. Applications of second order differential equations.

Unit-IV

Power series solutions of second order differential equations.

Unit-V

Laplace transforms and its applications to solutions of differential equations.

Part-II C – IV (Practical:Marks:25)

List of Practicals (Using any Software)

Practical /Lab work to be performed on a Computer.

1. Plotting of second order solution offamily ofdifferentialequations. 2. Growthmodel(exponentialcaseonly). 3. Decaymodel(exponentialcaseonly). 4. Oxygendebtmodel. 5. Economicmodel.

Book Recommended:

1.J.SinhaRoyandS.Padhy,ACourseofOrdinaryandPartialDifferentialEquations,KalyaniPublishers,NewDelhi.Chapters:1,2(2.1to2.7),3,4(4.1to4.7),5,7(7.1-7.4),9(9.1,9.2,9.3,9.4,9.5,9.10,9.11,9.13).

Books forReferences:

1.MartinBraun,DifferentialEquationsandtheirApplications,SpringerInternational.

2.M.D.Raisinghania-AdvancedDifferentialEquations,S.Chand &CompanyLtd.,NewDelhi.

3.G.DennisZill-AFirstCourseinDifferentialEquationswithModellingApplications,Cengage Learning India Pvt.Ltd.

4.S.L.Ross,DifferentialEquations,JohnWiley&Sons,India,2004

MATHMATICS- SEMESTER - 1 / III (Interdisciplinary):

GE-I: Calculus and Ordinary Differential Equations Time – 3 hrs. F.M.–100 [80(End Sem) + 20(Mid Sem))] Credit - 6

Unit-I Curvature, Asymptotes, Tracing of Curves (Cartenary, Cycloid, Folium of Descartes, Astroid, Limacon, Cissoid & loops), Rectification, Quardrature, Volume and Surface area of solids of revolution.

Unit-II

Sphere, Cones and Cylinders, Conicoid.

Unit-III

Explicit and Implicit functions, Limit and Continuity of functions of several variables, Partial derivatives, Partial derivatives of higher orders, Homogeneous functions, Change of variables, Mean value theorem, Taylors theorem and Maclaurins theorem for functions of two variables. Maxima and Minima of functions of two and three variables, Implicit functions, Lagranges multipliers. Multiple integrals.

Unit-IV

Ordinary Differential Equations of 1st order and 1st degree (Variables separable, homogenous, exactand linear). Equations of 1st order but higher degree.

Unit-V

Second order linear equations with constant coefficients, homogeneous forms, Second order equations with variable coefficients, Variation of parameters. Laplace transforms and its applications to solutions of differential equations.

Books Recommended:

1. Advanced Higher Calculus (Vidyapuri Dr. Ghanasyam Sen & Others) Ch- 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17

2. B. P. Acharya and D. C. Sahu-Analytical Geometry of Quadratic Surfaces, Kalyani Publishers, New Delhi, Ludhiana. Ch. (2,3,4)

3. J. SinharoyandS.Padhy-A Course of Ordinary and Partial Differential Equations, Kalyani Publishers.Chapters:2(2.1to2.7),3,4(4.1to4.7),5,9(9.1,9.2,9.3,9.4,9.5,9.10,9.11,9.13).

BooksforReferences:

1. Shanti Narayan and P.K.Mittal-Analytical Solid Geometry, S. Chand & Company Pvt. Ltd., NewDelhi.

2. David V.Weider-AdvancedCalculus,DoverPublications.

3. Martin Braun-DifferentialEquationsandtheirApplications-MartinBraun,SpringerInternational.

4. M. D. Raisinghania- Advanced Differential Equations, S. Chand & Company Ltd., NewDelhi 5. G. Dennis Zill-A First Course in Differential Equations with Modelling Applications, Cengage Learning India Pvt. Ltd.

MATH.-DSC-I, SEM-I IS SAME AS MATH.-CC-I,

SEM-I

MATHMATICS- SEMESTER – 1I / IV (Interdisciplinary):

GE-II: : Linear Algebra and Advanced Algebra

Time – 3 hrs. F.M. – 100 [80 (Th.) + 20 (Pr.)] Credit - 6 Unit-I

Vector space, Subspace, Span of aset, Linear dependence and Independence, Dimensions and Basis. Linear transformations, Range, Kernel, Rank, Nullity, Inverse of a linear map, Rank-Nullity theorem.

Unit-II

Matrices and linear maps, Rank and Nullity of a matrix, Transpose of a matrix, Types of matrices. Elementary row operations, System of linear equations, Matrix inversion using row operations, Determinant and Rank of matrices, Eigenvalues, Eigenvectors, Quadratic forms.

Unit-III

Group Theory: Definition and examples, Subgroups, Normal subgroups, Cyclic groups, Cosets, Quotient groups, Permutation groups, Homomorphism.

Unit-IV

Ring Theory: Definition and examples, Some special classes of Rings, Ideals, Quotientrings ,Ring homomorphism. Isomorphism theorems.

Unit-V

Zero divisors, Integral domain, Finite fields, Finite field Z/pZ, Field of quotients of an Integral domain, Polynomial ring, Division algorithm, Remainder theorem, Factorization of polynomials, irreducible and reducible polynomials, Primitive polynomials, Irreducibility tests,Eisenstein Criterion.

BooksRecommended:

1.V.Krishnamurty,V.P.Mainra,J.L.Arora-AnintroductiontoLinearAlgebra,AffiliatedEast-WestPressPvt.Ltd.,NewDelhi,Chapters:3,4(4.1to4.7),5(except5.3),6(6.1,6.2,6.5,6.6,6.8),7(7.4only).

2. I.N Iterstein, Topics in Algebra

Ch-1(1.3 only), 2 (2.1 to 2.6;2.7 excluding application, 2.10), 3 (3.1 to 3.6, 3.9, 3.10)

BooksforReferences:

1. I.H.Seth-Abstract Algebra, Prentice Hall of India Pvt.Ltd.,NewDelhi.Chapters:13,14,15,16, 17,18,19,20.

2.RaoandBhimasankaran-LinearAlgebra,HindustanPublishingHouse.

3.S.Singh-LinearAlgebra,VikasPublishingHousePvt.Ltd.,NewDelhi.

4.GilbertStrang-LinearAlgebra&itsApplications,CengageLearningIndiaPvt.Ltd.

5.Gallian-ContemporaryAbstractAlgebra,NarosapublishingHouse.

6.Artin-Algebra,PrenticeHallofIndia.

8.V.K.KhannaandS.K.Bhambri-A Course in Abstract Algebra, Vikas Publishing House Pvt.Ltd.,NewDelhi.

MATH.-DSC-II, SEM-II IS SAME AS MATH.-CC-IV, SEM-II

CHOICE BASED CREDIT SYSTEM SYLLABUS

Core Courses Semester-III MATHEMATICS ( Hons ) CC-V-3.1:Theory of Real Functions (Analysis-II)

TotalMarks:100

Theory: 80Marks+Mid-Sem:20Marks

5Lectures, 1Tutorial (per week per student)

Unit-I

Limits of functions(Ɛ -δapproach),sequential criterion for limits ,divergence criteria. Limit theorems, onesided limits. Infinite limits and limits at infinity. Continuous functions, sequential criterion for continuity and discontinuity.

Unit-II

Algebra of continuous functions. Continuous functions onan interval, intermediate value

theorem, location of roots theorem, preservation of intervals theorem. Uniform continuity,non-uniform continuity criteria, uniform continuity theorem.

Unit-III

Differentiability of a function at a point and in Zan interval, Caratheodorys theorem ,algebra of differentiable functions. Relative extrema ,interior extremum theorem.Rollestheorem,Meanvaluetheorem,intermediatevaluepropertyofderivatives.

Unit-IV

Darbouxs theorem. Applications of mean value theorem to inequalities and approximation of polynomials, Taylors theorem to inequalities.

Unit-V

Cauchys mean value theorem. Taylors theorem with Lagranges form of remainder, Taylors theorem with Cauchys form of remainder, application of Taylors theorem to convex functions, relative extrema.TaylorsseriesandMaclaurinsseriesexpansionsofexponentialandtrigonometricfunctions,ln(1+x),1/(ax+b)and(1+x)n.

Book Recommended:

1. G.DasandS .Pattanayak, Fundamentals of Mathematics Analysis, TMH Pub-lishing Co.,Chapters:6(6.1-6.8),7(7.1-7.7),

2. R.B Dash & D.K Dalai – A Course on Mathematical Analysis, Kalyani Publisher

Books for References:

1.R.BartleandD.R.Sherbert,IntroductiontoRealAnalysis,JohnWileyandSons,2003.

2.K.A.Ross,ElementaryAnalysis:TheTheoryofCalculus,Springer,2004.

3.A.Mattuck,IntroductiontoAnalysis,PrenticeHall,1999.

4.S.R.GhorpadeandB.V.Limaye,ACourseinCalculusandRealAnalysis,Springer,2006.

Core Courses Semester-III MATHEMATICS ( Hons ) CC-VI -3.2:Group Theory (Algebra-II)

Total Marks :100

Theory:80Marks+Mid-Sem:20Marks

5Lectures, 1Tutorial (per week per student)

Unit-I

Symmetries of a square,Dihedral groups,definition and examples of groups including permutation groups and quaternion groups(illustration through matrices),elementary properties of groups. Subgroups and examples of subgroups

Unit-II

Centralizer, normalizer, center of agroup, product of two subgroups, Properties of cyclic groups, classification of subgroups of cyclic groups. Unit-III

Cycle notation for permutations, properties of permutations, even and odd permutations, alternating group, properties of cosets, Lagranges theorem and consequences including FermatsLittle theorem. Unit-IV

Externaldirect product of a finite number of groups, normal subgroups, factorgroups, Cauchys theorem for finite abelian groups.

Unit-V

Group homomorphisms, properties of homomorphisms ,Cayleys theorem, properties of isomorphisms ,First, Second and Third isomorphism theorems.

Book Recommended:

1.JosephA.Gallian,ContemporaryAbstractAlgebra(4thEdn.),NarosaPublishingHouse,NewDelhi.

Books for References:

1.JohnB.Fraleigh, A First Course in Abstract Algebra, 7th Ed., Pearson, 2002.

2.M.Artin, Abstract Algebra, 2nd Ed., Pearson, 2011.

3.JosephJ.Rotman, AnIntroductiontotheTheoryofGroups,4thEd.,SpringerVerlag,1995.

4.I.N.Herstein, Topics in Algebra, Wiley Eastern Limited, India, 1975.

Ch 2 (2.1 to 2.7, 2.9,2.10,2.13)

Core Courses Semester-III MATHEMATICS ( Hons ) CC-VII -3.3:Partial Differential Equations and Systems of Ordinary Differential

Equations

(TotalMarks:100)

Part-I(Marks:75)

Theory:60Marks+Mid-Sem:15Marks

04Lectures(perweekperstudent)

Unit-I

Systems of linear differential equations, types of linear systems, differential operators, an

operator method for linear systems with constant coefficients, Basic Theory of linear systems in normal form ,homogeneous linear systems with constant coefficients(Two Equations in two unknown functions).

Unit-II

Simultaneous linear first order equations in three variables, methods of solution, Pfaffian differential equations, methods of solutions of Pfaffian differential equations int here variables.

Unit-III

Formation of first order partial differential equations, Linear and non-linear partial differential equationsoffirstorder,specialtypesoffirst-orderequations,Solutionsofpartialdifferentialequationsoffirstordersatisfyinggivenconditions.

Unit-IV

Linear partial differential equations with constant coefficients, Equations reducible to linear partial differential equations with constant coefficients, Partial differential equations with variable coefficients ,Separation of variables, Non-linear equation of the second order.

Unit-IV

Laplace equation, Solution of Laplace equation by separation of variables, One dimensional wave equation, Solution of the wave equation (method of separation of variables), Diffusion equation, Solution ofone-dimensionaldiffusionequation,methodofseparationofvariables.

Part-II (Practical:Marks:25)

List of Practicals (Using any Software)

Practical/Lab work to be performed on a Computer.

1.Tofindthegeneralsolutionofthenon-homogeneoussystemoftheform:

dxdy

dtdt

with given conditions.

2.PlottingtheintegralsurfacesofagivenfirstorderPDEwithinitialdata.

3.Solutionofwaveequation-c2=0 for the following associated conditions:

(a)u(x,0)=φ(x),ut(x,0)=ψ(x),x ∈R,t >0.(b)u(x,0)=φ(x),ut(x,0)=ψ(x),ux(0,t)= 0,x∈(0,∞), t>0.(c)u(x,0)=φ(x),ut(x,0)=ψ(x),u(0,t)=0,x∈(0,∞), t>0.(d)u(x,0)=φ(x),ut(x,0)=ψ(x),u(0,t)=0,u(1,t)=0,0<x<l,t>0.

4.Solutionofwaveequation-k2=0 for the following associated conditions:

(a)u(x,0)=φ(x),u(0,t)=a,u(l,t)=b,0<x<l,t>0.

(b) u(x,0)=φ(x),x ∈R,0<t<T.

=a1x+b1y+f1(t),=a2x+b2y+f2(t)

∂2u∂

2u

∂t2

∂x2

∂u∂2u

∂t∂x2

(c) u(x,0)=φ(x),u(0,t)=a,x ∈(0,∞),t ≥0.

Book Recommended: 1.J.Sinha RoyandS.Padhy,ACourseonOrdinaryandPartialDifferentialEquations,Kalyani

Publishers ,New Delhi,Ludhiana,2012.

Chapters:11,12,13(13.1-13.7),15(15.1,15.5),16(16.1,16.1.1), 17(17.1, 17.2, 17.3). Ch. 8 (8.1 to 8.4)

Books for References:

1.Tyn Myint-UandLokenathDebnath,LinearPartialDifferentialEquationsforScientistsandEn-

gineers, 4th edition, Springer, Indian reprint, 2006.

2.S.L.Ross,Differentialequations,3rdEd.,JohnWileyandSons,India,2004.

Core Courses Semester-IV MATHEMATICS ( Hons ) CC-VIII - 4.1:Numerical Methods

(TotalMarks:100)

Part-I(Marks:75)

Theory: 60Marks+ Mid-Sem: 15Marks

04Lectures (per week per student) (Using Scientific Calculator)

Unit-I

Algorithms,Convergence,Errors:Relative,Absolute,Roundoff,Truncation.TranscendentalandPolynomialequations:Bisectionmethod,Newtonsmethod,Secantmethod.Rateofconvergenceofthesemethods.

Unit-II

System of linear algebraic equations: Gaussian Elimination and Gauss Jordan methods. Gauss Jacobi method, Gauss Seidel method and their convergence analysis.

Unit-III

Interpolation: Lagrange and Newtons methods. Error bounds.Finite difference operators. Gregory forward and backward difference interpolation.

Unit-IV

Numerical Integration: Trapezoidalrule, Simpsons rule,Simpsons3/8th rule,Booles Rule.Midpointrule,CompositeTrapezoidalrule,CompositeSimpsonsrule.

Unit-V

Ordinary Differential Equations :Eulers method. Runge-Kutta methods of orders two and four.

Part-II (Practical: Mark s:25)

List of Practicals(Using any Software)

Practical/Lab work to be performed on a Computer.

1.Calculatethesum1/1+1/2+1/3+1/4+----------+1/N.

2.To find the absolute value of an integer.

3.Enter100integersintoanarrayandsorttheminanascendingorder.

4.BisectionMethod.

5.NewtonRaphsonMethod.

6.SecantMethod.

7.RegulaiFalsiMethod.

8.LUdecompositionMethod.

9.Gauss-JacobiMethod.

10.SORMethodorGauss-SiedelMethod.

11.Lagrange Interpolation or Newton Interpolation.

12.Simpsonsrule. Note:ForanyoftheCAS(Computeraidedsoftware)Datatypes-simpledatatypes,floatingdatatypes,character data types, arithmetic operators and operator precedence, variables and constant declarations,expressions,input/output,relationaloperators,logicaloperatorsandlogicalexpressions,controlstatementsandloopstatements,Arraysshouldbeintroducedtothestudents.

Book Recommended:

1.B.P.AcharyaandR.N.Das,ACourseonNumericalAnalysis,KalyaniPublishers,NewDelhi,Ludhiana.Chapters:1,2(2.1to2.4,2.6,2.8,2.9),3(3.1to3.4,3.6to3.8,3.10),4(4.1,4.2),5(5.1,5.2,5.3),6(6.1,6.2,6.3,6.10,6.11),7(7.1,7.2,7.3,7.4&7.7).

2.BrianBradie,AFriendlyIntroductiontoNumericalAnalysis,PearsonEducation,India,2007.

Books for References:

1.M.K.Jain,S.R.K.IyengarandR.K.Jain,NumericalMethodsforScientificandEngineeringComputation,6thEd.,NewageInternationalPublisher,India,2007.

2.C.F.GeraldandP.O.Wheatley,AppliedNumericalAnalysis,PearsonEducation,India,2008.

3.UriM.AscherandChenGreif,AFirstCourseinNumericalMethods,7thEd.,PHILearningPrivateLimited,2013.

4.JohnH.MathewsandKurtisD.Fink,NumericalMethodsusingMatlab,4thEd.,PHILearningPrivateLimited,2012.

Core Courses Semester-IV MATHEMATICS ( Hons ) CC-IX - 4.2:Riemann Integration and Series of Functions (Analysis-III)

TotalMarks:100

Theory:80Marks+Mid-Sem:20Marks

5Lectures,1Tutorial(perweekperstudent)

Unit-I

Riemannintegration; inequalities of upper and lower sums; Riemann conditions of integrability. Riemann sum and definition of Riemann integral through Riemann sums; equivalence of two definitions; Riemann integrability of monotone and continuous functions, Properties of the Riemann integral; definition and integrability of piecewise continuous and monotone functions.IntermediateValuetheoremforIntegrals;FundamentaltheoremsofCalculus.

Unit-II

Improper integrals; Convergence of Beta and Gamma functions.

Unit-III

Pointwiseanduniformconvergenceofsequenceoffunctions.Theorems on continuity, derivability andintegrabilityofthelimitfunctionofasequenceoffunctions.

Unit-IV

Seriesoffunctions;Theoremsonthecontinuityandderivabilityofthesumfunctionofaseriesoffunctions;CauchycriterionforuniformconvergenceandWeierstrassM-Test.

Unit-V

LimitsuperiorandLimitinferior.Powerseries,radiusofconvergence,CauchyHadamardTheorem,Differentiationandintegrationofpowerseries;AbelsTheorem;WeierstrassApproximationTheorem.

Book Recommended:

1. G.DasandS.Pattanayak-FundamentalsofMathematicsAnalysis, TMHPublishingCo., Chapters:8(8.1 to 8.6), 9 (9.1 to 9.8)

Books for References:

1.K.A.Ross,ElementaryAnalysis,TheTheoryofCalculus,UndergraduateTextsinMathematics,Springer (SIE), Indian reprint, 2004.

2.R.G.BartleD.R.Sherbert,IntroductiontoRealAnalysis,3rdEd.,JohnWileyandSons(Asia)Pvt.Ltd.,Singapore,2002.

3.CharlesG.Denlinger, Elements of Real Analysis, Jones & Bartlett (Student Edition), 2011.

4.S.C.MallikandS.Arora-Mathematical Analysis, New Age International Ltd., New Delhi.

5.Shanti Narayan and M.D.Raisinghania-ElementsofRealAnalysis,S.Chand&Co.Pvt.Ltd.

Core Courses Semester-IV MATHEMATICS ( Hons )

CC-X--4.3: Ring Theory and Linear Algebra-I (Analysis-III)

TotalMarks:100

Theory:80Marks+Mid-Sem:20Marks

5Lectures,1Tutorial(perweekperstudent)

Unit-I

Definitionandexamplesofrings,propertiesofrings,subrings,integraldomainsandfields,characteristicofaring.Ideal,idealgeneratedbyasubsetofaring,factorrings,operationsonideals,primeandmaximalideals.

Unit-II

Ringhomomorphisms,propertiesofringhomomorphisms,IsomorphismtheoremsI,IIandIII,fieldofquotients.

Unit-III

Vectorspaces,subspaces,algebraofsubspaces,quotientspaces,linearcombinationofvectors,linearspan,linearindependence,basisanddimension,dimensionofsubspaces.

Unit-IV

Lineartransformations,nullspace,range,rankand nullity of a linear transformation, matrix representation of a linear transformation, algebra of linear transformations.

Unit-V

Isomorphisms, Isomorphism theorems,invertibility and isomorphisms, change of coordinate matrix.

BookRecommended:

1.JosephA.Gallian,ContemporaryAbstractAlgebra(4thEdn.),NarosaPublishingHouse,NewDelhi.Chapters:12,13,14,15.

2.StephenH.Friedberg,ArnoldJ.Insel,LawrenceE.Spence,LinearAlgebra,4thEd.,Prentice HallofIndiaPvt.Ltd.,NewDelhi,2004.Chapters:1 (1.2-1.6), 2(2.1-2.5).

BooksforReferences:

1.JohnB.Fraleigh, A First Course in Abstract Algebra, 7th Ed., Pearson, 2002.

2.M.Artin, Abstract Algebra, 2nd Ed., Pearson, 2011.

3.S.Lang, Introduction to Linear Algebra, 2nd Ed., Springer, 2005.

4.Gilbert Strang, Linear Algebra and its Applications, Cengage Learning India Pvt.Ltd.

5.S.Kumaresan,LinearAlgebra-AGeometricApproach,PrenticeHallofIndia,1999.

6.KennethHoffman,RayAldenKunze,LinearAlgebra,2ndEd.,Prentice-HallofIndiaPvt.Ltd.,1971.

7.I.N.Herstein-Topics in Algebra, Wiley Eastern Pvt.Ltd.

Skills Enhancement Course (SEC) SEC – II Semester-IV Time - 2 hrs. F.M. - 50 [ 40 (Sem.) + 10 ( Int .) ] Credit - 2 Lectures - 30

-Logic and Sets

Introduction, propositions, truth table, negation, conjunction and disjunction. Implications, biconditional propositions, converse, contrapositive and inverse propositions and precedence of logical operators. Propositional equivalence: Logical equivalences. Predicates and quantifiers :Introduction, Quantifiers, Binding variables and Negations. Sets, subsets, Set operations and the laws of set theory and Venndiagrams. Examples of finite and infinite sets.Finite sets and counting principle. Empty set, properties of empty set. Standard set operations. Classes of sets. Power set of a set. Difference and Symmetric difference of two sets. Set identities, Generalized union and intersections. Relation: Product set, Composition of relations, Types of relations, Partitions, Equivalence Relations with example of

Congruence modulo relation, Partial ordering relations, naryrelations.

Books Recommended:

1.1.R.P.Grimaldi-DiscreteMathematicsandCombinatorialMathematics, PearsonEducation,1998.

2.P.R.Halmos-NaiveSetTheory,Springer,1974.

3.E.Kamke-TheoryofSets,DoverPublishers,1950

SEM-III Math DSC-III IS SAME AS CC-VII

SEM-IV Math DSC-IV IS SAME AS CC-VIII

Semester-V C-5.1:MultivariateCalculus(Calculus-II) TotalMarks:100 Theory:80Marks+Mid-Sem:20Marks 5Lectures,1Tutorial(per weekperstudent) Unit-I Functionsofseveralvariables,limitandcontinuityoffunctionsoftwovariablesPartialdifferentiation,totaldifferentiabilityanddifferentiability,sufficientconditionfordifferentiability.Chainruleforoneandtwoindependentparameters Unit-II Directionalderivatives,thegradient,maximalandnormalpropertyofthegradient,tangentplanes.Extremaoffunctionsoftwovariables,methodofLagrangemultipliers,constrained optimization problems, Definition of vector field, divergence and curl Unit-III

Extrema of functions of two variables, method of Lagrange multipliers, constrained optimization problems,Definitionofvector field, divergence and curl. Unit-IV

Doubleintegrationoverrectangularregion,doubleintegrationovernon-rectangularregion,Doubleintegralsinpolarco-ordinates,Tripleintegrals,Tripleintegraloveraparallelepipedandsolidregions.Volume by triple integrals, cylindrical and spherical co-ordinates.Changeofvariablesindoubleintegralsandtripleintegrals. Unit-V

Line integrals, Applications of line integrals:Mass and Work.Fundamental theorem for line integrals,conservativevectorfields,independenceofpath.Greenstheorem,surfaceintegrals,integralsoverparametricallydefinedsurfaces.Stokestheorem,TheDivergencetheorem. BooksRecommended:

1.M.J.Strauss,G.L.BradleyandK.J.Smith,Calculus,3rd Ed., Dorling Kindersley (India) Pvt.Ltd.(PearsonEducation),Delhi,2007.Chapters:11(11.1(Pages:541-543),11.2-11.6,11.7(Pages:598-605),11.8(Pages:610-614)),12(12.1,-12.3,12.4(Pages:652-660),12.5,12.6),13(13.2,13.3,13.4(Pages:712-716),13.5(Pages:723-726;729-730),13.6(Pages:733-737),13.7(Pages:742-745)). 2.Hiher calculus for degree student . G samal and others .(chapters:7,8,9,10,11,12,13,14) BooksforReference:

1.G.B.ThomasandR.L.Finney,Calculus,9thEd.,PearsonEducation,Delhi,2005.

2.E.Marsden,A.J.Tromba and A.Weinstein, Basic Multivariable Calculus, Springer (SIE), Indianreprint, 2005. 3.SantoshK.Sengar-AdvancedCalculus,CengageLearningIndiaPvt.Ltd.

C-5.2:ProbabilityandStatistics TotalMarks:100

Theory:80Marks+Mid-Sem:20Marks 5Lectures,1Tutorial(perweekperstudent)(scientific calculator

May be allowed) Unit-I Samplespace,probabilityaxioms,realrandomvariables(discreteandcontinuous),cumulativedistributionfunction,probabilitymass/densityfunctions Unit-II Mathematicalexpectation,moments,momentgeneratingfunction,characteristicfunction. Unit-III Discretedistributions:uniform,binomial,Poisson,geometric,negativebinomial,continuousdistributions:uniform,normal,exponential.Jointcumulativedistributionfunctionanditsproperties,jointprobability density functions, marginal and conditional distributions. Unit-IV

Expectationoffunctionoftworandomvariables,conditionalexpectations,independentrandomvariables, bivariate normal distribution, correlation coefficient, joint moment generating function (jmgf) andcalculation of covariance (from jmgf), linear regression for two variables. Unit-V

Chebyshevs inequality, statement and interpretation of (weak) law of large numbers and strong law oflargenumbers,CentralLimittheoremforindependentandidenticallydistributedrandomvariableswithfinitevariance,MarkovChains,Chapman-Kolmogorovequations,classificationofstates. BooksRecommended:

1.RobertV.Hogg,JosephW.McKeanandAllenT.Craig,IntroductiontoMathematicalStatistics,PearsonEducation,Asia,2007.Chapters:1(1.1,1.3.1.5-1.9),2(2.1,2.3-2.5). 2.IrwinMillerandMaryleesMiller,JohnE.Freund,MathematicalStatisticswithApplications,7thEd.,PearsonEducation,Asia,2006.Chapters:4,5(5.1-5.5,5.7),6(6.2,6.3,6.5-6.7),14(14.1,14.2) 3.SheldonRoss,IntroductiontoProbabilityModels,9thEd.,AcademicPress,IndianReprint,2007. Chapters:2(2.7),4(4.1-4.3).

BooksforReferences:

1.AlexanderM.Mood,FranklinA.GraybillandDuaneC.Boes,IntroductiontotheTheoryofStatistics,3rdEd.,TataMcGraw-Hill,Reprint2007.

2.S.C.GuptaandV.K.Kapoor-Fundamentals of Mathematical Statistics, S.ChandandCompanyPvt.Ltd.,NewDelhi. 3.S.Ross-AFirstCourseinProbability,PearsonEducation.

DSC-1 ProgramminginC++(Compulsory)

Part-I(Marks:75) (Theory:60Marks+Mid-Sem:15Marks)

Introductiontostructuredprogramming:datatypes-simpledatatypes,floatingdatatypes,characterdatatypes,stringdatatypes,arithmeticoperatorsandoperatorsprecedence,variablesandconstantdeclarations,expressions,inputusingtheextractionoperator¿¿andcin,outputusingtheinsertionoperator¡¡andcout,preprocessordirectives,increment(++)anddecrement(–)operations,creatingaC++program,input/output,relationaloperators,logicaloperatorsandlogicalexpressions,ifandif-elsestatement,switchandbreakstatements.for,whileanddo-whileloopsandcontinuestatement,nestedcontrolstatement,valuereturningfunctions,valueversusreferenceparameters,localandglobalvariables,onedimensionalarray,twodimensionalarray,pointerdataandpointervariables. BookRecommended:

1.D.S.Malik: C++ Programming Language, Edition-2009, Course Technology, Cengage Learning,IndiaEdition.Chapters:2(Pages:37-95),3(Pages:96-129),4(Pages:134-178),5(Pages:181-236),6,7(Pages:287-304),9(pages:357-390),14(Pages:594-600). BooksforReferences:

1.E.Balaguruswami:ObjectorientedprogrammingwithC++,fifthedition,TataMcGrawHillEducationPvt.Ltd. 2.R.JohnsonbaughandM.Kalin-Applications Programming in ANSI C, Pearson Education.

3.S.B.Lippman and J.Lajoie, C++ Primer,3rdEd.,AddisonWesley,2000.

4.Bjarne Stroustrup , The C++ Programming Language, 3rd Ed., Addison Welsley.

Part-II(Practical,Marks:25)

ListofPracticals(Usinganysoftware) Practical/LabworktobeperformedonaComputer.

1.CalculatetheSumoftheseries++..+for any positive integer N. 2.Writeauserdefinedfunctiontofindtheabsolutevalueofanintegeranduseittoevaluatethe function(-1)n/|n|,forn=-2,-1,0,1,2.

1111 123N

3.Calculate the factorial of anynaturalnumber.

4.Readfloatingnumbersandcomputetwoaverages:theaverageofnegativenumbersandtheaverageofpositivenumbers. 5.Writeaprogramthat promptstheusertoinputapositiveinteger.Itshouldthenoutputamessageindicatingwhetherthenumberisaprimenumber. 6.Writeaprogramthatpromptstheusertoinputthevalueofa,bandcinvolvedintheequationax2+bx+c=0andoutputsthetypeoftherootsoftheequation.Alsotheprogramshouldoutputsalltherootsoftheequation. 7.writeaprogramthatgeneratesrandomintegerbetween0and99.GiventhatfirsttwoFibonaccinumbersare0and1,generateallFibonaccinumberslessthanorequaltogeneratednumber 8.Writeaprogramthatdoesthefollowing:

a.Promptstheusertoinputfivedecimalnumbers. b.Printsthefivedecimalnumbers. c.Convertseachdecimalnumbertothenearestinteger. d.Addsthesefiveintegers. e.Printsthesumandaverageofthem.

9.Write a program that uses whileloops to perform the following steps:

a.Prompttheusertoinputtwointegers:firstNumandsecondNum(firstNumshoulbelessthansecondNum). b.OutputalloddandevennumbersbetweenfirstNumandsecondNum. c.OutputthesumofallevennumbersbetweenfirstNumandsecondNum. d.Output the sum of the square of the odd numbers firs tNumand second Num. e.OutputalluppercaseletterscorrespondingtothenumbersbetweenfirstNumandsecondNum, if any.

10.Write a programthat prompts the user to input fivedecimalnumbers.Theprogramshouldthenadd the five decimal numbers, convert the sum to the nearest integer, and print the result. 11.Writeaprogramthatpromptstheusertoenterthelengthsofthreesidesofatriangleandthenoutputsamessageindicatingwhetherthetriangleisarighttriangleorascalenetriangle. 12.Writeavaluereturningfunctionsmallertodeterminethesmallestnumberfrom a set of numbers.Usethisfunctiontodeterminethesmallestnumberfromasetof10numbers. 13.Writeafunctionthattakesasaparameteraninteger(asalongvalue)andreturnsthenumberofodd,even,andzerodigits.Alsowriteaprogramtotestyourfunction. 14.Enter100 integers into an array and short them in an ascending/ descending order and print thelargest/smallestintegers. 15.Enter10 integers into an array and then search for a particular integer in the array.

16.Multiplication/Additionoftwomatricesusingtwodimensionalarrays.

17.Usingarrays,readthevectorsofthefollowingtype:A=(12345678), B=(02340156)andcomputetheproductandadditionofthesevectors. 18.Readfromatextfileandwritetoatextfile.

19.Writeafunction,reverseDigit,thattakesanintegerasaparameterandreturnsthenumberwithitsdigitsreversed.For example, the value of function reverse Digit12345is54321andthevalueofreverseDigit-532is-235.

DSC-II TotalMarks:100

Theory:80Marks+Mid-Sem:20Marks 5Lectures,1Tutorial(perweekperstudent.

(Anyoneofthefollowing)

1-DiscreteMathematics Unit-I Logic,proportionalequivalence,predicatesandquantifiers,nestedquantifiers,methodsofproof,relationsandtheirproperties,n-aryrelationsandtheirapplications,Booleanfunctionsandtheirrepresentation. Unit-II Thebasiccounting,thePigeon-holeprinciple,GeneralizedPermutationsandCombinations,Recurrencerelations,Countingusingrecurrencerelations,

Unit-III Solvinglinearhomogeneousrecurrencerelationswithconstantcoefficients,Generatingfunctions,Solvingrecurrencerelationsusinggeneratingfunctions.

Unit-IV Partiallyorderedsets,Hassediagramofpartiallyorderedsets,mapsbetweenorderedsets,dualityprinciple,Latticesasorderedsets,Latticesasalgebraicstructures,sublattices,Booleanalgebraanditsproperties.Unit-V Graphs:Basicconceptsandgraphterminology,representinggraphsandgraphisomorphism.Distancein a graph, Cut-vertices and Cut-edges, Connectivity, Euler and Hamiltonian path. BookRecommended:

1.KennethH.Rosen,DiscreteMathematicsandApplications,TataMcGrawHillPublications,Chapters:1(1.1to1.5),4(4.1,4.2,4.5),6(6.1,6.2,6.5,6.6),7(7.1,7.2),8,10(10.1,10.2). BooksforReferences:

1.BA.DaveyandH.A.Priestley, Introduction to Lattices and Order, Cambridge University Press,Cambridge,1990. 2.EdgarG.GoodaireandMichaelM.Parmenter,DiscreteMathematicswithGraphTheory(2ndEdition), Pearson Education (Singapore) Pte.Ltd.,IndianReprint2003. 3.RudolfLidlandGnterPilz,AppliedAbstractAlgebra(2ndEdition),UndergraduateTextsinMathematics, Springer (SIE), Indian reprint, 2004. 4.D.S.Malik-Discrete Mathematics: Theory & Applications, Cengage Learning India Pvt.Ltd.

5.KevinFerland-DiscreteMathematicalStructures,CengageLearningIndiaPvt.Ltd. MATH SEM-V-SEC-III (for Gen) IS SAME AS MATH.-SEM-IV-SEC-II (for Hons) MATH SEM-V-DSE-I (for Gen) IS SAME AS MATH.-SEM-I-CC-II (for Hons)

Semester-VI C-6.1:MetricSpacesandComplexAnalysis(Analysis-IV)

TotalMarks:100 Theory:80Marks+Mid-Sem:20Marks

5Lectures,1Tutorial(perweekperstudent) Unit-I Metricspaces:definitionandexamples.Sequencesinmetricspaces,Cauchysequences.CompleteMetricSpaces.Openandclosedballs,neighbourhood,openset,interiorofaset.Limitpointofaset,closedset,diameterofaset,Cantorstheorem. Unit-II Subspaces,densesets,separablespaces.Continuousmappings, sequential criterion and other characterizations of continuity.Uniformcontinuity.Homeomorphism, Contraction mappings, Banach Fixed point Theorem.Connectedness,connectedsubsetsofR. Unit-III

Propertiesofcomplexnumbers,regionsinthecomplexplane,functionsofcomplexvariable,mappings.Derivatives,differentiationformulas,Cauchy-Riemannequations,sufficientconditionsfordifferentiability. Unit-IV

Analytic functions, examples of analytic functions, exponential function, Logarithmic function, trigonometricfunction,derivativesoffunctions,definiteintegralsoffunctions.Contours,Contourintegralsanditsexamples,upperboundsformoduliofcontourintegrals.Cauchy-Goursattheorem,Cauchyintegralformula. Unit-V

Liouvilles theorem and the fundamental theorem of algebra.Convergenceofsequencesandseries,Taylorseriesanditsexamples.Laurentseriesanditsexamples,absoluteanduniformconvergenceofpowerseries.

BooksRecommended:

1.P.K.Jain and K.Ahmad,MetricSpaces,NarosaPublishingHouse,NewDelhi.Chapters:2(1-9),3(1-4),4(1-4),6(1-2),7(1only).

2.JamesWardBrownandRuelV.Churchill, Complex Variables and Applications, 8th Ed., McGrawHillInternationalEdition,2009.Chapters:1(11only),2(12,13),2(15-22,24,25),3(29,30,34)4(37-41,43-46,50-53),5(55-60,62,63,66).

BooksforReferences:

1.SatishShiraliandHarikishanL.Vasudeva,MetricSpaces,SpringerVerlag,London,2006.

2.S.Kumaresan,TopologyofMetricSpaces,2ndEd.,NarosaPublishingHouse,2011.

3.S.Ponnusamy-FoundationsofComplexAnalysis,AlphaScienceInternationalLtd.

4.J.B.Conway-Functionsofonecomplexvariable,Springer.

5.N.Das- Complex Function Theory, Allied Publishers Pvt.Ltd.,Mumbai.

C-6.2:LinearProgramming TotalMarks:100

Theory:80Marks+Mid-Sem:20Marks 5Lectures,1Tutorial(perweekperstudent)

Unit-I ( Scientific Calculator may be allowed) Introduction to linear programming problem, Theory of simplex method, optimality and unboundedness,thesimplexalgorithm,simplexmethodintableauformat,introductiontoartificialvariables,twophasemethod, BigM method and their comparison. Unit-II

Duality, formulation of the dual problem, primal-dual relationships, economic interpretation of the dual.

Unit-III

Transportationproblem anditsmathematicalformulation,northwestcornermethodleastcostmethodandVogelapproximationmethodfordeterminationofstartingbasicsolution,algorithmforsolvingtransportation problem Unit-IV Assignmentproblem and itsmathematicalformulation,Hungarianmethodforsolvingassignmentproblem. Unit-V

Gametheory:formulationoftwopersonzerosumgames,solvingtwopersonzerosumgames,gameswithmixedstrategies,graphicalsolutionprocedure,linearprogrammingsolutionofgames. RecommendedBooks:

1.MokhtarS.Bazaraa,JohnJ.Jarvis and HanifD.Sherali, LinearProgrammingandNetworkFlows,2ndEd.,JohnWileyandSons,India,2004.Chapters:3(3.2-3.3,3.5-3.8),4(4.1-4.4),6(6.1-6.3). 2.F.S.HillierandG.J.Lieberman, Introduction to Operations Research, 9th Ed., Tata McGraw Hill,Singapore,2009.Chapter:14 3.HamdyA.Taha,OperationsResearch,AnIntroduction,8thEd.,PrenticeHallIndia,2006.Chapter:5(5.1,5.3,5.4). BooksforReference:

1.G.Hadley,LinearProgramming,NarosaPublishingHouse,NewDelhi,2002.

2.Kanti Swarup, P.K.GuptaandManMohan-OperationsResearch,S.ChandandCo.Pvt.Ltd.

3.N.V.R. Naidu,G. RajendraandT. KrishnaRao-OperationsResearch,I.K. InternationalPublishingHousePvt.Ltd., New Delhi, Bangalore. 4.R.Veerachamy and V.RaviKumar-OperationsResearch-I.K.InternationalPublishingHouse Pvt.Ltd., New Delhi, Bangalore. 5.P.K.GuptaandD.S.Hira-OperationsResearch,S.ChandandCompanyPvt.Ltd.,NewDelhi

DSC-III TotalMarks:100

Theory:80Marks+Mid-Sem:20Marks 5Lectures,1Tutorial(perweekperstudent.

(Anyoneofthefollowing)

1-DifferentialGeometry Unit-I TheoryofSpaceCurves:Spacecurves,Planercurves,Curvature,torsionandSerret-Frenetformulae.Osculatingcircles,Osculatingcirclesandspheres.Existenceofspacecurves.Evolutesandinvolutesofcurves. Unit-II

Osculatingcircles,Osculatingcirclesandspheres.Existenceofspacecurves.Evolutesandinvolutesofcurves. Unit-III

Developables:Developableassociatedwithspacecurvesandcurvesonsurfaces,Minimalsurfaces.

Unit-IV

TheoryofSurfaces:Parametriccurvesonsurfaces.Directioncoefficients.FirstandsecondFundamentalforms.PrincipalandGaussiancurvatures. Unit-V Linesofcurvature,Eulerstheorem.Rodriguesformula,ConjugateandAsymptoticlines. BookRecommended:

1.C.E.Weatherburn,DifferentialGeometryofThreeDimensions,CambridgeUniversityPress2003.Chapters:1(1-4, 7,8,10), 2(13, 14, 16,17),3,4(29-31,35,37, 38). BooksforReferences

1.T.J.Willmore, An IntroductiontoDifferentialGeometry,DoverPublications,2012.

2.S.Lang,FundamentalsofDifferentialGeometry,Springer,1999.

3.B.O’Neill,ElementaryDifferentialGeometry,2ndEd.,AcademicPress,2006.

4.A.N.Pressley-ElementaryDifferentialGeometry,Springer.

5.B.P.Acharya and R.N.Das-FundamentalsofDifferentialGeometry,KalyaniPublishers,Ludhiana,NewDelhi.

DSC-IV ProjectWork(Compulsory)

Dissertation: 60 Marks Presentation: 25 Marks Viva-voce: 15 Marks Projects submitted by the student are to be evaluated by the Internal Examiner and External Examiner appointed by University. Students should opt for Supervision of Dissertation from the internal faculties of his own college/Institution. The Supervisor in consultation with the concerned Head of the Department should decide the topic. The presentation should be open to all faculties as well as graduate students of the concerned Department

SEC-4 CombinartorialMathematics

Basiccountingprinciples,PermutationsandCombinations(withandwithoutrepetitions),Binomialtheorem,Multinomialtheorem,Countingsubsets,Set-partitions,StirlingnumbersPrincipleofInclusionandExclusion,Derangements,InversionformulaeGeneratingfunctions:Algebraofformalpowerseries,Generatingfunctionmodels,Calculatinggeneratingfunctions,Exponentialgeneratingfunctions.Recurrencerelations:Recurrencerelationmodels,Divideandconquerrelations,Solutionofrecurrencerelations,Solutionsbygeneratingfunctions.Integerpartitions,Systemsofdistinctrepresentatives. BooksRecommended:

1.J.H.vanLintandR.M.Wilson-ACourseinCombinatorics,2ndEd.,CambridgeUniversityPress,2001. 2.V.Krishnamurthy-Combinatorics, Theory and Application, Affiliated East-West Press 1985.

3.P.J.Cameron-Combinatorics,Topics,Techniques,Algorithms,CambridgeUniversityPress,1995.

4.M.Jr.Hall-Combinatorial Theory, 2nd Ed., John Wiley & Sons, 1986.

5.S.S.Sane-CombinatorialTechniques,HindustanBookAgency,2013.

6.R.A.Brualdi-IntroductoryCombinatorics,5thEd.,PearsonEducationInc.,2009.

MATH SEM-VI-DSE-II (for Gen) IS SAME AS MATH.-SEM-II-CC-III (for Hons)