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

Unit-I

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)

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,Prentic

eHall,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,KalyaniPublisher

s,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.R oss,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

Unit-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

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(7.4

only).

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.K hannaandS.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,NewDelh

i.

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

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

with given conditions.

2. PlottingtheintegralsurfacesofagivenfirstorderPDEwithinitialdata.

3. Solutionofwaveequation-c2=0∂

2 u∂

2 u

∂t2 ∂x2

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.

2 ∂u∂ 2 u 4.Solutionofwaveequation-k =0

∂t∂x2 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.

(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.TranscendentalandPoly

nomialequations:Bisectionmethod,Newtonsmethod,Secantmethod.Rateofconvergenceofthesem

ethods.

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,logicaloperatorsandlogicalexpression

s,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,NumericalMethodsforScientificandEngineeringComput

ation,6thEd.,NewageInternationalPublisher,India,2007.

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

3. UriM.AscherandChenGreif,AFirstCourseinNumericalMethods,7thEd.,PHILearningPrivateLi

mited,2013.

4. JohnH.MathewsandKurtisD.Fink,NumericalMethodsusingMatlab,4thEd.,PHILearningPrivat

eLimited,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;Theoremsonthecontinuityandderivabilityofthesumfunctionofaseriesoffunction

s;CauchycriterionforuniformconvergenceandWeierstrassM-Test.

Unit-V

LimitsuperiorandLimitinferior.Powerseries,radiusofconvergence,CauchyHadamardTheorem,Diffe

rentiationandintegrationofpowerseries;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,Sprin

ger (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,characteristic

ofaring.Ideal,idealgeneratedbyasubsetofaring,factorrings,operationsonideals,primeandmaximalid

eals.

Unit-II

Ringhomomorphisms,propertiesofringhomomorphisms,IsomorphismtheoremsI,IIandIII,fieldofquot

ients.

Unit-III

Vectorspaces,subspaces,algebraofsubspaces,quotientspaces,linearcombinationofvectors,linearsp

an,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

Functions of several variables, limit and continuity of functions of two variables. Partial

differentiation,tangent planes, Approximation and differentiability. Chain rule for one and two

independen tparameters.

Unit-II Directional derivatives and gradient, maximal property of the gradient, Normal Property

of the gradient, Tangent planes and the normal lines, Extreme of functions of two variables,

Method of Lagrange multipliers, Lagrange multipliers, constrained optimization problems, A

geometrical interpretation.

Unit-III

Double integration over rectangular region and over non-rectangular region. Double

integrals in polar co-ordinates, Triple integrals, Triple integral over a parallelpiped and solid

region, volume by triple integrals, cylindrical and spherical co-ordinates. Change of

variables in double integrals and triple integrals.

Unit-IV

Definition of vector field, Divergence and curl, Line integrals, Applications of line integrals

Mass and work, Fundamental teorem and path independence for line integrals.

Unit-V

Green’s teorem, Area as a line integral, Alternative forms of Green’s teorem, Normal

Derivatives, Surface integrals, Integrals over parametrically defined surface, Stokes

teorem, The Divergence teorem.

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.

4. 4.D.K. Dalai,-Multivariate Calculus

C-5.2:ProbabilityandStatistics

TotalMarks:100

Theory:80Marks+Mid-Sem:20Marks

5Lectures,1Tutorial(perweekperstudent)(scientific calculator

May be allowed)

Unit-I

Sample space, probability axioms, real random variables (discrete and continuous), cumulative

distributive function, probability mass density functions

Unit-II

Mathematical expectation, moments, moment generating function, characteristic function.

Unit-III

Discrete distributions: uniform, binomial, Poisson, geometric, negative binomial, continuous

distributions: uniform, normal, exponential. Joint cumulative distribution function and its

properties, joint probability density functions, marginal and conditional distributions.

Unit-IV

Expectation of function of two random variables, conditional expectations, independent

random variables, 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 of large numbers, Central Limit theorem for independent and identically distributed random

variables with finite variance, Markov Chains, Chapman-Kolmogorov equations, classification

of states.

BooksRecommended:

1. RobertV.Hogg,JosephW.McKeanandAllenT.Craig,IntroductiontoMathematicalStatistics,Pear

sonEducation,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.,P

earsonEducation,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,3

rdEd.,TataMcGraw-Hill,Reprint2007.

for any positive integer

2.S.C.GuptaandV.K.Kapoor-Fundamentals of Mathematical Statistics,

S.ChandandCompanyPvt.Ltd.,NewDelhi.

3.S.Ross-AFirstCourseinProbability,PearsonEducation.

DSC-1

Programming in C++ (Compulsory)

Part-I(Marks:75)

(Theory:60Marks+Mid-Sem:15Marks)

Introduction to structured programming: data types-simple data types, floating data types,

character data types, string data types, arithmetic operators and operators precedence, variables

and constant declarations, expressions, input using the extraction operator ¿¿ and cin, output using

the insertion operator ¡¡ and cout, preprocessor directives, increment(++) and decrement (–)

operations, creating a C++ program, input / output, relational operators, logical operators and

logical expressions, if and if-else statement, switch and break statements. for, while and do-

while loops and continue statement, nested control statement, value returning functions, value versus

reference parameters, local and global variables, one dimensional array, two dimensional array,

pointer data and pointer ariables.

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,TataMcGrawHillEducatio

nPvt.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++..+1111

N. 123N

2. Writeauserdefinedfunctiontofindtheabsolutevalueofanintegeranduseittoevaluatethe

function(-1)n/|n|,forn=-2,-1,0,1,2.

3. Calculate the factorial of anynaturalnumber.

4. Readfloatingnumbersandcomputetwoaverages:theaverageofnegativenumbersandtheaverageof

positivenumbers.

5. Writeaprogramthat

promptstheusertoinputapositiveinteger.Itshouldthenoutputamessageindicatingwhetherthenumb

erisaprimenumber.

6. Writeaprogramthatpromptstheusertoinputthevalueofa,bandcinvolvedintheequationax2+bx+c

=0andoutputsthetypeoftherootsoftheequation.Alsotheprogramshouldoutputsalltherootsoftheeq

uation.

7. writeaprogramthatgeneratesrandomintegerbetween0and99.GiventhatfirsttwoFibonaccinumber

sare0and1,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(firstNumshoulbelessthansecond

Num).

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. Writeaprogramthatpromptstheusertoenterthelengthsofthreesidesofatriangleandthenoutputs

amessageindicatingwhetherthetriangleisarighttriangleorascalenetriangle.

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,thattakesanintegerasaparameterandreturnsthenumberwithitsdigi

tsreversed.For example, the value of function reverse

Digit12345is54321andthevalueofreverseDigit-532is-235.

DSC-II

TotalMarks:100

Theory:80Marks+Mid-Sem:20Marks

5Lectures,1 Tutorial (per week perstudent. (Any

one of the following)

1-Discrete Mathematics

Unit-I

Logic, proportional equivalence, predicates and quantifiers, nested quantifiers, methods of proof,

relations and their properties, n-ary relations and their applications, Boolean functions and

their representation.

Unit-II

The basic counting, the Pigeon-holeprinciple, Generalized Permutations and Combinations,

Recurrence relations, Counting using recurrence relations,

Unit-III

Solving linear homogeneous recurrence relations with constant coefficients, Generating functions,

Solving recurrence relations using generating functions.

Unit-IV

Partially ordered sets, Hasse diagram of partially ordered sets, maps between ordered sets, duality

principle, Lattices as ordered sets, Lattices as algebraic structures, sublattices, Boolean algebra and its

properties

.Unit-V

Graphs: Basic concepts and graph terminology, representing graphs and graph isomorphism.

Distance in 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),UndergraduateTextsinMathem

atics, 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(per week per

student)

Unit-I

Metric spaces: definition and examples. Sequences in metric spaces, Cauchy sequences. Complete

Metric paces. Open and close dballs, neighbourhood, open set, interior of a set. Limit point of a set,

closed set, diameter of a set, Cantors theorem.

Unit-II

Subspaces, dense sets, separable spaces. Continuous mappings, sequential criterion and

other characterizations of continuity. Uniform continuity. Homeomorphism, Contraction

mappings, Banach Fixed point Theorem. Connectedness, connected subsets of R.

Unit-III

Properties of complex numbers, regions in the complex plane, functions of complex variable,

mappings. Derivatives, differentiation formulas, Cauchy-Riemann equations, sufficient

conditions for differentiability.

Unit-IV

Analytic functions, examples of analytic functions, exponential function, Logarithmic function,

trigonometric function, derivatives of functions, definite integrals of functions. Contours, Contour

integrals and its examples, upper bounds for moduli of contour integrals. Cauchy- Goursat

theorem, Cauchy integral formula.

Unit-V

Liouvilles theorem and the fundamental theorem of

algebra. Convergence of sequences and series, Taylor series and its examples. Laurent series and

its examples, absolute and uniform convergence of power series.

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

(per week per student)

Unit-I( Scientific Calculator may be allowed)

Introduction to linear programming problem, Theory of simplex method, optimality and

unboundedness, the simplex algorithm, simplex method in tableau format, introduction to artificial

variables, two phase method, BigM method and their comparison.

Unit-II

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

the dual.

Unit-III

Transportation problem and its mathematical formulation, north west corner method least

cost method and Vogel approximation method for determination of starting basic solution,

algorithm for solving transportation problem

Unit-IV

Assignment problem and Its mathematical formulation, Hungarian method for solving assignment problem.

Unit-V

Game theory: formulation of two person zero sum games, solving two person zero sum games, games

with mixed strategies, graphical solution procedure, linear programming solution of games.

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

Theory of Space Curves: Space curves, Planer curves, Curvature, torsion and Serret- Frenet

formulae. Osculating circles, Osculating circles and spheres. Existence of space curves. Evolutes and

involutes of curves.

Unit-II

Osculating circles, Osculating circles and spheres. Existence of space curves. Evolutes and involutes

of curves.

Unit-III

Developables: Developable associated with space curves and curveson surfaces, Minimal surfaces.

Unit-IV

Theory of Surfaces: Parametric curves on surfaces. Direction coefficients. First and second

Fundamenta lforms. Principal and Gaussian curvatures.

Unit-V

Lines of curvature, Eulers theorem. Rodrigues formula, Conjugate and Asymptotic lines.

BookRecommended:

1.C.E.Weatherburn,DifferentialGeometryofThreeDimensions,CambridgeUniversityPress2003.Ch

apters: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

Combinartorial Mathematics

Basic counting principles, Permutations and Combinations (with and without repetitions),

Binomialt heorem, Multinomial theorem, Counting subsets, Set- partitions, Stirling numbers

Principle of Inclusion and Exclusion, Derangements, Inversion formulae Generating functions:

Algebra of formal power series, Generating function models, Calculating generating functions,

Exponential generating functions. Recurrence relations: Recurrence relation models, Divide and

conquer relations, Solution of recurrence relations, Solutions by generating functions. Integer

partitions, Systems of distinct representatives.

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)