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B.A./B.Sc. Mathematics COURSE STRUCTURE
Semester
Paper
Subject
Hrs.
Credits
IA
ES
Total
6
5
25
75
100
6
5
25
75
100
6
5
25
75
100
6
5
25
75
100
6
5
25
75
100
Paper-VI
Electives:
1) Integral Transforms
2) Numerical Analysis - I
3) Number Theory - I
4) Fluid Mechanics - I
5) Graph Theory – I
&
Elective
Problem Solving Sessions
6
5
25
75
100
Paper-VII
Linear Algebra
&
Linear Algebra
Problem Solving Sessions
6
5
25
75
100
Paper-VIII
Electives:
1) Integral Transforms &
Fourier Series
2) Numerical Analysis - II
3) Number Theory - II
4) Fluid Mechanics - II
5) Graph Theory - II
&
Elective
Problem Solving Sessions
6
5
25
75
100
FIRST YEAR
SEMESTER I
Paper-I
SEMESTER II
Paper-II
Differential Equations
&
Differential Equations
Problem Solving Sessions
Solid Geometry
&
Solid Geometry
Problem Solving Sessions
SECOND YEAR
SEMESTER
III
SEMESTER
IV
Paper-III
Paper-IV
Abstract Algebra
&
Abstract Algebra
Problem Solving Sessions
Real Analysis
&
Real Analysis
Problem Solving Sessions
THIRD YEAR
Paper-V
Ring Theory & Vector
Calculus
&
Ring Theory & Vector Calculus
Problem Solving Sessions
SEMESTER
V
SEMESTER
VI
B.A./B.Sc. FIRST YEAR MATHEMATICS SYLLABUS PAPER - I
(SEMESTER –I)
DIFFERENTIAL EQUATIONS
UNIT – I (12 Hours), Differential Equations of first order and first degree :
Linear Differential Equations; Differential Equations Reducible to Linear Form; Exact Differential Equations;
Integrating Factors; Change of Variables.
UNIT – II (12 Hours), Orthogonal Trajectories.
Differential Equations of first order but not of the first degree :
Equations solvable for p; Equations solvable for y; Equations solvable for x; Equations that do not contain. x (or y);
Equations of the first degree in x and y – Clairaut’s Equation.
UNIT – III (12 Hours), Higher order linear differential equations-I :
Solution of homogeneous linear differential equations of order n with constant coefficients; Solution of the nonhomogeneous linear differential equations with constant coefficients by means of polynomial operators.
General Solution of f(D)y=0
General Solution of f(D)y=Q when Q is a function of x.
1
is Expressed as partial fractions.
f  D
P.I. of f(D)y = Q when Q= beax
P.I. of f(D)y = Q when Q is b sin ax or b cos ax.
UNIT – IV (12 Hours), Higher order linear differential equations-II :
Solution of the non-homogeneous linear differential equations with constant coefficients.
P.I. of f(D)y = Q when Q= bx k
P.I. of f(D)y = Q when Q= e ax V
P.I. of f(D)y = Q when Q= xV
P.I. of f(D)y = Q when Q= x m V
UNIT –V (12 Hours), Higher order linear differential equations-III :
Method of variation of parameters; Linear differential Equations with non-constant coefficients; The Cauchy-Euler
Equation.
Prescribed Text Book :
Scope and treatment as in Differential Equations and Their Applications by Zafar Ahsan, published by PrenticeHall of India Learning Pvt. Ltd. New Delhi-Second edition :
Sections :- 2.5 to 2.9, 3.1, 3.2, 4.2, 5.1, 5.2, 5.3, 5.5, 5.6, 5.7.
Reference Books :
1. N. Krishna Murthy & others “A text book of mathematics for BA/BSc Vol 1 S. Chand & Company, New Delhi.
2. Rai Singhania, “Ordinary and Partial Differential Equations”, S. Chand & Company, New Delhi.
3. Differential Equations with applications and programs – S. Balachandra Rao & HR Anuradha-universities press.
B.A./B.Sc. FIRST YEAR MATHEMATICS SYLLABUS PAPER - II
(SEMESTER – II)
SOLID GEOMETRY
UNIT – I (12 hrs) : The Plane :
Equation of plane in terms of its intercepts on the axis, Equations of the plane through the given points,
Length of the perpendicular from a given point to a given plane, Bisectors of angles between two planes,
Combined equation of two planes, Orthogonal projection on a plane.
UNIT – II (12 hrs) : The Line :
Equation of a line; Angle between a line and a plane; The condition that a given line may lie in a given
plane; The condition that two given lines are coplanar; Number of arbitrary constants in the equations of straight
line; Sets of conditions which determine a line; The shortest distance between two lines; The length and equations
of the line of shortest distance between two straight lines; Length of the perpendicular from a given point to a
given line; Intersection of three planes; Triangular Prism.
UNIT – III (12 hrs) : Sphere :
Definition and equation of the sphere; Equation of the sphere through four given points; Plane sections of a
sphere; Intersection of two spheres; Equation of a circle; Sphere through a given circle; Intersection of a sphere and
a line; Power of a point; Tangent plane; Plane of contact; Polar plane; Pole of a Plane; Conjugate points; Conjugate
planes; Angle of intersection of two spheres; Conditions for two spheres to be orthogonal; Radical plane; Coaxial
system of spheres; Simplified from of the equation of two spheres.
UNIT – IV (12 hrs) : Cones :
Definitions of a cone; vertex; guiding curve; generators; Equation of the cone with a given vertex and
guiding curve; Enveloping cone of a sphere; Equations of cones with vertex at origin are homogenous; Condition
that the general equation of the second degree should represent a cone; Condition that a cone may have three
mutually perpendicular generators; Intersection of a line and a quadric cone; Tangent lines and tangent plane at a
point; Condition that a plane may touch a cone; Reciprocal cones; Intersection of two cones with a common vertex;
Right circular cone; Equation of the right circular cone with a given vertex; axis and semi-vertical angle.
UNIT – V (12 hrs) Cylinders :
Definition of a cylinder; Equation to the cylinder whose generators intersect a given conic and are parallel
to a given line; Enveloping cylinder of a sphere; The right circular cylinder; Equation of the right circular cylinder
with a given axis and radius.
Prescribed Text Book : Scope as in Analytical Solid Geometry by Shanti Narayan and P.K. Mittal Published by
S. Chand & Company Ltd. Seventeenth Edition.
Sections :- 2.4, 2.7, 2.9, 3.1 to 3.8, 6.1 to 6.9, 7.1 to 7.8.
Reference Books : 1. V Krishna Murthy & Others “A text book of Mathematics for BA/B.Sc Vol 1, Published
by S. Chand & Company, New Delhi.
2. P.K. Jain and Khaleel Ahmed, “A text Book of Analytical Geometry of Three
Dimensions”, Wiley Eastern Ltd., 1999.
3. Co-ordinate Geometry of two and three dimensions by P. Balasubrahmanyam,
K.Y. Subrahmanyam, G.R. Venkataraman published by Tata-MC Gran-Hill Publishers
Company Ltd., New Delhi.
Note : Concentrate on Problematic parts in all above units.
B.A./B.Sc. SECOND YEAR MATHEMATICS SYLLABUS PAPER - III
SEMESTER – III
ABSTRACT ALGEBRA
UNIT – 1 : (10 Hrs) GROUPS : Binary Operation – Algebraic structure – semi group-monoid – Group definition and elementary
properties Finite and Infinite groups – examples – order of a group. Composition tables with examples.
UNIT – 2 : (14 Hrs) SUBGROUPS : Complex Definition – Multiplication of two complexes Inverse of a complex-Subgroup definition
– examples-criterion for a complex to be a subgroups.
Criterion for the product of two subgroups to be a subgroup-union and Intersection of subgroups.
Co-sets and Lagrange’s Theorem :Cosets Definition – properties of Cosets–Index of a subgroups of a finite groups–Lagrange’s
Theorem.
UNIT –3 : (12 Hrs) NORMAL SUBGROUPS : Definition of normal subgroup – proper and improper normal subgroup–Hamilton group –
criterion for a subgroup to be a normal subgroup – intersection of two normal subgroups – Sub
group of index 2 is a normal sub group – simple group – quotient group – criteria for the existence
of a quotient group.
UNIT – 4 : (10 Hrs) HOMOMORPHISM : Definition of homomorphism – Image of homomorphism elementary properties of
homomorphism – Isomorphism – aultomorphism definitions and elementary properties–kernel of a
homomorphism – fundamental theorem on Homomorphism and applications.
UNIT – 5 : (14 Hrs) PERMUTATIONS AND CYCLIC GROUPS : Definition of permutation – permutation multiplication – Inverse of a permutation – cyclic
permutations – transposition – even and odd permutations – Cayley’s theorem.
Cyclic Groups :Definition of cyclic group – elementary properties – classification of cyclic groups.
Prescribed Text Book :
A. First course in Abstract Algebra, by J.B. Fraleigh Published by Narosa Publishing house.
Chapters : 1 to 7 and 11 to 13.
Reference Books :
1. A text book of Mathematics for B.A. / B.S. by B.V.S.S. SARMA and others Published by S.Chand &
Company New Delhi.
2. Modern Algebra by M.L. Khanna.
B.A./B.Sc. SECOND YEAR MATHEMATICS SYLLABUS PAPER-IV
(SEMESTER – IV)
REAL ANALYSIS
UNIT – I (12 hrs) : REAL NUMBERS :
The algebraic and order properties of R, Absolute value and Real line, Completeness property of R,
Applications of supreme property; intervals. No. Question is to be set from this portion.
Real Sequences : Sequences and their limits, Range and Boundedness of Sequences, Limit of a sequence and
Convergent sequence.
The Cauchy’s criterion, properly divergent sequences, Monotone sequences, Necessary and Sufficient condition
for Convergence of Monotone Sequence, Limit Point of Sequence, Subsequences and the Bolzano-weierstrass
theorem – Cauchy Sequences – Cauchey’s general principle of convergence theorem.
UNIT –II (12 hrs) : INFINITIE SERIES :
Series : Introduction to series, convergence of series. Ceanchy’s general principle of convergence for series tests
for convergence of series, Series of Non-Negative Terms.
1. P-test
2. Canchy’s nth root test or Root Test.
3. D’-Alemberts’ Test or Ratio Test.
4. Alternating Series – Leibnitz Test.
Absolute convergence and conditional convergence, semi convergence.
UNIT – III (12 hrs) : CONTINUITY :
Limits : Real valued Functions, Boundedness of a function, Limits of functions. Some extensions of the
limit concept, Infinite Limits. Limits at infinity. No. Question is to be set from this portion.
Continuous functions : Continuous functions, Combinations of continuous functions, Continuous Functions on
intervals, uniform continuity.
UNIT – IV (12 hrs) : DIFFERENTIATION AND MEAN VALUE THEORMS :
The derivability of a function, on an interval, at a point, Derivability and continuity of a function,
Graphical meaning of the Derivative, Mean value Theorems; Role’s Theorem, Lagrange’s Theorem, Cauchhy’s
Mean value Theorem - Generalized Mean value Theorems - Taylor’s Theorem, Maclaurin’s Theorem, Expansion
of functions with different forms of remainders, Taylor’s Maclaurins Seriess, power series representation of
functions.
UNIT – V (12 hrs) : RIEMANN INTEGRATION :
Riemann Integral, Riemann integral functions, Darboux theorem. Necessary and sufficient condition for R –
integrability, Properties of integrable functions, Fundamental theorem of integral calculus, integral as the limit of a
sum, Mean value Theorems.
TEXT BOOK :REAL NUMBERS
“Introduction to Real Analysis” by RABERT g BARTELY and .D.R. SHERBART Published by John
Wiley. (Chapters 3.1 to 3.7, 5.1 to 5.4, 6.1 to 6.4, 7.1 to 7.3, 9.1 to 9.3)
REFERENCE TEXT BOOKS :
1. A Text Book of B.Sc mathematics by B.V.S.S. Sarma and Published by . S. Chand & Company Pvt. Ltd.,
New Delhi.
2. Elements of Real Analysis on per UGC Syllabus by Shanthi Narayan and Dr. M.D. Raisingkania Published by
S. Chand & Company Pvt. Ltd., New Delhi.
B.A./B.Sc. THIRD YEAR MATHEMATICS SYLLABUS – PAPER-V
SEMESTER – V
RING THEORY & VECTOR CALCULUS
UNIT – 1 (12 hrs) RINGS-I : Definition of Ring and basic properties, Boolean Rings, divisors of zero and cancellation laws
Rings, Integral Domains, Division Ring and Fields, The characteristic of a ring - The characteristic of an
Integral Domain, The characteristic of a Field.
UNIT – 2 (12 hrs) RINGS-II : Definition of Homomorphism – Homorphic Image – Elementary Properties of Homomorphism –
Kernel of a Homomorphism – Fundamental theorem of Homomorhphism –
Maximal Ideals – Prime Ideals.
UNIT –3 (12 hrs) VECTOR DIFFERENTIATION : Vector Differentiation, Ordinary derivatives of vectors, Differentiability, Gradient, Divergence,
Curl operators, Formulae Involving these operators.
UNIT – 4 (12 hrs) VECTOR INTEGRATION : Line Integral, Surface Integral, Volume integral with examples.
UNIT – 5 (12 hrs) VECTOR INTEGRATION APPLICATIONS : Theorems of Gauss and Stokes, Green’s theorem in plane and applications of these theorems.
Prescribed Text Books :
1. First course in Abstract Algebra by J. Fralieh Published by Narosa Publishing house.
Sections :- 23.1, 23.2, 24.1, 24.2, 24.3, 25.1, 25.4, 29.1, 29.2.
2. Vector Calculus by Santhi Narayana Published by S. Chand & Company Pvt. Ltd., New Delhi.
Sections :- Chapter 6 (6.1 to 6.17), Chapter 7 (7.1 to 7.11).
Reference Books :- 1. A text Book of B.Sc., Mathematics by B.V.S.S.Sarma and others publisher by
S. Chand & Company Pvt. Ltd., New Delhi.
2. Vector Calculus by R. Gupta Published by Laxmi Publications.
3. Vector Calculus by P.C. Matthews Published by Springer Vector Publicattions.
4. Rings and Linear Algebra by Pundir & Pundir Published by Pragathi Prakashan.
B.A./B.Sc. THIRD YEAR MATHEMATICS SYLLABUS PAPER - VI
SEMESTER – V (ELECTIVE-1)
INTEGRAL TRANSFORMS
UNIT – 1 (12 hrs) Laplace Transform I : Definition of - Integral Transform – Laplace Transform Linearity, Property, Piecewise continuous
Functions, Existence of Laplace Transform, Functions of Exponential order, and of Class A, First Shifting
Theorem. Second Shifting Theorem, Change of Scale Property, Laplace Transform of the derivative of f(t),
UNIT – 2 (12 hrs) Laplace Transform II : Initial Value theorem and Final Value theorem, Laplace Transform of Integrals – Multiplication by t,
Multiplication by tn – Division by t. Laplace transform of Bessel Function, Laplace Transform of Error
Function, Laplace Transform of Sine and cosine integrals.
UNIT –3 (12 hrs) Inverse Laplace Transform I : Definition of Inverse Laplace Transform. Linearity, Property, First Shifting Theorem, Second Shifting
Theorem, Change of Scale property, use of partial fractions, Examples.
UNIT – 4 (12 hrs) Inverse Laplace Transform II : Inverse Laplace transforms of Derivatives–Inverse Laplace Transforms of Integrals – Multiplication
by Powers of ‘P’– Division by powers of ‘P’– Convolution Definition – Convolution Theorem – proof and
Applications – Heaviside’s Expansion theorem and its Applications.
UNIT – 5 (12 hrs) Application of Laplace Transform to solutions of Differential Equations : Solutions of ordinary Differential Equations.
Solutions of Differential Equations with constants co-efficient
Solutions of Differential Equations with Variable co-efficient
Prescribed Text Books : Integral Transforms by A.R. Vasistha and Dr. R.K. Gupta Published by Krishna
Prakashan Media Pvt. Ltd. Meerut.
Sections :- 1.1 to 1.13 and 1.16 to 1.20. 2.1 to 2.9, 2.10 to 2.16 and 3.1, 3.2.
Reference Books :1. Fourier Series and Integral Transforms by Dr. S. Sreenadh S.Chand and Co., Pvt. Ltd., New Delhi.
2. Lapalce and Fourier Transforms by Dr. J.K. Goyal and K.P. Gupta, Published by Pragathi Prakashan,
Meerut.
3. Integral Transforms by M.D. Raising hania, - H.C. Saxsena and H.K. Dass Published by S. Chand and
Company Pvt. Ltd., New Delhi.
FINAL YEAR B.A/B.SC MATHEMATICS: PAPER VI (ELECTIVE – 2)
SEMESTER – V
NUMERICAL ANALYSIS
60 Hrs
UNIT- I : (12 hours)
Errors in Numerical computations : Errors and their Accuracy, Mathematical Preliminaries, Errors and
their Analysis, Absolute, Relative and Percentage Errors, A general error formula, Error in a series
approximation.
Solution of Algebraic and Transcendental Equations: The bisection method, The iteration method,
The method of false position, Newton Raphson method, Generalized.
UNIT – II : (12 hours) Interpolation - I
Interpolation : Errors in polynomial interpolation, Finite Differences, Forward differences, Backward
differences, Central Differences, Symbolic relations, Detection of errors by use of Differences Tables,
Differences of a polynomial, Newton’s formulae for interpolation.
UNIT – III : (12 hours) Interpolation - II
Central Difference Interpolation Formulae, Gauss’s central difference formulae, Stirling’s central
difference formula, Bessel’s Formula, Everett’s Formula. Relation between Bessel’s and Everett’s
Formulae.
UNIT – IV : (12 hours) Interpolation - III
Interpolation with unevenly spaced points, Lagrange’s formula, Error in Lagrange’s formula, Derivation
of governing equations, End conditions, Divided differences and their properties, Newton’s general
interpolation Formula, Inverse interpolation, Method of successive approximations.
Unit – V (12 Hours)
Curve Fitting : Least – Squares curve fitting procedures, fitting a straight line, nonlinear curve fitting,
Curve fitting by a sum of exponentials, approximation of functions, Chebyshev polynomials,
Economization of power series.
Prescribed Text Book : Scope as in Introductory Methods of Numerical Analysis by S.S.Sastry,
Prentice Hall of India Pvt. Ltd., New Delhi. (Latest Edition) Chapter 1,2,3 and 4.
Reference Books :
1. G. Sankar Rao New Age International Publishers, New – Hyderabad.
2. Finite Differences and Numerical Analysis by H.C Saxena S. Chand and Company, Pvt. Ltd., New Delhi.
3. Numerical methods for scientific and engineering computation by M.K.Jain, S.R.K.Iyengar, R.K. Jain.
FINAL YEAR B.A/B.SC MATHEMATICS : PAPER - VI (ELECTIVE - 3 )
SEMESTER - V
NUMBER THEORY
60 H
UNIT – I (12 hours)
Divisibility , Greatest common divisor , Euclidean Algorithm , The Fundamental Theorem of
arithmetic , congruence’s , Special divisibility tests , Chinese remainder theorem , Fermat’s little
theorem , Wilson’s theorem , residue classes and reduced residue classes.
UNIT – II (12 hours)
Euler’s theorem , An Application to cryptography, ( n ) , Mobius inversion Formula , The greatest
integer function,   n  ,   n  , d (n),  Arithmetic functions perfect numbers , Mersenne primes and
Fermat numbers Primitive roots and indices Quadratic residues.
UNIT – III (12 hours)
Legendre symbol, Quadratic reciprocity law , Jacobi symbol, Binary quadratic forms and their
reduction
sums
of two
and
four
squares , positive
definite
binary
quadratic
ax  by  c, 2  y 2  z 2, 4  y 4  z 2. Farey sequences , Continued fractions , Approximation of reals
by rationals , Pell’s equations , Minkowski’s theorem in Geometry of Number s and its applications.
UNIT – IV (12 hours)
Order of magnitude and average order o f arithmetic functions. Euler summation formula , Abel’s
Identify , Elementary results on distribution of primes , Congruences Elementary properties of prime
numbers . Residue classes and Euler’s function. Linear congruences and congruences of higher
degree.
UNIT – IV (12 hours)
Congruence’s with prime moduli. The theorems of Fermat, Euler and Wilson. Number – Theoretic
Functions 22 Mobius function. The function [] , The symbols O and their basic properties.
Prescribed Text Books :
1. ‘ Introduction to the Theory of Numbers’ , 5th Edition, by Niven , Zuckerman &
Montgomery.
2. “ Elementary Number Theory ” , 2nd Edition, by David M. Burton
References Books :
1. Elementary Number Theory , 2nd Edition (UBS Publishers) by David, M. Burton.
2. Introduction to Theory of Numbers, 5th Edition (John Wiley & Sons) Niven, Zuckerman
& Montgomery.
3. (Camb, Univ, Press) by Davenport H., Higher Arithmetic.
4. Number Theory (Oxford Univ, Press) Hardy & Wright.
5. Element of the Theory of Numbers (Academic Press) Dence, J. B & Dence T.P.
FINAL YEAR B.A/B.SC MATHEMATICS: PAPER VI (ELECTIVE – 4)
SEMESTER – V
FLUID MECHANICS
60 Hrs
Unit – I : (12 hours)
Conservation of Matter Introduction. Fields and continuum concepts. Lagrangian and Eulerian
specifications. Local, Convective and total rates of charge. Conservation of mass. Equation of continuity.
Unit – II (12 hours)
Boundary conditions. Nature of Forces and Fluid Flow Surface and body forces. Stress at a point.
Viscosity and Newton’s viscosity law. Viscous and inviscid flows.
Unit – III (12 hours)
Laminar and turbulent flows. Compressible and incompressible flows. Bernoulli’s theorem and its
applications. Circulation, rate of change of circulation (kelvin’s theorem). Aaxially symmetric motion.
Unit – IV (12 hours)
Kelvin’s minimum energy theorem. Conservation of linear momentum. Bernoulli’s theorem and its
applications. Circulation, Rate of change of circulation (Kelvin’s theorem). Aaxially symmetric motion.
Unit – V (12 hours)
Stokes’s stream function. Two dimensional Motion stream function. Complex potential and Complex
velocity, Uniform flows. Sources, Sinks and vortex flows. Flow in a sector. Flow around a sharp edge.
Flow due to a doublet.
Prescribed Text Book :
1. Classical Mechanics by Herbert Goldstin, published by Narosa Publications, New Delhi.
2. A Text Book of Fluid Dynamic by F. Charlton Published by CBS Publications, New Delhi.
Reference Book :
1. T. Allen and I.L. Ditsworth : Fluid Mechanicsm, (McGraw Hill, 1972)
2. I.G. Currie : Fundamentals of Mechanics of fluids, (CRC, 2002)
3. Chia-shun Yeh : Fluid Mechanics : An Introduction to the theory, (McGraw Hill, 1974)
4. F.M. White : Fluids Mechanics, (McGraw Hill, 2003)
5. R.W. Fox, A.T Mc Donald and P.J. Pritchard : Introduction to Fluid Mechanics, (John Wiley and Sons
Pvt. Ltd., 2003)
B.A./B.Sc. THIRD YEAR MATHEMATICS SYLLABUS PAPER - VI
SEMESTER – V (ELECTIVE – 5)
GRAPH THEORY-1
UNIT – I (12 hrs) Graphs and Sub Graphs :
Graphs and Simple graph, graph isomorphism, the incidence and adjacency matrices, sub graphs, vertex degree,
Hand shaking theorem, paths and connection, cycles Applications, the shortest path problem, Sperner’s lemma.
UNIT – II (12 hrs) Trees :
Trees, cut edges and Bonds, cut vertices, cayley’s formula and applications, the connector problem.
UNIT – III (12 hrs) :
Connectivity, Blocks and Applications, construction of reliable communication Networks, Euler tours, Hamilton
cycles and Applications, the Chinese postman problem, the travelling salesman problem.
UNIT – IV (12 hrs) :
Matching’s, Matching’s and coverings in Bipartite graphs, perfect matching’s and Applications, the perfect
Assignment problem, the optical Assignment problem.
UNIT –V (12 hrs) :
Edge Coloring’s :- Edge Chromatic Number, vizing’s theorem and applications, the timetabling problem.
Prescribed Text Book : Scope and standard as in chapters 1 to 6 of graph theory with Applications, by J.A. Bondy
and U.S.R. Murthy, of M.C. Millan Press.
Reference Books : 1. S. Arumugham and S. Ramachandran, Introduction to Graph theory of scitech
Publications, Chennai-17.
2. A Text Book of Discrete Mathamatics by Dr. Swapan Kumar Sankar, S.Chand & Co.
Publishers, New Delhi.
3. Graph theory and combinations by H.S. Govinda Rao of Galgotia Publications.
B.A./B.Sc. THIRD YEAR MATHEMATICS SYLLABUS – PAPER-VII
SEMESTER – VI
LINEAR ALGEBRA
UNIT – I (12 hrs) : Vector Spaces-I :
Vector Spaces, General properties of vector spaces, n-dimensional Vectors, addition and scalar multiplication of
Vectors, internal and external composition, Null space, Vector subspaces, Algebra of subspaces, Linear Sum of
two subspaces, linear combination of Vectors, Linear span Linear independence and Linear dependence of
Vectors.
UNIT –II (12 hrs) : Vector Spaces-II :
Basis of Vector space, Finite dimensional Vector spaces, basis extension, co-ordinates, Dimension of a Vector
space, Dimension of a subspace, Quotient space and Dimension of Quotientspace.
UNIT –III (12 hrs) : Linear Transformations :
Linear transformations, linear operators, Properties of L.T, sum and product of LTs, Algebra of Linear Operators,
Range and null space of linear transformation, Rank and Nullity of linear transformations – Rank – Nullity
Theorem.
UNIT –IV (12 hrs) : Matrix :
Matrices, Elementary Properties of Matrices, Inverse Matrices, Rank of Matrix, Linear Equations, Characteristic
Roots, Characteristic Values & Vectors of square Matrix, Cayley – Hamilton Theorem.
UNIT –V (12 hrs) : Inner product space :
Inner product spaces, Euclidean and unitary spaces, Norm or length of a Vector, Schwartz inequality, Triangle in
Inequality, Parallelogram law, Orthogonality, Orthonormal set, complete orthonormal set, Gram – Schmidt
orthogonalisation process. Bessel’s inequality and Parseval’s Identity.
Prescribed Text Books :1. Linear Algebra by J.N. Sharma and A.R. Vasista of Krishna Prakashan Mandir, Meerut-250002.
2. Matrices by Shanti Narayana (S.Chand Publications).
Reference Books : 1. Linear Algebra by Kenneth Hoffman and Ray Kunze, Pearson Education (low priced
edition), New Delhi.
2. Linear Algebra by Stephen H. Friedberg et al Prentice Hall of India Pvt. Ltd. 4th Edition
2007.
B.A./B.Sc. THIRD YEAR MATHEMATICS SYLLABUS PAPER - VIII
SEMESTER – VI (ELECTIVE-1)
INTEGRAL TRANSFORMS AND FOURIER SERIES
UNIT – 1 (12 hrs) Application of Laplace Transform : Solution of simultaneous ordinary Differential Equations.
Solutions of partial Differential Equations.
UNIT – 2 (12 hrs) Application of Laplace Transforms to Integral Equations : Definitions : Integral Equations-Abel’s, Integral Equation-Integral Equation of Convolution Type, Integro
Differential Equations.
Application of L.T. to Integral Equations.
UNIT –3 (12 hrs) Fourier Transforms-I : Definition of Fourier Transform – Fourier’s in Transform – Fourier cosine Transform – Linear Property of
Fourier Transform – Change of Scale Property for Fourier Transform – sine Transform and cosine transform
shifting property – modulation theorem.
UNIT – 4 (12 hrs) Fourier Transform-II : Convolution Definition – Convolution Theorem for Fourier transform – parseval’s Indentify –
Relationship between Fourier and Laplace transforms – problems related to Integral Equations.
Finte Fourier Transforms : Finte Fourier Sine Transform – Finte Fourier Cosine Transform – Inversion formula for sine and cosine
Transforms only statement and related problems.
UNIT –5 (12 hrs) Fourier Series : Fourier Series, Theorems, Dirichlet’s Conditions, Fourier series for even and odd functions, Half range
Fourier series, Other forms of Fourier series.
Prescribed Text Books :
1. Integral Transforms by A.R. Vasistha and Dr. R.K. Gupta Published by Krishna Prakashan Media Pvt. Ltd. Meerut.
Sections :- 3.3, 3,4, 4.1, 4,2 and 6.4 to 6.14 and 6.17 to 6.20 and 7.1 to 7.4.
2. A Course of Mathematical Analysis by Shanthi Narayana and P.K. Mittal, by S. Chand and Company pvt. Ltd., New
Delhi.
Sections :- 10.1, 10.5, 10.6, 10.7, 10.8.
Reference Books :- 1. Fourier Series and Integral Transforms by Dr. S. Sreenadh S.Chand and Company Pvt. Ltd.,
New Delhi.
2. Lapalce and Fourier Transforms by Dr. J.K. Goyal and K.P. Gupta, Published by Pragathi
Prakashan, Meerut.
3. Integral Transforms by M.D. Raising hania, - H.C. Saxsena and H.K. Dass Published by
S. Chand and Company pvt. Ltd., New Delhi.
THIRD YEAR B.SC MATHEMATICS: PAPER VIII (ELECTIVE – 2)
SEMESTER – VI
NUMERICAL ANALYSIS
60 Hrs
UNIT- I : (12 hours)
Numerical Differentiation and Numerical Integration : Numerical differentiation, Errors in numerical
differentiation, Maximum and minimum values of a tabulated function.
Numerical integration, Trapozoidal rule, Simpson’s 1/3 – rule, Simpson’s 3/8 – rule, Boole’s and
Weddle’s rules, Romberg Integration, Euler – Maclaurin Formula, Gaussian Integration, Numerical
Evaluation of Singular integrals, Trapezoidal Rule.
UNIT – II: (12 hours)
Matrices and Linear Systems of Equations:
Basic Definition, Matrix Operations, Transpose and Rank of a Matrix, Consistency of a Linear System of
Equations, Inverse of Matrix, Linear systems of equations, Solution of linear systems – Direct methods,
Matrix inversion method, Gaussian elimination methods, Method of factorization, Solution of
Tridiagonal Systems, Ill-conditioned linear systems. Iterative methods. Jacobi’s method, Gauss-siedal
method.
UNIT – III (12 Hours)
Numerical solution of ordinary differential equations: Introduction, Solution by Taylor’s Series,
Picard’s method of successive approximations, Euler’s method, Modified Euler’s method, Runge – Kutta
methods, Predictor – Corrector methods, Adams – Moulton Method, Milne’s method. Simultaneous and
Higher Order Equations, Boundary vale problems, Finite-Difference Method.
UNIT – IV (12 Hours)
Numerical solution of Partial differential equations: Introduction, Finite – Difference approximations
to Derivatives, Laplace’s Equation, Jacobi’s method, Gauss – Seidal method, Successive overRelaxation or S.O.R method, Iterative methods for the solution of equations.
UNIT – V (12 Hours)
Numerical solution of integral equations : Introduction, finite difference methods, chebyshev series
method, method of invariant inbedding, method using generalized quadrature, A method for degenerate
kernels.
Prescribed text Book: Scope as in Introductory Methods of Numerical Analysis by S.S.Sastry, Prentice
Hall India (Latest Edition). Chapters :- 5, 6, 7, 8 and 9.
Reference Books :
1. G. Sankar Rao New Age International Publishers, New – Hyderabad.
2. Finite Differences and Numerical Analysis by H.C Saxena S. Chand and Company, Pvt. Ltd., New Delhi.
3. Numerical methods for scientific and engineering computation by M.K.Jain, S.R.K.Iyengar, R.K.
Jain.
THIRD YEAR B.SC MATHEMATICS: PAPER – VIII (ELECTIVE – 3)
SEMESTER – VI
NUMBER THEORY
60 Hrs
UNIT-I (12 hours)
Primitive roots and indices Integers belonging to a given exponent ( mod P ) . Primitive roots
and composite modull . Determination of integers having primitive roots. Indices, Solutions , of
Higher Congruences by Indices .
UNIT- II (12 hours)
Diophantine Equations Equations and fermat’s conjecture for n = 2, n = 4. Quadratic Residues
Composite moduli , Legendre symbol . Law of quadratic reciprocity .
UNIT-III (12 hours)
The Jacobi symbol. Algebraic Number Theory Polynomials over a field . Divisibility properties
of Polynomials. Gauss’s lemma. The Eisenstein’s irreducibility criterion. Symmetric Polynomials.
UNIT-IV (12 hours)
Extensions of a field. Algebraic and transcendental numbers. Bases and finite extensions, Properties of
finite extensions. Conjugates and discriminants. Al gebraic integers in a quadratic field ,
UNIT-V (12 hours)
Integral bases . Units and primes in a quadratic field . Ideals , Arithmetic of ideals in an algebraic
number Field . The horm of an ideal , Prime ideals .
Prescribed Text Book :
[ Scope as in 6 & 7 of ‘ Introduction to the Theory of Numbers’ , 5th Edition, by Niven ,
Zuckerman & Montgomery.]
[ Scope as in Chapters 2- 8 , 10 of “ Elementary Number Theory ”, 2nd Edition, by David M.
Burton, Chapters 3, 5 ( sections 5.1 , 5.3 , 5.4 ) of ‘ Introduction to the Theory of Numbers ‘, 5th
Edition , by Niven , Zuckerman & Montgomery . )
References Text Books :
1. David, M. Burton, Elementary Number Theory, 2nd Edition (UBS Publishers).
2. Niven, Zuckerman & Montgomery Introduction to Theory of Numbers, 5th Edition (John Wiley & Sons).
3. Davenport H., Higher arithmetic (Camb, Univ, Press).
4. Hardy & Wright, Number Theory (Oxford Univ. Press).
5. Dence, J.B & Dence T.P, Element of the Theory of Numbers (Academic Press).
THIRD YEAR B.SC MATHEMATICS: PAPER VIII (ELECTIVE – 4)
SEMESTER – VI
FLUID MECHANICS
60 Hrs
Unit – I : (12 hours)
Two and Three – Dimensional Potential Flows Circular cylinder without circulation. Circular Cylinder
with circulation. 24 Blasius theorem. Kutta condition and the flat – plate airfoil. Joukowski airfoil.
Vortex motion.
Unit – II
(12 hours)
Karman’s vortex street. Method of Images. Velocity Potential. Stoke’s stream function. Solution of the
Potential equation. Uniform flow. Source and sink.
Unit – III (12 hours)
Flow due to a doublet. Viscous Flows of incompressible fluids Constitutive equations. Navier–Stokes
equations and their exact solutions. Steady unidirectional flow. Poiseuille flow.
Unit – IV (12 hours)
Steady unidirectional flow. Poiseuille flow. Couette flow. Flow between rotating cylinders. Stoke’s first
problem. Stoke’s second problem.
Unit – V (12 hours)
Approach to Fluid Flow Problems Similarity from a differential equation. Dimensional analysis. One
dimensional, Steady compressible flow.
Prescribed Text Book :
1. Classical Mechanics by Herbert Goldstin, published by Narosa Publications, New Delhi.
2. A Text Book of Fluid Dynamic by F. Charlton Published by CBS Publications, New Delhi.
Reference Text Books :
1. T. Allen and I.L. Ditsworth : Fluid Mechanics, (McGraw Hill, 1972)
2. I.G. Currie : Fundamentals of Mechanics of fluids, (CRC, 2002)
3. Chia-shun Yeh : Fluid Mechanics : An Introduction to the theory, (McGraw Hill, 1974)
4. F.M White : Fluids Mechanics, (McGraw Hill, 2003)
5. R.W Fox, A.T Mc Donald and P.J. Pritchard : Introduction to Fluid Mechanics, (John Wiley and Sons
Pvt. Ltd., 2003)
B.A/.B.Sc. THIRD YEAR MATHEMATICS SYLLABUS PAPER - VIII
SEMESTER – VI (ELECTIVE – 5)
GRAPH THEORY-2
UNIT –1 (12 hrs) :
Independent sets, Ramsey’s theorem, Turom’s theorem and Applications, sehur’s theorem. A Geometry problem
chromatic Number, Broke’s theorem, Hajos conjecture, Chromatic polynomials, Girth and chromatic Number and
Applications, A storage problem.
UNIT –2(12 hrs) :
Plane and planar graphs, Dual graphs, Euler’s Formula, Bridges, Kuratowster’s, Theorem the five colour theorem
and the four-colour-conjedure, Non-hamiltonian planar graphs and Applications.
UNIT –3 (12 hrs) :
Directed graphs, directed paths, Directed cycles and Applications, A Job sequencing problem, Designing an
Efficient Computer Drum, Marking a Road System one way, Ranking the participants in a tournaments.
UNIT –4 (12 hrs) : Networks :- Flows, Cuts, the Max-flow, Min-cut theorem and Applications, Menger’s
theorems, Feasible Flows.
UNIT –5 (12 hrs) :
The cycle space and sand space :- Circulations and potential differences, the number of spanning trees and
Applications, perfect sources.
Prescribed Book :- Scope and standard as in chapters 7 to 12 of graph theory with Applications by J.A. Bondy and
U.S.R. Murthy, MC. Millan Press.
Reference Books :1. S. Arumugham and S. Ramachandran : Invitation to graph theory, SciTech publications, Chennai-17.
2. A text book of Discrete Mathematics by Dr. Swapan Kumar Sarkar, S. Chand Publishers.
3. Graph theory and combinations by H.S. Govinda Rao, Galgotia Publications.
Revised Common Framework of CBCS for Colleges in Andhra Pradesh
(A.P. State Council of Higher Education)
Table-7: B.Sc., SEMESTER – I
Sno
1
2
3
4
5
6
7
8
9
10
Total Mid Sem Sem End Teaching
Credits
Marks
Exam*
Exam
Hours
Course
First Language
(Tel/Hin/Urdu/Sans…)
Second Language
English
Foundation Course - 1
HVPE (Human Values &
Professional Ethics)
Foundation course -2
Communication & Soft Skills -1
DSC 1 A
(Group Sub- 1)
DSC 1 A Lab Practical
DSC 2 A
(Group Sub- 2)
DSC 2 A Lab Practical
DSC 3 A
(Group Sub- 3)
DSC 3 A Lab Practical
Total
100
25
75
4
3
100
25
75
4
3
50
0
50
2
2
50
0
50
2
2
100
25
75
4
3
50
0
50
2
2
100
25
75
4
3
50
0
50
2
2
100
25
75
4
3
50
0
50
2
2
750
-
-
30
25
Note: For Science Domain Subjects which had no lab practical component earlier (eg. Mathematics) the
following format is applicable. They, however, will have co-curricular activities (eg. Problem solving sessions
etc.). The total marks will change accordingly for such combinations. For example for Maths, Physics and
Chemistry the total marks will be 700.
DSC (without Lab Practical)
100
25
75
6
5
 Mid sem exam at the college (The marks split between Formal Test and Co-curricular activities may be
decided by the University concerned). End Sem Exam by the Univ.
 Practical component will not be applicable to those science subjects which had no such component earlier
(ex. Mathematics)
**Syllabus size shall be in accordance with the number of teaching hours
Sno
1
2
3
4
5
6
7
8
9
10
Course
First Language
(Tel/Hin/Urdu/Sans…)
Second Language
English
Foundation course - 3
Environmental Sci
Foundation course – 4A
ICT – 1 (Information &
Communication Technol)
DSC* 1 B
(Group Sub- 1)
DSC 1 B Lab Practical
Table-8: B.Sc., SEMESTER – II
Total Mid Sem Sem End
Marks
Exam
Exam
100
25
75
Teaching Credits
Hours
4
3
100
25
75
4
3
50
0
50
2
2
50
0
50
2
2
100
25
75
4
3
50
0
50
2
2
DSC 2 B
(Group Sub- 2)
DSC 2 B Lab Practical
100
25
75
4
3
50
0
50
2
2
DSC 3 B
(Group Sub- 3)
DSC 3 B Lab Practical
100
25
75
4
3
50
0
50
2
2
Total
750
-
-
30
25
B.Sc. Table-9: B.Sc., SEMESTER – III
Sno
1
2
3
4
5
6
7
8
9
10
Course
Total
Marks
First Language
100
(Tel/Hin/Urdu/Sans…)
Second Language
100
English
Foundation Course - 5
50
Entrepreneurship
Foundation course -2B
50
Communication &
Soft Skills -2
DSC 1 C
100
(Group Sub- 1)
DSC 1 C Practical
50
Mid Sem
Exam
25
Sem End
Exam
75
Teaching Credits
Hours
4
3
25
75
4
3
0
50
2
2
0
50
2
2
25
75
4
3
0
50
2
2
DSC 2 C
(Group Sub- 2)
DSC 2 C Practical
100
25
75
4
3
50
0
50
2
2
DSC 3 C
(Group Sub- 3)
DSC 3 C Practical
100
25
75
4
3
50
0
50
2
2
Total
750
-
-
30
25
Sno
1
2
3
4
5
6
7
8
9
10
Table-10: B.Sc., SEMESTER – IV
Course Total Mid Sem Sem End Teaching Credits
Marks
Exam*
Exam
Hours**
Foundation Course – 2C*
50
0
50
2
2
Communication &
Soft Skills -3
Foundation Course – 6*
50
0
50
2
2
Analytical Skills
Foundation Course - 7 **
50
0
50
2
2
CE (Citizenship Education)
Foundation course – 4B
50
0
50
2
2
ICT – 2 (Information &
Communication Technol)
DSC 1 D
100
25
75
4
3
(Group Sub- 1)
DSC 1 D Lab Practical
50
0
50
2
2
DSC 2 D
(Group Sub- 2)
DSC 2 D Lab Practical
100
25
75
4
3
50
0
50
2
2
DSC 3 D
(Group Sub- 3)
DSC 3 D Lab Practical
100
25
75
4
3
50
0
50
2
2
Total
750
30
*To be taught by English Teachers (and partly by Maths/Stat Teachers)
** To be taught by Telugu Teachers
25
Table-11: B.Sc., SEMESTER – V
Sno
Total
Marks
1 Skill Development Course – 1
50
(University’s Choice)
2
DSC 1 E
100
(Group Sub- 1)
3
DSC 1 E Lab Practical
50
4
Course
Mid Sem
Exam
0
Sem End
Exam
50
Teaching Credits
Hours
2
2
25
75
4
3
0
50
2
2
DSC 2 E
(Group Sub- 2)
DSC 2 E Lab Practical
100
25
75
4
3
50
0
50
2
2
DSC 3 E
(Group Sub- 3)
DSC 3 E Lab Practical
100
25
75
4
3
50
0
50
2
2
Elective -1*: DSC 1 F /
Inter-disp
Elective-1 Lab Practical
100
25
75
4
3
50
0
50
2
2
Elective*-2: DSC 2 F /
Inter-disp
Elective-2 Lab Practical
100
25
75
4
3
50
0
50
2
2
100
25
75
4
3
13
Elective*-3: DSC 3 F /
Inter-disp
Elective-3 Lab Practical
50
0
50
2
2
14
Total
950
-
-
38
32
5
6
7
8
9
10
11
12
6th (F) paper of each of the domain specific subjects (2nd paper of semester V) may preferably be an
Elective. More than one Elective may be offered giving choice to students. The Electives may be of
Domain (applied/specialization) or Inter-disciplinary in nature. The number of Electives may be decided
(along with the syllabus) by the University concerned keeping the feasibility of conduct of University
examinations in view.
Table-12: B.Sc., SEMESTER – VI
Sno
Course
Total
Marks
50
Mid Sem
Exam
0
100
25
75
4
3
50
0
50
2
2
DSC 2 G
(Group Sub- 2)
DSC 2 G Lab Practical
100
25
75
4
3
50
0
50
2
2
DSC 3 G
(Group Sub- 3)
DSC 3 G Lab Practical
100
25
75
4
3
50
0
50
2
2
Elective -4*: DSC 1 H /
Inter-disp/Gen Elec
Elective-4 Lab Practical
100
25
75
4
3
50
0
50
2
2
Elective*-5: DSC 2 H /
Inter-disp/Gen Elec
Elective-5 Lab Practical
100
25
75
4
3
50
0
50
2
2
100
25
75
4
3
13
Elective*-6: DSC 3 H /
Inter-disp/Gen Elec
Elective-3 Lab Practical
50
0
50
2
2
14
Total
950
-
-
38
32
1 Skill Development Course – 2
(University’s Choice)
2
DSC 1 G
(Group Sub- 1)
3
DSC 1 G Lab Practical
4
5
6
7
8
9
10
11
12
Sem End
Teaching Hours
Exam
50
2
Credits
2
*8th (H) paper of each of the domain specific subjects (2nd paper of semester VI) may preferably be an
Elective. More than one Elective may be offered giving choice to students. The Electives may be of
Domain (applied/specialization) or Inter-disciplinary or General in nature. The number of Electives may be
decided (along with the syllabus) by the University concerned keeping the feasibility of conduct of
University examinations in view.
Total Credits for a B.Sc. Course: 164