RASHTRASANT TUKDOJI MAHARAJ NAGPUR
UNIVERSITY, NAGPUR
BOARD OF STUDIES IN MATHEMATICS
B. Sc. Three Years (SIX SEMESTER) DEGREE
COURSE
B.Sc. Part I (Semester I & II)
B. Sc. Part II and Final (Semester III, IV.
V & VI)
## B.
Sc. Part I ( Semester I )
M-1 Algebra and Trigonometry
Unit I
Rank of matrix , Normal form , Some theorems
on rank(without proof) , Linear equation,
Solution of non- homogeneous linear
equations , Homogeneous linear equations.
Eigen values, Eigen vectors , Characteristic
value problem , Cayley-Hamilton theorem
(without proof) and it’s use in finding
Inverse of a matrix.
Unit II
Relation between the roots and coefficients
of general polynomial equation in one
variable , Transformation of equations ,
Reciprocal equations , Descartes rules of
signs , Solution of cubic equations (Cardon
method) , Biquadratic equations (Ferrari’s
method).
Unit III
De Moivre’s theorem & its applications ,
Inverse circular and hyperbolic functions ,
Logaritham of a complex quantity , Expansion
of trigonometrical functions , Gregory’s
series.
Unit IV
Definition of a group with examples and
properties , Subgroups , Cosets ,
Langrange’s theorem , Permutation groups ,
Even and odd permutations.
Text Books:
Mathematics For Degree Students (B.Sc. First
Year)
Dr. P. K. Mittal , S. Chand and Co. Ltd, New
Delhi , 2010.
For Unit –I: Scope: Chapters 6, 7, 8 of
Algebra with articles 6.2, 6.3, 6.3.1
(without proofs), Examples, Exercise, 7.1,
7.3, 7.4 (without proofs), 7.5, Examples,
Exercise, 8.1, Examples, Exercise,
8.3(without proof), 8.3.1, Examples,
Exercise.
For Unit-II: Scope: Chapter 9 of Algebra
with articles 9.1, 9.2( without proofs) 9.3,
9.4, Example, Exercise, 9.5, Examples,
Exercise, 9.6, Example, Exercise, 9.7,9.8,
9.8.1, Example, Exercise, 9.11, 9.12,
Examples, Exercise, 9.13.1, Example,
Exercise.
For Unit-III: Scope: Chapters 1,2,3,4,5 of
Trigonometry with articles 1.1, 1.2, 2.1,
2.2, 2.3, 2.4, 2.5, 2.9, 3.1, 3.2, 3.3, 3.4,
4.1, 4.1.1, 4.12, 5.1, 5.2, 5.3
For Unit-IV: Scope: Chapters 10, 11 of
Algebra with articles 10.1, 10.1.1, 10.3,
10.3.1, 10.4 (theorems 1-6) 10.7, 10.10,
10.10.1, 10.10.2, 10.12, 10.13, 11.1, 11.2,
11.3, 11.4, 11.5, 11.6.
Reference Books:
1. I. N. Herstein: Topics in Algebra, Wiley
Eastern Ltd., New Delhi, 1975
2. K. B. Datta: Matrix and Linear Algebra,
Prentice Hall of India Pvt. Ltd., New Delhi-
2000.
3. P. B. Bhattacharya, S. K. Jain and S. R.
Nagpaul: First course in linear algebra,
Willey western New Delhi, 1983.
4. P. B. Bhattacharya, S. K. Jain and S. R.
Nagpaul: Basic Abstract Algebra (2nd
edition) Cambridge University Press, Indian
Edition, 1997.
5. S. K. Jain, A. Gunawardena and P. B.
Bhattacharya: Basic Linear Algebra with
Matlab, Key College Publishing
(Springer-Verlag), 2001.
6. H. S. Hall and S. R. Knight: Higher
Algebra, S.Chand & Co. Ltd.,New Delhi,2008.
7. R. S. Verma & K. S. Shukla: Text Book on
Trigonometry, Pothishala Pvt. Ltd. Allahbad.
8. Ayres Jr. Frank: Matrices, Schaum’s
Outline Series, Mcgraw Hill Book Company,
Singapre, 1983.
9. Hohn Franz E: Elementary Matrix Algebra,
Amerind Publishing Co. Ltd., 1964.
10. McCoyNeal H: Introduction to Modern
Algebra, Allyn & Bacon inc., 1965.
11. Spiegel M. R.: Complex Variables,
Scaum’s Outline Series, McGraw- Hill, 1981.
12. Shanti Narayan: A Text Book of Matrices,
S. Chand & Co. Ltd., New Delhi.
13. S.L.Loney:Plane Trigonometry(Part II),S.
Chand & Co. Ltd.,New Delhi,
14. Chandrika Prasad:Text Book on Algebra
and Theory of Equations, Pothishala Private
Ltd., Allahabad.
##
B. Sc. Part I ( Semester I )
M-2 Calculus
Unit
I
Definition of the limit of the function of
one variable and basic properties of limit ,
Continuous function of one variable and
classification of discontinuities (only
definitions with examples) , Differentiation
, Successive differentiation , Lebinitz’s
theorem.
Unit II
Maclaurin and Taylor series expansions ,
Curvature , Asymptotes , Indeterminate forms
and L’Hospital’s rule.
Unit III
Partial differentiation ,Differential and
Chain rules (Definitions and Theorems
without Proof) , Euler’s theorem on
homogeneous functions and its applications ,
Jacobians
Unit IV
Integrations of irrational algebraic
functions, Reduction formulae , Definite
integrals (Examples on properties of
definite integral).
Text Books:
1. Differential Calculus
Shanti Narayan & P. K. Mittal , S.Chand and
Co. Ltd, 2005.
Scope : Chapters 3, 4 (articles 4.1 to 4.1.4
), 5,6,10,11,12,14 (articles 14.1 to
14.3.2), 15(articles 15.1 to 15.3.2).
2. Integral Calculus
Gorakh Prasad, Pothishala Private Ltd. 1999,
Scope : Chapters 3,4,5 (Art
3.1,3.2,3.3,3.4,3.5,3.7,4.11,4.13,4.15,4.2,4.21,5.1,5.2
(without proof), 5.3).
Reference Books:
1. Gabriel Klamballel: Mathematical
Analysis, Marcel Dekkar Inc., New York, 1975
2. N. Piskunov: Differential and Integral
Calculus, Peace Publishers, Moscow.
3. P. K. Jain and S. K. Kaushik: An
Introduction to Real Analysis, S. Chand and
Co.Ltd., New Delhi, 2002.
4. Gorakh Prasad: Differential Calculus,
Pothishala Private Ltd., Allahbad.
5. Ayres F. Jr.: Calculus, Schaum’s Outline
Series, McGraw- Hill, 1981
6. Edward J.: Differential Calculus for
Beginners, MacMillan and Co. Ltd., 1963
7. Edward J.: Integral Calculus for
Beginners, AITBS Publishers and
Distributors, 1994
8. Greenspan D: Introduction to Calculus,
Harper and Row, 1968
9. Erwin, Kreyszig: Advanced Engineering
Mathematics, John Wiley and Sons, 1999.
10. Dr. P. K. Mittal: Mathematics for Degree
Students (B. Sc. First Year), S. Chand and
Co. Ltd., New Delhi,2010.
11. Murray R. Spiegel: Theory and Problems
of Advanced Calculus, Schaum’s Outline
Series, Schaum Publishing Co; New York.
##
B. Sc. Part I ( Semester II )
M-3 Geometry, Differential and Difference
Equations
Unit I
Sphere ,Plane section of sphere
,Intersection of two spheres ,Sphere through
a given circle ,Tangent line ,Tangent plane
, Right circular cone , Right circular
cylinder.
Unit II
First order exact differential equation,
Integrating factor, First order linear
differential equation and Bernoulli’s
differential equation, First order higher
degree equations (solvable for x , y, p) ,
Clairaut’s form.
Unit III
Higher order linear differential equation
with constant coefficients , Operator method
to find particular integral , Euler’s
equidimensional equation , To find unknown
solution by using known solution , Method of
variation of parameters
Unit IV
Difference equation , Formation of
difference equation , Order of difference
equation, Linear difference equation ,
Homogeneous linear equation with constant
coefficients, Non-homogeneous linear
equation ,Particular integrals.
Text Books:
1. Analytical Solid Geometry
Shanti Narayan, S.Chand & Co. Ltd, New
Delhi, 1997.
Scope: Chapters 6,7 (Art 6.11, 6.12, 6.13,
6.2, 6.31, 6.32, 6.33, 6.4, 6.41, 6.5, 6.6,
6.7, 6.71, 7.61, 7.62, 7.81, 7.82).
2. Ordinary and Partial Differential
Equations (Theory and Applications)
Nita H. Shah, PHI, 2010,
Scope: Chapters 2, 3,4,5,7
3. Calculus of Finite Differences and
Numerical Analysis,
H.C. Saxena, S. Chand and Co. Ltd, New
Delhi, 1976,
Scope: Chapter 8
Reference Books:
1. D. A. Murray: Introducing Course on
Differential Equations, Orient Longman
(India), 1967
2. E.A. Codington: An Introduction to
Ordinary Differential Equations and their
Applications, CBS Publisher and
Distribution, New Delhi, 1985
3. H. T. H. Piaggio: Elementary Treatise on
Differential Equations and Their
Applications, CBS Publisher and
Distribution, New Delhi, 1985
4. W. E. Boyee and P. C. Diprima: Elementary
Treatise on Differential Equations and
Boundry Value Problems, John Willey, 1986
5. Erwin Kreyszig: Advanced Engineering
Mathematics, John Wiley and sons, 1999
6. Gorakh Prasad and H. C. Gupta: Text Book
on Coordinate Geometry, Pothishala Pvt.
Ltd., Allahbad.
7. R.J.T. Bell: Elementary Treatise on
Coordinate Geometry of Three Dimensions,
Wiley Eastern Ltd.,1994
8. P. K. Jain and Khalil Ahmad: A Text Book
of Analytical Geometry of Three Dimensions,
Wiley Eastern Ltd.,1994
9. N. Saran and R. S. Gupta: Analytical
Geometry of Three Dimensions, Pothishala
Pvt. Ltd., Allahbad.
10. Dr. P. K. Mittal: Mathematics for Degree
Students (B. Sc. First Year), S. Chand and
Co. Ltd., New Delhi,2010.
11. G. F. Simmions: Differential Equations
with Historical Notes (Second Edition)
McGraw- Hill International Edition, 1991.
12. S. L .Loney: The Elements of Coordinate
Geometry, MacMilan & Co., London.
##
B. Sc. Part I ( Semester II )
M-4 Vector Calculus and Improper Integrals
Unit I
Vector differentiation, Gradient, Divergence
and Curl, Solenoidal and Irrotational vector
fields, Line integral , Workdone
Unit II
Double integration and its evaluation, Area
by double integration, Change of order of
integration, Transformation of double
integral in polar form, Evaluation of triple
Integral.
\
Unit III
Surface integral, Volume integral, Green’s
theorem in a plane, Stoke’s theorem
(statement only), Gauss divergence theorem
(statement only)
Unit IV
Improper integrals and their convergence,
Comparision tests, Beta and Gamma functions.
Text Books
1. Vector Analysis (Second Edition)
Murray R Spiegel, Schaum’s outlines, 2009
Scope: Chapters 3,4,5,6.
2. Theory and Problems of Advanced Calculus
Murray R Spiegel, Schaum Publishing Company,
New York, 1974.
Scope: Chapters 12,13.
3. Integral Calculus
Gorakh Prasad, Pothishala Pvt. Ltd;
Allahabad, 1999,
Scope: Chapter 10.
Reference Books:
1. N. Saran and S. N. Nigam: Introduction to
Vector Analysis, Pothishala Pvt. Ltd.,
Allahbad.
2. Erwin Kreyszig: Advanced Engineering
Mathematics, John Wiley and Sons, 1999
3. N. Piskunov: Differential and Integral
Calculus, Peace Publishers, Moscow.
4. Shanti Narayan: A course of Mathematical
analysis, S. Chand & Co. Ltd., New Delhi
5. D. Somasundaram and B. Choudhary: A First
Course in Mathematical Analysis, Narosa
Publishing House,New Delhi,1977.
6. Dr. P. K. Mittal: Mathematics for Degree
Students (B. Sc. First Year), S. Chand and
Co. Ltd., New Delhi, 2010.
##
B. Sc. Part II ( Semester III )
M-5 Advanced Calculus , Sequence and Series
Unit I
Mean value theorems, Limit and Continuity of
function of two variables, Taylor’s theorem
for function of two variables.
Unit II
Envelopes, Maxima, Minima and Saddle Points
of functions of two variables, Langrange’s
Multiplier Method.
Unit III
Sequences and theorems on limit of sequence,
Bounded and Monotonic sequences ,
Cauchy’s sequence, Cauchy’s Convergence
Criterion.
Unit IV
Series of non-negative terms ,Comparision
test ,Cauchy’s integral test ,Ratio test ,
Root test , Alternating series ,Absolute and
Conditional convergence.
Text Books:
1. Mathematics for Degree Students (B. Sc.
Second Year)
Dr. P. K. Mittal , S. Chand & Co. Ltd, New
Delhi, 2011.
Scope: Chapters 3,6,8,9,14,15,16 of Advanced
Calculus.
2. Differential Calculus
Shanti Narayan & P. K. Mittal , S.Chand & Co
Ltd,New Delhi, 2012.
Scope: Chapter 8.
Reference Books:
1. Earl D. Rainvile: Infinite Series, Inc
MacMillan Co., New York.
2. Gabriel Klambauel: Mathematical Analysis,
Marcel Dekkar, Inc., New York,1975
3. I.M. Apostol: Mathematical Analysis,
Narosa Publishing House, New Delhi,1985.
4. R.R. Goldberg: Real Analysis, Oxford &
I.B.M Publishing Co., New Delhi,1970.
5. D.Somasundarran And B. Choudhary: A First
Course in Mathematical Analysis, Narosa
Publishing House, New Delhi,1977.
6. P.K. Jain & Kaushik: An Introduction to
Real Analysis, S. Chand & Co.Ltd., New
Delhi,2000.
7. Murray R. Spiegel: Theory and Problems of
Advanced Calculus, Schaum Publishing Co.,
New York,1974.
##
B. Sc. Part II ( Semester III )
M-6 Differential Equations & Group
Homomorphism
Unit I
Bessel’s and Legendre’s equations, Bessel’s
and Legendre’s functions with their
properties, Recurrence relations and
generating functions, Orthogonality of
functions.
Unit II
Laplace transform of some elementary
functions, Linearity of Laplace transform,
Laplace transforms of derivatives and
integrals, Shifting theorem, Differentiation
and integral of transform, Convolution
theorem.
Unit III
Solutions of Ordinary differential equations
with constant and variable coefficients,
Solutions of simultaneous ordinary and
partial differential equations, Fourier
transform, Sine and Cosine transforms.
Unit IV
Normal subgroup ,Quotient Group ,Cyclic
Group ,Group Homomorphism and Isomorphism
,The fundamental theorem of homomorphism.
Text Books:
1. Ordinary and Partial Differential
Equations (Theory and Applications)
Nita H Shah, PHI, New Delhi, 2010.
Scope: Chapter 14 (Articles 14.2, 14.5,
14.6, 14.7, 14.8), Chapter 15(articles 15.2,
15.4, 15.7, 15.9, 15.10, 15.11) and Chapters
16, 18.
2. Mathematics for Degree Students (B. Sc.
First Year)
Dr. P. K. Mittal S. Chand & Co. Ltd., New
Delhi,2010.
Scope: Chapters 10, 12, 13 of Algebra
(Articles 10.9, 12.1, 12.2, 12.3, 12.4,
12.7, 13.1, 13.3)
Reference Books:
1. Erwin kreyzig: Advanced Engineering
Mathematics , John Willey and Son’s , Inc.
New York,1999.
2. A.R. Forsyth: A Treatise on Differential
Equations, McGraw-Hill Book Company,1972.
3. B. Courant and D. Hilbert: Methods of
Mathematical Physics( Vol I and
II),Willey-interscience,1953.
4. I.N. Sneddon: Fourier Transforms, Mc Graw
-Hill Book Co.
5. P.B. Bhattachaya, S.K. Jain and S.R.
Nagpaul: First Course in Linear Algebra
,Willey Eastern, New Delhi,1983.
6. P.B. Bhattachaya, S.K. Jain and S.R.
Nagpaul: Basic Abstract Algebra,(2nd
Edition) Cambridge University Press India
Edition.
7. H.S. Hall and S.R. Knight: Higher
Algebra,S.Chand & Co. Ltd., New Delhi, 2008.
8. Goel & Gupta: Integral Transforms,
Pragati Prakashan, Meerut , 2001.
9. I. N. Herstein: Topic in Algebra, Willey
Eastern Ltd., New Delhi, 1975
##
B. Sc. Part II (Semester IV)
M-7 Partial Differential Equations &
Calculus of Variation
Unit I
Simultaneous differential equations of the
first order and the first degree in three
variables, Methods of solution of ⁄ ⁄ ⁄ ,
Pfaffian differential equation, Solution of
Pfaffian differential equation in three
variables, Partial differential equations
and origins of first order partial
differential equation, Formation of partial
differential equations by eliminating
arbitrary function and arbitrary constants.
Unit II
Lagrange’s equation, Integral surface
passing through given curve, Compatible
system of first order equation, Charpit’s
method, Jacobi’s method.
Unit III
Partial differential equation of second and
higher order, Linear partial differential
equation of second order with constant
coefficients, Homogeneous and
Non-homogeneous linear partial diffrential
equations with constant coefficients,
Partial differential equations reducible to
equations with constant coefficients.
Unit IV
Functional, Continuity of functional, Linear
functional, Extremum of a functional,
Euler’s differential equation and
applications, Invariance of Euler’s
equations.
Text Books:
1. Elements of Partial Differential
Equations:
IAN N. Sneddon, McGraw- Hill Book Company,
1986
Scope: Chapter 1 (articles 2,3,5,6) and
Chapter 2 (articles 1, 2, 4,5,9,10,13).
2. Mathematics for Degree Students (B.Sc.
Second year)
Dr. P. K .Mittal , S.Chand and Co. Ltd, New
Delhi, 2011.
Scope : Chapters 10, 11, 13 of Differential
Equations.
Reference Books:
1. Erwin Kreyszig: Advanced Engineering
Mathematics, John Willey and
Son’s,Inc.,1999.
2. D.A.Murray: Introductory Course on
Differential Equations, Orient
Longman(India),1967.
3. A.R.F0rsyth: A Treatise on Differential
Equations, Macmillan and Co. Ltd, London.
4. Francis B.Hilderbrand: Advance Calculus
for Applications , Prentice Hall of India
Pvt.ltd , New Delhi , 1977.
5. Jane Cronin: Differential Equations,
Marcel Dekkar, Inc.New Yark, 1994.
6. Richard Bronson: Theory and Problems of
Differential Equations,McGraw Hill,
Inc.,1973.
7. B. Courant and D. Hilbert : Methods of
Mathematical Physics,(Volumes I & II),Willey
Interscience,1953.
8. I.M. Gelfand and S.V. Fomin: Calculus of
Variables ,Prentice Hill, Englewood Cliffs
(New Jersey),1963.
9. A.M. Arthurs: Complementary Variational
Principles, Clarendon Press,Oxford,1970.
10. V.Komkav: Variational Principles of
Continuum Mechanics with Engineering
Applications, (Volume I), Reidel Pup.
Dordrecht,Holland,1985.
11. J.I. Oden and J.N Reddy: Variational
Methods in Theoretical Mechanics,
Springer-Veriag, 1976.
12. G. F. Simmons: Differential Equations
with Applications & Historical Notes (Second
Editions) McGraw-Hill,1991.
13. Frank Ayres: Theory and Problems of
Differential Equations, McGraw-Hill Book
Co., 1998.
14. A. S. Gupta: Calculus of Variations with
Applications, Prentice Hall of India Pvt.
Ltd., New Delhi, 1997.
##
B. Sc. Part II (Semester IV)
M-8 Mechanics
Unit I
Analytical condition of equilibrium of
coplanar forces, Virtual work, Catenary.
Unit II
Velocities and Accelerations along radial
and transverse directions, and along
tangential and normal directions, Simple
harmonic motion.
Unit III
Mechanics of a particle and a system of
particles , Constraints, D’Alembert’s
Principle and Lagrange’s equations
,Velocity-dependant potential and the
Dissipation function , Applications of
Lagrangian.
Unit IV
Reduction to the equivalent one-body problem
, Equations of motion and First integrals ,
Virial theorem, Central orbits.
Text Books:
1. Mathematics for Degree Students (B.sc
Second Year )
Dr. P. K. Mittal , S. Chand & Co Ltd , New
Delhi , 2011.
Scope: Chapters 1,2,3 of Statics and
Chapters 1,2,3 of Dynamics.
2. Classical Mechanics (Second Edition)
Herbert Goldstein , Narosa Publishing House
, New Delhi , 1998.
Scope : Chapter 1 and Chapter 3 (articles
3.1,3.2,3.4,3.5)
Refernce Books:
1. R. S. Verma: A Text Book on Statics,
Pothishala Pvt. Ltd. Allahbad.
2. S.L. Loney: Statics , Macmillan and
Company, London.
3. S.L. Loney: An Elementary Treatise on the
Dynamics of a Particle and of Rigid Bodies,
Cambridge University Press, 1956.
##
B. Sc. Final (Semester V)
M-9 Analysis
Unit I
Fourier series, Even & Odd functions,
Dirichlet’s condition, Half Range Fourier
Sine and Cosine series.
Unit II
The Riemann-Stieltjes integral, Existence
and Properties of the integral, The
Fundamental theorem of calculus.
Unit III
Differentiability of complex function,
Analytic function, Cauchy- Reimann
equations,
Harmonic functions, Constructions of
analytic functions.
Unit IV
Elementary function, Mapping of elementary
function, Mobius transformation, Cross
ratio, Fixed points, Inverse points and
Critical points of mappings, Conformal
mapping.
Text Books:
1. Differential Equations with Applications
and Historical Notes (Second Edition)
G. F. Simmons, McGraw-Hill International
Editions,1991.
Scope: Chapter 6 (articles 33,34,35,36)
2. Principles of Mathematical Analysis
(Third Edition)
Walter Rudin, McGraw-Hill International
Edition,1976.
Scope: Chapter 6(articles 6.1-6.13 and
6.20-6.22.)
3. Functions of a Complex Variable
Goyal & Gupta, Pragati Prakashan,2010.
Scope: Chapters 3 and 5.
Reference Books:
1. I. M. Apastol: Mathematical Analysis,
Narosa Publishing house, New Delhi, 1985
2. R. R. Goldberg: Real Analysis, Oxford &
IBH Publishing Co., New Delhi, 1970
3. S. Lang: Undergraduate Analysis,
Springer-Verlag, New York, 1983
4. D. Somasundaram and B. Chaudhary: A First
Course in Mathematical Analysis , S. Chand
Co. New Delhi, 2000
5. P. K. Jain and S. K. Kaushik: An
Introduction to Real Analysis, S. Chand &
Co., New Delhi. 2000.
6. R. V. Churchil and J. W. Brown: Complex
Variables and Applications (5th Edition),
McGraw Hill, New York, 1990
7. Monk J. Ablowitz and A. S. Fokas: Complex
Variables (Introduction and Applications),
Cambridge University Press, South Asian
Edition, 1998.
8. Shanti Narayan: A Course of Mathematical
Analysis, S. Chand & Co. Ltd., New Delhi.
9. Shanti Narayan: Theory of Complex
Variables, S. Chand & Co. Ltd., New Delhi.
##
B. Sc. Final (Semester V)
M-10 Metric Space, Complex integration &
Algebra
Unit I
Countable and uncountable sets, Definition
and examples of metric space, Neighbourhood,
Limit points, Interior points, Open and
closed sets, closure and interior.
Unit II
Completeness, Compactness, Connectedness
Unit III
Ring, Integral domain, Ideals, Fields,
Quotient ring, Ring-Homomorphism.
Unit IV
Complex integration: Cauchy’s integral
theorem and formula, Singularity, Residue
theorem, Evaluation of integrals.
Text Books:
1. Principles of Mathematical Analysis
(Third Edition)
Walter Rudin, McGraw Hill International
Editions, 1976.
Scope: Chapter 2
2. Functions of a Complex Variable
Goyal & Gupta, Pragati Prakashan,2010.
Scope: Chapters 7 ,8, 9.
3. Topics in Algebra ( Second Edition)
I.N.Herstein , Wiley Eastern Ltd. , New
Delhi , 1975.
Scope:Chapter 3 (articles 3.1-3.4).
Reference Books:
1. R. V. Churchil and J. W. Brown: Complex
Variables and Applications (5th Edition),
McGraw Hill, New York, 1990
2. Mark J. Ablowitz and A. S. Fokas: Complex
Variables (Introduction and Applications),
Cambridge University Press, South Asian
Edition, 1998
3. G. F. Simmons: Introduction to Topology
and Modern Analysis, McGraw Hill Book
Company,
4. E. C. Tichmarsh: Theory of Functions.
5. N. Jacobian: Basic Algebra (Volume I &
II), W. H. Freeman, 1980 (Also Published by
Hindustan Company)
6. K. B. Datta: Matrix and Linear Algebra,
Prentice Hall of India Pvt., New Delhi,
2000.
7. K. Hoffman and R. Kunze: Linear Algebra
(2nd Edition), Prentice-Hill Englewood
Cliffs (New Jercy) 1971.
8. Shanti Narayan: Theory of Complex
Variables, S. Chand & Co. Ltd., New Delhi.
9. P. K. Jain and K. Ahmad: Metric Spaces,
Narosa Publishing House, New Delhi, 1968.
##
B. Sc. Final (Semester VI)
M-11 Abstract Algebra
Unit I
Group Automorphisms, Inner Automorphisms,
Cayley’s theorem, Conjugacy relation,
Normalizer.
Unit II
Definition and examples of Vector Spaces,
Subspaces, Sum and direct sum of subspaces,
Linear span, Linear Dependence and
independence, Basis, Dimensions.
Unit III
Algebra of linear transformation, Range,
Rank, Kernel, Nullity of a linear map,
Inverse of linear transformation,
Composition of linear maps.
Unit IV
Matrix associated with a linear map, Linear
map associated with a matrix, Rank and
nullity of a matrix, Inner product space,
Gram-Schmidt orthogonalisation process,
Orthogonal and unitary matrices.
Text Books:
1. Topics in Algebra
I. N. Herstein, Wiley Eastern Ltd., New
Delhi, 1975.
Scope: Chapter 2 (Art. 2.8, 2.9, 2.11)
2. An Introduction to Linear Algebra
V. Krishnamurthy, V. P. Mainra and J. L.
Arora
Affiliated East-West Pres Pvt. Ltd., 1976.
Scope: Chapters 3, 4, 5, 7
Reference Books:
1. I. S. Luther and I. B.S. Passi: Algebra
[Volume-I Groups, Volume II-Rings], Narosa
Publishing House, New Delhi , (Vol. I-1996 ,
Vol. II-1999).
2. Vivek Sahani and Vikas Bist: Algebra,
Narosa Publishing House, New Delhi, 1997
3. S. Kumaresan: Linear Algebra (A
Geometrical Approach), Prentice Hall of
India, 2000
4. S. K. Jain, A. Gunawardena and P. B.
Bhattacharya: Basic Linear Algebra with
MATLAB, Key College Publishing
(Springer-Verlag) 2001.
5. K. Hoffman and R. Kunze: Linear Algebra
(2nd Edition), Prentice-Hall, Englewood
Cliffs (New Jersey), 1971.
6. K. B. Datta: Matrix and Linear Algebra,
Prentice Hall of India Pvt., New Delhi, 2000
7. Shanti Narayan: A Text Book of Modern
Abstract Algebra, S. Chand & Co.Ltd., New
Delhi.
8. N. Jacobson: Basic Algebra (Volumes I &
II), W. H. Freeman, 1980 (Also Published by
Hindustan Publishing Company).
9. D. S. Malik, J. N. Mordeson and M. K.
Sen: Fundamentals of Abstract Algebra,
McGraw- Hill International Edition,1997.
10. P. B. Bhattacharya, S. K. Jain and S. R.
Nagpaul: Basic Abstract Algebra (2nd
Edition), Cambridge University Press, India
Edition, 1997.
##
B. Sc. Final (Semester VI)
##
M - 12 Discrete Mathematics and Elementary
Number Theory ( Optional Paper )
Unit
I
Binary Relations, Equivalence Relation,
Partitions, Partial Order Relation,
Lattices, Duality, Distributive and
Complemented Lattices, Boolean Algebra,
Graph, Multigraph, Weighted Graphs,
Isomorphisms of Graphs, Node
Unit II
Divisibility, Division Algorithm, G.C.D.,
Euclidean Algorithm L.C.M., Primes,
Properties of Congruences, Theorems of
Fermat, Euler & Wilson, Conguence of degree
1, Chinese Remainder Theorem, The Function
φ(n).
Unit III
Quadratic Residues and Reciprocity, The
Jacobi’s symbol, Greatest integer function ,
Arithmetic functions, Moebius inversion
formula.
Unit IV
The Diophantine equations, , Positive
solutions, Other linear equations, The
equations & , Farrey sequences.
Text Books:
1. Discrete Mathematical Structures with
Applications to Computer Science
J. P. Tremblay and R. Manohar, Tata
McGraw-Hill Edition, 1997.
Scope: Chapters 2, 4, 5 ( Art. 2.3.1, 2.3.2,
2.3.4, 2.3.5, 2.3.8, 2.3.9, 4.1.1, 4.1.2,
4.1.3, 4.1.4, 4.1.5, 4.2.1, 5.1.1, 5.1.2,
5.1.4)
2. An Introduction to the theory of Numbers
(Third Edition)
I. Niven and H. S. Zuckerman, John Wiley,.
Scope: Chapters 1,2, 3, 4, 5, 6
Reference Books:
1. C. L. Liu: Elements of Discrete
Mathematics (Secoond Edition), McGraw Hill
International Edition, Computer Sciences
Series, 1986.
2. David M. Burton: Elementary Number
Theory, Win. C. Brown Publishers,
Dubugaelawa, 1989.
3. K. Irland and M. Rosen: A Classical
Introduction to Modern Number Theory, GTM
Volume 84, Springer-Verlag 1972
4. G. A. Jones and I. M. Jones: Elementary
Number Theory, Springer, 1998
5. W. Slerpinski: Elementary Theory of
Number, North-Holland, 1988, Ireland.
6. K. Rosen and M. Rosen: A Classical
Introduction to Modern Number Theory, GTM
Volume 94, Springer-Verlag, 1972.
##
B. Sc. Final (Semester VI)
M-12 Differential Geometry (Optional Paper)
Unit I
Parametric representation and definition of
curve in space , Special Curves and their
representation, length of arc, Tangent at a
given point to a curve, Oscillating plane,
Normal Plane , Principal normal and binormal
, Rectifying plane , Fundamental planes,
Curvature of curve, Torsion of curve,
Serret-Frenet Formulae, Helices, Locus of
the centre of curvature, Oscillating sphere
(Sphere of Curvature), Locus of centre of
spherical curvature.
Unit II
Involute & Evolute , The curvature and
torsion of the evolute, Bertrand curves,
Fundamental theorem for space curves,
Envelopes and characteristics relating to
one parameter family of surfaces and planes,
Developable surfaces, Ruled surfaces.
Unit III
Curves on a surface, Parametric Curves, Two
fundamental forms, Positive definiteness,
Fundamental magnitudes for some important
surfaces, Direction coefficients,
Orthrogonal trajectories of given curves ,
The formulas of Gauss , Meusnier’s theorem ,
Lines of curvature as parametric curves ,
Euler’s theorem on normal curvature,
Rodrigues’ formula, Third fundamental form.
Unit IV
Definition and the differential equations of
Geodesic, Canonical equations for Geodesics,
Geodesics on a surface of revolution, Normal
property of Geodesics, Tortion of Geodesic,
Curvature of Geodesic, Bonnet’s Theorem,
Geodesic parallels, Geodesic polar
coordinates, Theorems on geodesic parallels,
Geodesic ellipse and hyperbolas,
Gauss-Bonnet Theorem, Gaussian Curvature.
Text Books:
1. Differntial Geometry (Third Revised
Edition)
H. D. Singh & P. K. Singh, Ram Prasad and
Sons, Agra-3, 1995.
Scope: Chapter 2 (Art. 2.1 to 2.24) and
Chapter 7 (Art. 7.1 to 7.24)
2. Differntial Geometry (Fifth Edition)
Bansilal & Sanjay Arora, Atma Ram & Sons.
Scope: Chapters 2, 3(Art. 2.60 to 2.90,
3.20, 3.40, 3.50 to 3.63, 3.70 to 3.81) and
Chapters 4, 5 (Art. 4.00 to 4.31,
4.33,4.40,4.41,4.73,4.74, 5.20,5.21, 5.30,
5.35)
Reference Books:
1. I. M. Singer and J. A. Thorpe, Lecture
Notes on Elementry Topology and Geometry.
Springer-verlag, 1967
2. B. O. Nell. Elementary Differential
Geometry, Academic Press, 1966.
3. S. Sternberg, Lectures on Differential
Geometry, Prentice-Hall. 1976.
4. M. Docarmo : Differential Geometry of
Curves and Surfaces, Prentice Hall, 1976.
##
B. Sc. Final (Semester VI)
M-12 Special Theory of Relativity (Optional
Paper)
Unit I
Newtonian Relativity , Galilean
Transformations , The theory of Ether,
Michelson – Morley experiment , Lorentz
transformation equations , Geometrical
interpretation of Lorentz transformations ,
Group properties of Lorentz transformations.
Unit II
Event and Particle , Simultaneity,
Relativistic formulae for composition of
velocities (Transformation of particle
velocities), Relativistic addition law for
velocities ,Relativistic formulae for
composition of accelerations of a particle,
Transformation of Lorentz contraction factor
, length contraction , time dilation.
Unit III
Tensors , Riemannian metric, metric tensor
or fundamental tensor, Minkowski space ,
Space and Timelike intervals , Light cone or
null cone , world points and world lines ,
Events occurring at the same point and the
same time , four vector , four tensors .
Unit IV
Equivalence of mass and energy i. e. E= mc2,
Transformation formula for mass ,
Transformation formula for momentum and
energy ,Energy momentum four vector, four-
velocity , four- acceleration ,Relativistic
equations of motins, The energy momentum
tensor Tij , Maxwell’s equations of
electromagnetic theory in vacuo ,
Propagation of electric and magnetic field
strengths , four potential , Transformations
of electromagnetic four potential vector.
Text Books:
1. The Theory of Relativity
C. Moller, Oxford Claredon Press , 1952.
2. Theory of Relativity
Goyal and Gupta, Krishna Prakashan, Meerut ,
Delhi.
Reference Books:
1. Murray R.. Spiegel, Theory and Problems
on Vector Analysis SIJ Metrics and
Introduction to Tensor Theory, Schaum’s
outline Series, Mcgraw -Hill Book Company.
2. Sriranjan Banerji and Asip Banerjee, The
Theory of Relativity. PHI, New Delhi 2010.
BCA Bachelor Of Computer Application, BCCA, Bachelor of Commerce & Computer Application BE IT/CS, Information technology/Computer Science MCA, Master of Computer Application MCM, Master of Computer Management Diploma , Polytechnic B.Arch Bachelor Of Architecture, Others,
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