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1
COURSES OF STUDIES
for
Bachelor Degree Examination: 2014-2017
MATHEMATICS
Ravenshaw University
Cuttack
2
B.Sc.(Mathematics) Pass/Hons
UG 1st year :
First Semester Subject Code
Calculus I (Pass and Hons) 1.1.1 Pass/Hons
Matrix Algebra (Pass and Hons) 1.1.2 Pass/Hons
Number Theory ( Hons) 1.1.3 Hons
Second Semester
Calculus II (Pass and Hons) 1.2.3(Pass) & 1.2.4(Hons)
Ordinary Differential Equation (Pass and Hons) 1.2.4(Pass) & 1.2.5(Hons)
Discrete Mathematics (Hons) 1.2.6.( Hons)
UG 2nd
year :
Third Semester
Calculus III (Pass and Hons) 2.3.5(Pass) & 2.3.7(Hons)
Algebra-I (Pass and Hons) 2.3.6(Pass) & 2.3.8(Hons)
Probability Theory (Hons) 2.3.9(Hons)
Fourth Semester
Numerical Analysis-I (Pass and Hons) 2.4.7(Pass) & 2.4.10(Hons)
Linear Programming (Pass and Hons) 2.4.8(Pass) & 2.4.11(Hons)
Algebra-II (Hons) 2.4.12(Hons)
3
B.Sc.(Mathematics)
UG 3rd
year :
Fifth Semester Subject Code
Algebra-III (Hons) 3.5.13(Hons)
Mechanics (Hons) 3.5.14(Hons)
Differential Geometry (Hons) 3.5.15(Hons)
Programming in C (Hons) 3.5.16(Hons)
Topology of Metric Spaces (Hons) 3.5.17(Hons)
Practical using C-Language (Hons) 3.5.18(Hons)
Sixth Semester
Analysis of Several Variables (Hons) 3.6.19(Hons)
Numerical Analysis-II (Hons) 3.6.20(Hons)
Calculus of Variations (Hons) 3.6.21(Hons)
Ordinary and Partial Differential Equation (Hons) 3.6.22(Hons)
Complex Analysis (Hons) 3.6.23(Hons)
Problem Comprehensive (Hons) 3.6.24(Hons)
4
UG 1st year
MATHEMATICS (PASS /HONS)
First Semester Paper-1.1.1 (P & H)
CALCULUS-I
Full marks: 10+40 Time : 2 Hours
Unit-I
Real numbers : Algebra of real numbers, order, Completeness(continuum), Upper and
Lower Bounds, Least Upper Bounds and Greatest Lower Bounds, Density, Archimedean
Principle, One- to- one correspondence, Cardinality, Countability, Uncountability.
Unit-II
Convergence of Sequence and Series: Convergence, Limit theorem, Weierstrass’s
completeness principle, Cantor’s completeness principle, Subsequences and Bolzano-
Weierstrass theorem, Cauchy’s completeness principle, Convergence of series, Series of
positive terms, Absolute convergence, Conditionally convergent series.
Unit-III
Limit and Continuity : Limit of function, Left and right limit, Continuity, Discontinuity,
Further properties of continuous function defined on closed intervals, Uniform continuity.
Differentiation : Left and right derivative, Mean value theorems, Higher derivatives and
Taylor’s theorem.
Book Prescribed:
Fundamentals of Mathematical Analysis by G.Das and S.Pattanayak ( Tata McGraw-Hill
Publishing Company LTD).
Chapters : 2(2.1-2.4, 2.6), 3(3.1-3.4), 4(4.1-4.7, 4.10-4.13), 6(6.1-6.6, 6.9),
7(7.1-7.4, 7.6, 7.7).
Book for reference:
Mathematical Analysis by S. C. Mallick and S. Arora (New Age International).
5
Paper-1.1.2 (P & H)
MATRIX ALGEBRA Full marks: 10+40 Time : 2 Hours
Unit-I
Introduction to vectors: Vectors and linear combinations, length and dot products,
matrices, Solving linear equations: Vectors and linear equations, the idea of elimination,
elimination using matrices, Rules of matrix operations, inverse matrices, factorization,
Transposes and Permutations
UNIT-II
Vector spaces and subspaces: The space of vectors, the nullspace of , solving
homogeneous system of linear equations, rank and row reduced form, complete solution
to non-homogeneous system of linear equations, independence, basis and dimension,
Determinants: properties of determinants, permutations and cofactors, Cramer’s rule,
inverses and volumes
UNIT-III
Eigen values and Eigen vectors: Introduction to Eigen values, Diagonalizing a matrix,
Applications to differential equations, symmetric matrices, positive definite matrices, similar
matrices, Complex matrices: Hermitian and Unitary matrices
1. Book Prescribed: An Introduction to Linear Algebra, 4th Ed., Gilbert Strang (Wellesley
Cambridge Press). Chapters: 1, 2, 3(3.1-3.5), 5, 6(6.1-6.6), 10(10.1-10.2)
Books for reference:
2. Linear Algebra, S. Kumarsen (PHI).
3. Hoffman and Kunze, Linear Algebra, 2nd
Ed., (PHI).
4. V. Krishnamurthy, Linear Algebra (East West Press)
6
Paper-1.1.3 (Hons)
NUMBER THEORY
Full marks: 10+40 Time : 2 Hours
Unit-I
Divisibility theory in the integers, Primes and their distribution, Theory of congruences, Euler’s
Phi-function, Euler’s theorem.
Unit-II
Quadratic reciprocity, Number theoretic functions, Arithmatic functions, Mobius inversion
formula, Greatest Integer function.
Unit-III
Diophantine equations, The equations x2+y
2=z
2, the equation x
4+y
4=z
2, the sum of four and five
squares, Fermat’s last theorem, sum of fourth powers.
Book Prescribed:
Elementary number theory- David M. Burton (Tata MC-Graw Hill).
Chapters: 2 (2.2-2.5), 3 (3.1, 3.2), 4 (4.2-4.4), 6 (6.2, 6.3), 7 (7.2, 7.3), 9 (9.2-9.4), 12 (12.1, 12.2)
Book for reference:
An Introduction to the Theory of Numbers- Ivan Niven & H. S. Zuckerman (Wiley).
7
UG 1st year
MATHEMATICS (PASS /HONS)
Second Semester
Paper-1.2.3(Pass) & 1.2.4(Hons)
CALCULUS-II
Full marks: 10+40 Time : 2 Hours
Unit-I
Riemann Integration : Riemann integral, Continuity and Integrability, Properties
of Riemann integral, Fundamental theorem of calculus.
Unit-II
Sequences and Series of Functions : Pointwise convergence, Uniform
convergence, Uniform convergence and continuity, Term-by-term integration of
series, Term-by-term differentiation of series, Power series, Taylor’s series.
Unit-III
Improper integrals, Beta and Gamma functions.
Books Prescribed:
1. Fundamentals of Mathematical Analysis by G.Das and S.Pattanayak ( Tata McGraw-
Hill Publishing Company LTD).
Chapters : 8(8.1-8.5), 9(9.1-9.7)
2. Mathematical Analysis by S. C. Mallick and Sabita Arora (New Age International).
Chapter : 11, Appendix-I
8
Paper-1.2.4(Pass) & 1.2.5(Hons)
ORDINARY DIFFERENTIAL EQUATIONS
Full marks: 10+40 Time : 2 Hours
Unit-I
Ordinary differential equations of first order and first degree, Exact equation, Integrating factor,
Equation reducible to linear form, Equation of first order of higher degree, Equation solvable for
p, y and x, Equation homogeneous in x and y, Clairaut’s and Lagrange’s Equation.
Unit-II
Linear equations with Constant Co-efficient and with variable co-efficient.
Unit-III
Laplace transforms and Applications to ODE.
Book Prescribed:
A course on Ordinary and Partial Differential Equation with Application- J. Sinha Roy and
S. Padhy (Kalyani Publishers).
Chapters: 2 (2.4 to 2.7), 3, 4 (4.1 to 4.7), 5, 9 (9.1, 9.2, 9.5, 9.10, 9.11, 9.13)
9
Paper-1.2.6(Hons)
DISCRETE MATHEMATICS
Full marks: 10+40 Time : 2 Hours
Unit –I
The Foundations: Logic and proofs: Propositional logic, Propositional equivalences, predicates
and quantifiers, nested Quantifiers, rules of inference, Introduction to proofs, Normal forms.
Mathematical Induction: Mathematical Induction, strong Induction and well ordering
Unit-II
Advanced Counting techniques: recurrence relations, solving linear recurrence relations,
generating functions, Inclusion-Exclusion, Applications of Inclusion-Exclusion
Unit-III
Relations: Relations and their properties, n-ary relations and their applications, representing
relations, closure of relations, equivalence relations, partial orderings
Book Prescribed:
1. Discrete Mathematics and its Applications, Kenneth H. Rosen, Tata Mc-Graw Hill
Education private Limited, Seventh Edition (Indian adaptation by Kamala Krithivasan),
2012
Chapter 1 (1.1-1.7), 4 (4.1-4.2), 6(6.1-6.6, Excluding 6.3), 7(7.1-7.6)
Book Recommended:
1. Discrete Mathematical structures with Application to Computer Science – J. P. Tremblay
and R. Monohar.
10
UG 2nd
year MATHEMATICS (PASS /HONS)
Third Semester
Paper-2.3.5(Pass) & 2.3.7(Hons)
CALCULUS-III Full marks: 10+40 Time : 2 Hours
Unit-I
Analytical Properties of R and C : Open sets, Closed sets, Limit points, Bolzano-Weirstrass
theorem, Closure, Interior and Boundary, Compactness, Sequential compactness, Heine-Borel
theorem(Statement only).
Unit-II
Function of several variable, Limit of a function, Algebra of limits, Repeated limits, Partial
derivatives, Differentiability, Equality of cross derivatives, Derivatives of composite functions,
Derivatives of implicit functions, Change of variables, Homogeneous functions, Mean value
theorem, Taylor’s theorem, Maclaurin’s theorem, Jacobians, Maxima/Minima, Lagrange’s
multipliers.
Unit-III
Differential operator : Scalar and vector point functions, Gradient, Tangent plane and normal line,
Divergence and curl.
Integral Theorem : Line integrals, Surface integrals, Volume integrals, Volume integrals, Integral
theorems (Gauss, Stoke’s, Green’s theorems).
Books Prescribed:
1. Topics in Calculus – by R. K. Panda & P. K. Satapathy
Chapters: 3, 4(4.1-4.6, 4.10, 4.12-4.16), 5, 9
2. Fundamental of Mathematical Analysis- by G. Das & S. Pattanayak
Chapters- 5(5.1-5.6)
11
Paper-2.3.6(Pass) & 2.3.8(Hons)
ALGEBRA-I
Full marks: 10+40 Time : 2 Hours
Unit-I
Symmetries: Motivation for the definition of a group, Dihedral groups.
Groups : Binary operation on a set, Axiomatic definition of a group, examples, these to include
general and special linear groups over familiar fields, the group of units of integers modulo n and
connection with Euler’s function.
Elementary properties : Uniqueness of identity and inverse, cancellation, etc; subgroups; order of
a group; order of an element; finite groups.
Unit-II
Cyclic groups : Classification; the structure of subgroups of a cyclic group. Permutation groups :
Definition; cycle notation; representation as a product of disjoint cycles; generation by transpositions;
sign of a permutation; alternating group. Cayley’s theorem.
Unit-III
Cosets : Lagrange’s theorem; theorem of Fermat and Euler from number theory to be seen as
special cases. Homomorphism, Isomorphisms, Normal subgroups, kernels and images; kernels
and normal subgroups. Quotient groups. Conjugation; inner and outer automorphisms.
Books Prescribed:
Contemporary Abstract Algebra- Gallian, 4th edition, Narosa.
Chapters : 1,2,3,4,5,6,7,9,10
12
Paper-2.3.9(Hons)
PROBABILITY THEORY
Full marks: 10+40 Time : 2 Hours
Unit –I
Probability: Examples of probability, Deduction from the axioms, independent events,
arithmetical density, Counting-fundamental rules, Diverse ways of sampling, allocation models,
Binomial coefficients.
Unit-II
Random variables-What is random variable. How do random variables come about , Distribution
and expectation, integer valued random variables with densities, Conditioning and independence,
Example of conditioning , basic formulas, sequential sampling.
Unit-III
Mean variance,: basic properties of expectation, the density case, multiplication theorems,
Poisson and normal distribution, Models for Poisson’s distributions, normal distributions.
Book Prescribed:
Elementary Probability Theory with Stochastic process by Kai Lai Chung.
Chapters: 2(2.1-2.5), 3(3.1-3.4), 4(4.1-4.5), 5(5.1-5.3), 6(6.1-6.3), 7(7.1-7.4).
13
UG 2nd
year MATHEMATICS (PASS /HONS)
Fourth Semester
Paper-2.4.7(Pass) & 2.4.10(Hons)
NUMERICAL ANALYSIS-I
(Scientific non-programmable calculators are allowed in Examination Hall)
Full marks: 10+40 Time : 2 Hours
Unit-I
Computer Arithmetic, Octal and Hectadecimal systems, Floating point Arithmetic, Errors,
Significant digits and Numerical Stability. Transcendental and Algebraic Equations, Bisection
Methods, Iterative Methods based on First Degree Approximations, Rates of Convergence.
Unit-II
System of Linear Algebraic Equations, Direct Method, Crammer’s Rule, Gauss Elimination
Method, Gauss Jordan Method, Triangularisation Method, Cholesky Method.
Unit-III
Interpolation, Lagrange’s and Newton’s Interpolations, Finite Differences, Numerical
Integrations, Methods based on Undetermined Co-efficient.
Book Prescribed:
Numerical Methods for Scientific and Engineering Computation by M. K. Jain, S. R. K. Iyegar
and R. K. Jain (Willy Eastern Ltd.)
Chapters: 1(1.2, 1.3), 2(2.1, 2.2, 2.3, 2.5, 2.6), 3(3.2), 4(4.1 to 4.4), 5(5.6, 5.7, 5.8.1)
14
Paper-2.4.8(Pass) & 2.4.11(Hons)
LINEAR PROGRAMMING
Full marks: 10+40 Time : 2 Hours
Unit- I
Pre-requisites: Vectors, Vector-inequalities, Linear combination of vectors, Hyper plane
and Hyper spheres, Convex sets and their properties, supporting and separating hyper
planes, convex functions, local and global extrema, Quadratic form.
LPP- Mathematical formulation, graphical solutions.
Unit-II
General LPP, Canonical and standard forms, Simplex method- Introduction, Simple
algorithm, Use of artificial variables, Two phase method, Big M method or method of
penalties.
Unit-III
Duality in LPP : Introduction, primal dual conversion, duality and simplex method, dual
simplex method.
Transportation problem: Introduction, basic concepts, finding initial basic feasible
solutions by NWC, LCM and VAM, Test of optimality by MODI method, some
exceptional cases.
Assignment problem: Introduction, Hungarian’s method, special cases.
Book Prescribed:
Operation Research- Kanti Swarup, P.K Gupta & Man Mohan (Sultan Chand & Sons),
Ninth Edition 2001.
Chapters : 0(0.9-0.17), 2(2.2), 3(3.2, 3.4, 3.5), 4(4.1, 4.3, 4.4, 4.5, 4.6), 5(5.1-5.4, 5.7,
5.9), 10(10.1-10.6, 10,8-10.12, 10.14), 11(11.1-11.4).
15
Paper-2.4.12(Hons)
ALGEBRA-II
Full marks: 10+40 Time : 2 Hours
Unit-I
Rings : Motivation, definition, examples, basic properties; sub rings.
Ideals and factor rings : one and two sided ideals; factor rings; prime and maximal
ideals in commutative rings: characterization in terms of properties of quotients.
Zero divisors : integral domains; fields; finite domains are fields; the finite field
Z/pZ Field of quotients of an Integral domain.
Unit II
Ring homeomorphisms : kernels are ideals; first isomorphism theorem.
Polynomial ring over a ring : division algorithm; remainder theorem; polynomial in F[X]
of degree n has at most n roots.
Principal ideal domains : Z and F[X]; prime ideals and maximal ideals in PIDs.
Unit-III
Factorization of polynomials: irreducible; Z[X]: content of a polynomial, primitive
polynomials, Gauss’s lemma: product of primitives is primitive; primitive and irreducible
over Q is irreducible; Irreducibility tests: reading modulo primes; Eisenstein’s criterion;
cyclotomic polynomials; irreducibility of a polynomial over F[X] being equivalent to it
generating a prime ideal; Unique factorization in Z[X].
Book Prescribed:
Contemporary Abstract Algebra-Gallian, 4th
edition, Narosa.
Chapters : 12 to 18.
16
UG 3rd
year MATHEMATICS (HONS)
Fifth Semester
Paper-3.5.13(Hons)
ALGEBRA-III
Full marks: 10+40 Time : 2 Hours
Unit –I
Vector spaces : fields and vector spaces; linear combinations; subspaces; span of a set of vectors;
linear independence; finite dimensional vector spaces, bases and dimension; coordinates.
Linear transformation: Definition, linear functional, composition of transformations; the
endomorphism algebra; invertible transformations; isomorphism: there is only one vector space
up to isomorphism of a given dimension; representation of transformations by matrices;
correspondence between the algebra of transformations and the algebra of matrices; how the
representing matrix changes with choice of bases: similarity transformations; the rank-nullity
theorem: row-rank equals column-rank.
Unit-II
Dual Vector Space: The dual V* of a vector space V, dimension of the dual and the dual basis,
the double dual and isomorphism of a vector space with its double dual. Linear equations and
matrices Revisited in the light of the abstract definition of vector spaces and transformations.
Eigenvalue: review of the material under this heading in the Matrix algebra course: eigenvalues,
eigenvectors of a linear transformation, linear independence of eigenvectors corresponding to
different eigenvalues, characteristic polynomial. Cayley-Hamilton theorem, Minimum
polynomial.
Unit-III
Geometry of Inner Product Spaces : (over the fields of real numbers and complex numbers,
separately) inner products, Schwarz inequality, Gram-Schmidt orthogonalization, linear
functional on an inner product space, orthogonality, existence of orthogonal basis, orthogonal
projections, Linear functional and Adjoints.
Book Prescribed:
Linear Algebra by Kenneth Hoffman and Ray Kunze. PHI Learning PVT. Ltd.
Chapters: 2(2.1-2.6), 3(3.1-3.7), 6(6.1-6.4), 8(8.1-8.3)
17
Paper-3.5.14(Hons)
MECHANICS
Full marks: 10+40 Time : 2 Hours
Unit –I
Methods of Plane Statics, Applications of plane statics, Plane kinematics.
Unit-II
Methods and applications in plane Dynamics
Unit-III
Plane impulsive motion. The equation of Lagrange and Hamilton
Book Prescribed:
Principles of Mechanics, J.L. Synge and B. A. Griffith, Mc. Graw Hill
Chapters: 2(2.1-2.4), 3(3.1,3.2), 4(4.1),8,15
18
Paper-3.5.15(Hons)
DIFFERENTIAL GEOMETRY
Full marks: 10+40 Time : 2 Hours
Unit –I
Curve with torsion, Tangent, Principal normal, Curvature, Binomial torsion, Locus of
center of spherical curvature, Intrinsic equation Heloics, Spherical indicatrix.
Unit-II
Involute, Evolute, Bertrand curves, Surfaces, Tangent plane, Normal, Envelope
characteristics, Edge of regression, Developable surfaces.
Unit-III
Curvilinear co-ordinates, First order magnitudes, Direction on a surface, Second order
magnitude, Derivation of n.
Books Prescribed:
Differential Geometry of Three Dimensional by C. E. Weatherbun (ELDS)
Chapter : 1, 2(2.13-2.17), 3(3.22-3.27)
19
Paper-3.5.16(Hons)
PROGRAMMING IN ‘C’
Full marks: 10+40 Time : 2 Hours
Unit –I
Overview of C, Constants, variables and data types, operations and Expressions, Managing Input
and output operation. Writing simple programs
Unit-II
Decision making and branching.
Unit-III
Arrays
Recommended Books
Programming in ANSIC(2nd
Edn.), E.Balaguruswamy(Tata Mc. Graw Hill) Ch.1,2,3, 4, 5,6,7
20
Paper-3.5.17(Hons)
TOPOLOGY OF METRIC SPACES
Full marks: 10+40 Time : 2 Hours
Unit –I
Introductory concepts: Definition and examples of metric space, Open sphere and Closed
sphere, Neighborhoods’, Open sets, Equivalent metrics, Interior points, closed set,
Limit points and Isolated points, Closure of a set, Boundary points, Distance between sets
and diameter of a set, Subspace of a metric space, Product metric spaces and basis.
Unit-II
Completeness: Convergent Sequences, Cauchy-Sequence, Complete spaces, Dense
sets and separable spaces, no-where dense sets, Baire’s theorem, Completion.
Continuous functions: Definition and Characteristics, Extension theorem, Uniform
continuity, Homeomorphism, Uniformly equivalent metrics
Unit-III
Compactness: Compact space and Compact set, Sequentially Compactness, totally
boundedness, Equivalence of compactness and sequential compactness, Compactness
and finite intersection property, Continuous functions and compact spaces, Fixed point
theorems.
Recommended Books:
1. Metric spaces by Pawan K. Jain and Khalil Ahmad (Narosa Publishing House), 2nd
Edition, 2012
Chapter : 2(2.1-2.14), 3(3.1-3.7), 4 (4.1-4.5), 5(5.1-5.6), 7(7.1)
21
Paper-3.5.18(Hons)
Full marks: 10+40 Time : 2 Hours
Practical Using ‘C’LANGUAGE
1. Solutions of a Quadratic Equation.
2. Generate Fibonacci sequence.
3. Find factorial of a number using recursion.
4. Sorting.
5. Test whether a number is an Armstrong Prime.
6. Evaluation of Integrals by Trapezoidal and Simpson’s method.
7. Solving an ODE by Runge-Kutta method.
8. Matirx multiplication.
9. Count number of vowels in a word.
10. Reverse a string or a number.
22
UG 3rd
year MATHEMATICS (HONS)
Sixth Semester
Paper-3.6.19(Hons)
ANALYSIS OF SEVERAL VARIABLES
Full marks: 10+40 Time : 2 Hours
Unit-I
Mutivariate Differential Calculus
Introduction,The directional derivative, Directional derivatives and continuity, total derivative, total
derivative expressed in terms of partial derivatives, an application to complex-valued functions, matrix of
a linear function, Jacobian matrix , chain rule, Matrix form of the chain rule
Unit-II
Implicit Functions and Extremum Problems
Introduction , Functions with nonzero Jacobian determinant , The inverse function theorem , The implicit
function theorem, Extrema of real-valued functions of one variable, Extremum problems with side
conditions
Unit-III
Multiple Riemann Integrals
Introduction, The measure of a bounded interval in Rn , The Riemann integral of a bounded function
defined on a compact interval in R", Sets of measure zero and Lebesgue's criterion for existence of a
multiple Riemann integral, Evaluation of a multiple integral by iterated integration
Recommended Books:
1. Tom M Apostol: Mathematical Analysis, 2nd
edition, Narosa Publishing House
Chapter 12 (12.1-12.10), Chapter 13, Chapter 14 (14.1-14.5)
23
Paper-3.6.20.(Hons)
NUMERICAL ANALYSIS-II
Full marks: 10+40 Time : 2 Hours
(Scientific non-programmable calculators are allowed in Examination Hall)
Unit-I
System of Linear Algebraic Equations, Iteration Methods, Jacobi Iteration Method, Gauss-Seidel Method,
SOR Method, Eigen values and Eigen vectors, Given’s Method, Power method.
Unit-II
Ordinary differential Equations, Euler method, Backward Euler Method, Mid-Point Method, Single Step
Method, Taylor series Method.
Unit-III
Runge-Kutta method, Second Order Method, Fourth Order Method, Explicit and Implicit Runge Kutta
Method.
Recommended Books:
1. Numerical Methods for Scientific and Engineering Computation by M. K. Jain, S. R. K. Iyengar
and R. K. Jain (Wiley Eastern Ltd.)
Chapter 3 (3.4, 3.5), 6 (6.2, 6.3, 6.4)
24
Paper-3.6.21(Hons)
CALCULUS OF VARIATIONS
Full marks: 10+40 Time : 2 Hours
Unit –I
The concept of variation and its properties, Euler’s equation, variational problem for functional of
the form, Functional dependent on higher order derivatives, Functional dependent on functions of
several independent variables.
Unit-II
Variational problem in parametric form, some applications to problems of mechanics, variational
problems leading to an integral equation or a differential difference equation. Theorem of du
bois-Reymond strocastic Calculus of variations.
Unit-III
Variational problem with moving Boundaries, Functional of the form [y(x)=x], variational
problem with a movable boundary for a functional dependent on two functions one sided
variation reflection and refraction of internals, diffraction of light rays.
Recommended Books:
Calculus of variation with Application by A.S. Gupta.
Chapters: 1(1.1-1.10), 2(2.1-2.5)
25
Paper-3.6.22(Hons)
ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS
Full marks: 10+40 Time : 2 Hours
Unit-I
Ordinary differential Equation in more than two variables, Partial differential Equations of First
order.
Unit-II
Partial differential equation of the second and higher order.
Unit-III
Series solutions and Special functions.
Recommended Books:
1. A course on Ordinary and Partial Differential Equation with Application- J. Sinha Roy and S.
Padhy (Kalyani Publishers)
Chapter 7, 11, 12, 13
26
Paper-3.6.23(Hons)
COMPLEX ANALYSIS
Full marks: 10+40 Time : 2 Hours
Unit –I
Analytic functions : Functions of a complex variable, Limit, Continuity,
Differentiability, Analytic functions, Cauchy-Riemann Equations, Harmonic
functions, Periodic functions, Exponential functions, Trigonometric functions,
Hyperbolic functions.
Unit-II
Complex Integration : Contour integral, Primitives, Cauchy-Goursat theorem and its
extensions, Winding number, Cauchy-integral formula, some subsequences of the
Cauchy integral formula, Maximum moduli of functions.
Unit-III
Series Expansions : Taylor series, Zero’s of Analytic functions, Laurent series.
Singularities and Residue : Classification of singularities, Residue, Poles and Zeros.
Recommended Books
1. Complex variables : Theory and Applications by H. S. Kasana
Chapter 2(2.1-2.7), 3(3.1-3.4), 4(4.1-4.9), 6(6.3-6.5), 7(7.1-7.3)
27
Paper-3.6.24(Hons)
Problem Comprehensive
Full marks: 10+40 Time : 2 Hours
Unit-I
Problems pertaining to Differential Equations
Unit-II
Problems pertaining to Calculus-I, Calculus-II, Calculus-III
Unit-III
Problems pertaining to Algebra-I, Algebra-II, Algebra-III
Reference Books:
1. Mathematical Analysis by S. C. Mallick and S. Arora (New Age International).
2. Contemporary Abstract Algebra- Gallian, 4th edition, Narosa.
3. Hoffman and Kunze, Linear Algebra, 2nd
Ed., (PHI).
4. V. Krishnamurthy, Linear Algebra (East West Press)
5. Fundamentals of Mathematical Analysis by G.Das and S.Pattanayak ( Tata McGraw-Hill
Publishing Company LTD).
6. A course on Ordinary and Partial Differential Equation with Application- J. Sinha Roy and
S. Padhy (Kalyani Publishers).