Here I am providing the syllabus of M.Sc Applied Mathematics of the Andhra University:

**Andhra University M.Sc Applied Mathematics Syllabus**
Basic Topology: finite, countable and uncountable sets, metric spaces, compact sets,

perfect sets, connected sets. (One question is to be set)

Continuity: limits of functions, continuous functions, continuity and compactness,

continuity and connectedness, discontinuities, monotone functions, infinite limits and

limits at infinity. (Chapters 2 and 4 of Ref.1). (One question is to be set)

The Riemann - Stieltjes integral: linearity properties, integration by parts, change of

variable, reduction to a Riemann integral, monotonically increasing integrators,

Riemann’s condition, comparison theorems, integrators of bounded variation, sufficient

conditions for existence of R-S. integrals, necessary conditions for existence of R-S

integrals, mean-value theorems for R-S integrals, integral as a function of interval,

second fundamental theorem of integral calculus, second mean-value theorem for

Riemann integrals. (Sections: 7.1 to 7.7 and 7.11 to 7.22 of Ref.2)

(One question is to be set)

Multivariable Differential Calculus: directional derivative, total derivative, Jacobian

matrix, chain rule, mean-value theorem for differentiable functions, sufficient conditions

for differentiability and for equality of mixed partial derivatives, Taylor’s formula for

real valued functions in n real variables. (Chapter 12 of Ref.2).

(One question is to be set)

Sequences and series of functions: uniform convergence, uniform convergence and

continuity, uniform convergence and integration, uniform convergence and

differentiation. equicontinuous families of functions, the Stone – Weierstrass theorem.

(Chapter 7 of Ref.1) (Two questions are to be set)

Linear equations with variable coefficients, the wronskian and linear independence,

reduction of the order of a homogeneous equations, the non-homogeneous equations.

Homogeneous equations with analytic coefficients. Linear equations with regular singular

points, Eulers equations and series solutions. Existence and uniqueness of solutions of 1st

order equations, exact equations, Picard’s method of successive approximations,

existence & uniqueness of solution to systems. (Chapter 3 (excluding section 8 & 9),

chapter 4 (excluding sections 5, 7 & 8), chapter 5 (excluding section 7) and chapter 6

(sections 1,3,5,6) of Text book.1.

(Three questions are to be set)

(Three questions are to be set)

Calculus of variations : Euler’s equations, functions of the form

.... , , .... , 2 1 , 2 , 1 dx. Functional dependence on higher order derivatives,

variational problems in parametric form and applications (chapter VI of Text book.2).

Tensor Analysis: N-dimensional space, covariant and contravariant vectors, contraction,

second & higher order tensors, quotient law, fundamental tensor, associate tensor, angle

between the vectors, principal directions, christoffel symbols, covariant and intrinsic

derivatives geodesics (chapter 1 to 4 of Text book.3).

(Three questions are to be set)

Text books:

1. E.A. Coddington. An Introduction to ordinary differential equations, Prentice Hall of

India Pvt. Ltd., New Delhi, 1987.

2. L. Elsgolts: Differential equations and calculus of variations, Mir Publishers, Moscow,

M.Sc. FIRST SEMESTER APPLIED MATHEMATICS

Duration: 3 hours Maximum Marks: 85

(A total of seven questions are to be set and the student has to answer 5 (five) questions.

All questions carry equal marks. The first question which is compulsory carries 17

marks. It consists of 4 short answer sub questions covering the entire syllabus. The

remaining six questions each carrying 17 marks are to be set as suggested in the body of

the syllabi.)

Lagrangian Formulation: Mechanics of a particle, mechanics of a system of particles,

constraints, generalized cordinates generalized velocity, generalized force and potential.

D’Alembert’s principle and Lagranges equations, some applications of Lagrangian

formulation, Hamilton’s principle, derivation of Lagrange’s equations from Hamilton’s

principle, extension of Hamilton’s principle to non-holonomic systems, Advantages of

variational principle formulation, conservation theorems and symmetry properties (scope

and treatment as in Art.1.1 to 1.4 and Art 1.6 to 2.6 of Text book.1).

(Two questions are to be set)

Hamiltonian formulation: Legendre transformations and the Hamilton equations of

motion, cyclic coordinates and conservation theorems, derivation of Hamilton’s

equations from a variational principle, the principle of least action, the equation of

canonical transformation, examples of canonical transformation, Poisson and Lagrange

brackets and their invariance under canonical transformation. Jacobi’s identity; Poisson’s

Theorem. Equations of motion infinitesimal canonical transformation in the poisson

bracket formulation. Hamilton Jacobi Equations for Hamilton’s principal function, The

harmonic oscillator problem as an example of the Hamilton – Jacobi method. (Art. 8.1,

8.2, 8.5, 8.6, 9.1, 9.2, 9.4, 9.5, 10.1, 10.2 of Text book.1)

(Three questions are to be set)

New concept of space and Time, postulates of special theory of relativity, Lorentz

transformation equations, Lorentz contraction, Time dilation, simultaneity, Relativistic

formulae for composition of velocities and accelerations, proper time, Lorentz

transformations form a group (Scope and treatment is as in chapter 1 and 2 of Text

book.2).

(One question is to be set)

Text books:

1. Classical mechanics by H.Goldstein, 2nd edition, Narosa Publishing House.

2. Relevant topics from Special relativity by W.Rindler, Oliver & Boyd, 1960.

M.Sc. FIRST SEMESTER APPLIED MATHEMATICS

M.Sc. FIRST SEMESTER APPLIED MATHEMATICS

Duration: 3 hours Maximum Marks: 85

(A total of seven questions are to be set and the student has to answer 5 (five) questions.

All questions carry equal marks. The first question which is compulsory carries 17

marks. It consists of 4 short answer sub questions covering the entire syllabus. The

remaining six questions each carrying 17 marks are to be set as suggested in the body of

the syllabi.)

Algebraic systems: some simple algebraic systems – semi groups and monoids,

homomorphism of semi-group and monoids, groups, subgroups and homomorphism,

cosets and Lagranges theorem, normal subgroups. Sections 3-1, 3-2, 3-5.1, 3-5.2, 3-5.3

and 3-5.4 Chapter 3 of Text book.1).

Binary group codes, binary symmetric channels, encoding and decoding, block codes,

matrix encoding techniques, group codes, decoding tables, and Hamming codes (chapter

8 of Text book.2)

(Three questions are to be set)

Relations and ordering: partially ordered relations, Partially ordered sets, representation

and associated terminology. (Sections 2-3.1,2-3.2, 2-3.8, 2-3.9 of Chapter 2 in Text

book1)

Lattices, Lattices as partially ordered sets, some properties of Lattices, Lattices as

algebraic systems, sub-Lattices, direct product and homomorphism some special Lattices.

(Sections: 4-1.1 to 4-1.5 of chapter 4 of Text book.1).

Boolean Algebra, subalgebra, direct product and Homomorphism, Boolean forms and

free Boolean Algebras, values of Boolean expressions and Boolean functions (Sections:

4-2.1, 4-2.2, 4-3.1, 4-3.2 of chapter of Text book 1)

(Three questions are to be set)

Text books:

1. Discrete Mathematical structures with Applications to Computer Science by

J.P. Trembly and R.Manohar, Tata Mc.Grawhill Edition.

2. Modern Applied Algebra by G.Birkhoff. and Thomas C.Bartee

M.Sc. FIRST SEMESTER APPLIED MATHEMATICS

Duration: 3 hours Maximum Marks: 85

(A total of seven questions are to be set and the student has to answer 5 (five) questions.

All questions carry equal marks. The first question which is compulsory carries 17

marks. It consists of 4 short answer sub questions covering the entire syllabus. The

remaining six questions each carrying 17 marks are to be set as suggested in the body of

the syllabi.)

Fortran 77 programming: Introduction, Flowcharts, Fortran programming preliminaries,

Fortran constants and variables, Arithmetic expressions, Input-output statements, control

statements, Do statements, Subscripted variables, Elementary format specifications.

Logical variables and logical expressions, function subprograms, subroutine

subprograms, simple examples on these topics (Scope and treatment as in chapters 3 to

12 and 14 of Text book.1).

(Three questions are to be set)

Numerical techniques of solving transcendental and polynomial equations: Bisection

methods, secant method, Newton-Raphson method, Chebyshev method, Rate of

convergence, Iteration methods of first and second orders. Methods for multiple roots.

Numerical techniques of solving system of lineal Algebraic equations: Triangularization

method, Gauss elimination method, Gauss-jordan method, Iterative methods: Jacobi

method, Gauss-Seidel method. Numerical techniques of determining the eigen values

and eigen vectors of a matrix: Jacobi method, power method and Rutishausher method

(Scope and treatment as in chapters 2 and 3 of Text book.2).

(Three questions are to be set)

Text books:

1. V. Rajaraman, Computer programming in Fortran-77, 4th edition Prentice Hall of

India Private Ltd.

2. Jain, S.R.K. Iyangar, R.K. Jain - Numerical Methods for Scientific and Engg.

Computation, 3rd Edition, New Age international (P) Ltd. Publishers.

M.Sc. SECOND SEMESTER APPLIED MATHEMATICS

Duration: 3 hours Maximum Marks: 85

(A total of seven questions are to be set and the student has to answer 5 (five) questions.

All questions carry equal marks. The first question which is compulsory carries 17

marks. It consists of 4 short answer sub questions covering the entire syllabus. The

remaining six questions each carrying 17 marks are to be set as suggested in the body of

the syllabi.)

Functions of a complex variable: Analytic functions and Harmonic functions, Cauchy –

Riemann equations, Sufficient conditions.

Complex integration: Contour integration, Cauchy – Goursat theorem, antiderivatives,

Integral representation for analytic functions, Theorems of Morera and Liouville and

some applications.

Series: Uniform convergence of series, Taylor and Laurent series representations,

singularities, Zeros and poles, Applications of Taylor and Laurent series.

(Three questions are to be set)

Residue theory: Residue theorem, calculus of Residues, evaluation of Improper real

integral, Inderned contour integrals, Integrals with Branch point. Rouche’s theorem.

Conformal mapping : Basic properties of conformal mapping, Bilinear transformations,

mappings involving elementary functions.

(Three questions are to be set)

Text book: Complex analysis for Mathematics and Engineering – 3rd Edition by John H.

Mathews and Russel W, Howell. Narosa publishing house (chapters: 3, 6, 7, 8 and 9).

M.Sc. SECOND SEMESTER APPLIED MATHEMATICS

Duration: 3 hours Maximum Marks: 85

(A total of seven questions are to be set and the student has to answer 5 (five) questions.

All questions carry equal marks. The first question which is compulsory carries 17

marks. It consists of 4 short answer sub questions covering the entire syllabus. The

remaining six questions each carrying 17 marks are to be set as suggested in the body of

the syllabi.)

Partial differential equations: Equations of the form , dx dy dz

P Q R

= = Orthogonal

trajectories, Pfaffian differential equations, 1st order partial differential equations;

Charpit’s method and some special methods. Jacobi’s method. Second order Partial

differential equations with constant & Variable coefficients, canonical forms, separation

of variables method, Monge’s method (Chapter 1 (excluding sections 7 & 8), chapter-II

(excluding section 14), chapter III (excluding section 10) of Text book.1).

(Three questions are to be set)

Integral equations: Basic concepts, solutions of integral equations, Volterre’s integral

equations and Fredholm’s integral equations (Chapters: 1 & 2 of Text book 2)

(One question is to be set)

Fourier and Laplace Transforms with applications to ordinary, partial differential

equations and Integral equations ( Chapters 1,2,3,4,5,6 and 8 (section 8.1 & 8.2 only) of

Text book 3)

(Two questions are to be set)

Text books:

1. I.N. Sneddon, Elements of partial differential equations. Mc Graw Hill International

student Edition, 1964.

2. Shanti Swarup- Integral equations, Krishna Prakashan Media (P) Ltd, Meerut, 2003.

3. A.R.Vasishtha & R.K.Gupta, Integral transforms, Krishna Prakashan Media (P) Ltd,

Meerut, 2003.

M.Sc. SECOND SEMESTER APPLIED MATHEMATICS

Duration: 3hrs. Max.Marks:85

A total of seven questions are to be set and student has to answer 5(Five) questions. All

questions carry equal marks. The first question which is compulsory carries 17 marks. It

consists of 4 short answer sub questions covering the entire syllabus. The remaining six

questions each carrying 17 marks are to be set as suggested in the body of the syllabus.

Analysis of strain, deformation, affine deformation, infinitesimal affaine deformation,

geometrical interpretation of the components of strain, principal directions, invariants,

general infinitesimal deformation, Examples of strain, questions of compatibility

(Chapter 1 of Text book 1)

Analysis of stress, body and surface forces, stress tensor, equations of equilibrium,

transformations of coordinates, stress quadric of Cauchy, Mohr’s diagram, examples of

stress. (Chapter 2 of Text book 1)

(Three questions are to be set)

Kinematics of fluids, real and ideal fluids, velocity of fluid at a point, streamlines and

path lines, velocity potential, velocity vector, local and particle rates of change, equation

of continuity, Acceleration of fluid conditions at a rigid boundary. General analysis of

fluid motion.

(Chapter 2 of Text book 2)

Equation of motion of a fluid, pressure at a point in a fluid at rest and in a moving fluid

conditions at a boundary of two in viscid immiscible fluids, Euler’s equations of motion,

Bernoulli’s equation. Discussion of the case of steady motion under conservative body

forces. Some potential theorems. Flows involving axial symmetry. Impulsive motion.

Vortex motion, Kelvin’s circulation theorem. Some further aspects of vertex motion.

(Chapter 3 of Text book 2)

(Three questions are to be set)

__________________________________________________ ______________________

Text books:

1. Mathematical theory of Elasticity, by I.S.SOKOLNIKOFF

2nd edition; Tata Mc Graw Hill-New Delhi

2. Text book of Fluid dynamics by F.Chorlton, CBS publishers and

distributors, New Delhi..

M.Sc. SECOND SEMESTER APPLIED MATHEMATICS

Duration: 3 hours Maximum Marks: 85

(A total of seven questions are to be set and the student has to answer 5 (five) questions.

All questions carry equal marks. The first question which is compulsory carries 17

marks. It consists of 4 short answer sub questions covering the entire syllabus. The

remaining six questions each carrying 17 marks are to be set as suggested in the body of

the syllabi.)

Mathematical logic: statements structures and notation, connectives, well formed

formulas, tautologies, equivalences, implications, normal forms – Disjunctive and

conjunctive, Principle disjunctive and conjunctive normal forms.

Theory of Inference: Theory of inferences for statement calculus, validity using truth

tables, values of Inference. Predicate calculus: predicates, predicate formulas, quantifiers,

free and bound variables, Inference theory of predicate calculus. (Scope and treatment as

in Sections: 1.1 to 1.6 of Text book.1)

(Three questions are to be set)

Theory of Recursion: Recursive functions, primitive recursive functions, partial recursive

functions and Ackerman’s functions (scope and treatment as in Section 2-6.1 of Ref.1)

Graph Theory: Graphs and multigraphs, subgraphs, Isomorphism and homomorphism,

paths, connectivity, traversable multigraph, labeled and weighted graphs; complete,

regular and bipartite graphs, tree graphs, planar graphs.

Directed graphs: sequential representation of Directed graphs, shortest path, Binary trees,

Complete and extended binary trees, Representation of binary trees; traversing binary

trees and binary search tree (Scope as in Sections 8.2 to 8.9 of chapter 8, 9.2 to 9.7 of

chapter 9 and 10.1 to 10.6 of chapter 10 of Text book.2).

(Three questions are to be set)

Text books:

1. Discrete Mathematical structures with Applications to Computer Science by

J.P.Tremblay and R.Manohar Tata Mc Graw-Hill Edition.

2. Discrete Mathematics, Schaum’s outline series, second edition, by Seymour

Lipschutz and Marc Lipson Tata Mc Graw-Hill.

M.Sc. SECOND SEMESTER APPLIED MATHEMATICS

Duration: 3 hours Maximum Marks: 85

(A total of seven questions are to be set and the student has to answer 5 (five) questions.

All questions carry equal marks. The first question which is compulsory carries

marks. It consists of 4 short answer sub questions covering the entire syllabus. The

remaining six questions each carrying 17 marks are to be set as suggested in the body of

the syllabi.)

Interpolation and Approximation: Lagrange interpolation, Hermite interpolation, Spline

interpolation, Least squares approximation.

Numerical techniques for evaluating derivatives and integrals: Differentiation methods

based on interpolation formulae, methods based on finite differences, extrapolation

methods, partial differentiation. Numerical Integration methods based on interpolation

formulae, Newton – Cote’s methods, Trapezoidal and Simpsons formulae, Methods

based on undetermined coefficients – Gauss Legendre, Gauss-Chebyshev integration

methods, Lobatto integration, Composite integration methods – Trapezoidal rule,

simpsons rule and Romberg integration. (Chapter 4 and 5 of Text book.1).

(Three questions are to be set)

Numerical techniques for solving ordinary differential equations: Euler method,

backward Euler method, Midpoint method. Single step methods: Taylor series method,

Runge-Kutta methods. Multistep methods: Predictor-corrector method, Adams

Bashforth method, Adams –Moultan method, Convergence and stability analysis of

single – step methods. (Chapter 6 of Text book.1)

Numerical methods for solving elliptic partial differential equations: Difference

methods, Dirichlet problem, Laplace and Poisson equations. (Chapter 1.1, 1.2, 4.1 to 4.2

of Text book.2).

(Three questions are to be set)

Text books:

1. Numerical method for Scientific and Engineering Computation, M.K.Jain, S.R.K.

Iyengar and R.K. Jain, 3rd edition, 1993, New Age International Pvt.Ltd.

2. Computational methods for partial differential equations by M.K. Jain,

S.R.K.Iyengar and R.K. Jain, New Age International Pvt. Ltd. (1993).

M.Sc. THIRD SEMESTER APPLIED MATHEMATICS

Duration: 3 hours Maximum Marks: 85

(A total of seven questions are to be set and the student has to answer 5 (five) questions.

All questions carry equal marks. The first question which is compulsory carries 17

marks. It consists of 4 short answer sub questions covering the entire syllabus. The

remaining six questions each carrying 17 marks are to be set as suggested in the body of

the syllabi.)

Lebesgue Measure: Introduction, Outer measure, Measurable sets and Lebesgue

measure, A nonmeasurable set, Measurable functions, Littlewood’s three principles.

The Lebesgue Integral: The Riemann integral, The Lebesgue integral of a bounded

function over a set of finite measure. The integral of a nonnegative function. The

general Lebesgue integral, Convergence in measure. (Cahpters 3 and 4 of the Text

book).

(Three questions are to be set)

Differentiation and Integration: Differentiation of Monotone functions, Functions of

bounded variation, Differentiation of an integral, Absolute continuity, Convex functions.

The classical Banach Spaces: The Lp spaces, The Holder and Minkowski inequalities,

Convergence and completeness, Bounded linear functionals on the P L spaces. (Chapters

(Three questions are to be set)

Text Book: Real Analysis, H.L. Royden – Macmillan publishing Cp.

M.Sc. THIRD SEMESTER APPLIED MATHEMATICS

Duration: 3 hours Maximum Marks: 85

(A total of seven questions are to be set and the student has to answer 5 (five) questions.

All questions carry equal marks. The first question which is compulsory carries

marks. It consists of 4 short answer sub questions covering the entire syllabus. The

remaining six questions each carrying 17 marks are to be set as suggested in the body of

the syllabi.)

The vibrating string, Boundary value problems of Mathematical Physics, Eigenvalues

and Eigenfunctions, Eigenfunction Expansions, Upper and lower bounds of

eigenfunctions. (Article: 3.5 to 3.9 of the Text book)

(One question is to be set)

Orthogonal co-ordinate systems, Separation of variables. Sturm – Liouville Problems

Series Solutions of boundary value problems. (Article: 4.1 and 4.2 of the Text book)

(Two questions are to be set)

Greens functions: Non/homogenous boundary value problems. One dimensional

Green’s function. Generalized functions. Green’s function in higher dimensions.

Problems in unbounded regions. (Article: 5.1 to 5.5 of the Text book)

(Three questions are to be set)

Text Book: John W.Dettman, Mathematical Methods in Physics and Engineering,

Mc.Graw Hill Book Company, Second edition. (1969)

M.Sc. THIRD SEMESTER APPLIED MATHEMATICS

Duration: 3 hours Maximum Marks: 85

(A total of seven questions are to be set and the student has to answer 5 (five) questions.

All questions carry equal marks. The first question which is compulsory carries 17

marks. It consists of 4 short answer sub questions covering the entire syllabus. The

remaining six questions each carrying 17 marks are to be set as suggested in the body of

the syllabi.)

Data types, Operators and Some statements: Identifiers and key words, Constants, C

operators, Type conversion.

Writing a Program in C: Variable declaration, Statements, Simple C Programs, Simple

input statement, Simple output statement, Featutre of stdio.h.

Control statements: Conditional expressions: If statement, if-else statement, Switch

statement, Loop statements: For loop, While loop, Do – while loop, Breaking control

statements: Break statement, Continue statement, goto statement.

Functions and Program Structures: Introduction, Defining a function, Return statement,

Types of Functions, Actual and formal arguments, Local Global variables. The scope of

variables: Automatic Variables, Register Variables, Static Variables, External wariables,

Recursive functions.

(Four questions are to be set)

Arrays: Array Notation, Array declaration, Array initialization, Processing

Remaining syllabus are in the attachment .........

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