Thursday, July 5, 2018

AUTHOR MESSAGE


                    I am very very fond of the great mathematician SRINIVASA RAMANUJAN because of his intelligence. Not only me, most of the professors and teachers are following his brilliant methodology in their teaching style. Because of his wide knowledge in mathematics we can learn small and toughest concepts in mathematics very easily. There by this world always praise his superior intelligence.

Seven Wonder About My GURU Srinivasa Ramanujan - The Man Who Knew Infinity

Ramanujan was a child prodigy, and a mathematical genius. Irrespective of having little or no access to having exposure to advance mathematics, he turned out to be one hell of a genius as a kid! As most child prodigies, if not all, Ramanujan could not survive for long. He died at the age of 33 on April 20, 1920, but even in his short-lived life, he became Mozart of mathematics! 
1. When Ramanujan was thirteen, he could work out Loney’s Trigonometry exercises without any help!
By the age of 13, he had completed advanced trigonometry and discovered complex theorems on his own! Feeling numerically dyslexic already? 

2. He never had any friends in school because his peers rarely understood him at school & were always in awe of his mathematical acumen!

It is an obvious fact that being a nerd, it became difficult for Ramanujan to gel with kids of his age. While others were involved in sports, it was math which caught his interest at a tender age. 

3. As a young man, he failed to get a degree, as he did not clear his fine arts courses, although he always performed exceptionally well in mathematics. 

Who says failure should be the dead-end in a person's life! This man is an inspiration for many that temporary failure can't ever decide your future. Don't let minor disappointments come your way and keep following your passion. 

4. Because paper was expensive, poor Ramanujan often used to derive his results on a 'slate' to jot down results of his derivations.

After he died, people close to him found a treasure! This treasure was nothing materialistic but something which was even more precious! He left behind a 'notebook' with merely summaries and results in it, with little or no proofs - his personal notebook

      The first notebook had 351 pages with 16 organised chapters and some unorganized material. The second notebook has 256 pages in 21 chapters and 100 unorganized pages, and his third notebook had 33 unorganized pages. The results in his notebooks inspired numerous papers by mathematicians! 

5. G.H. Hardy brought Ramanujam with him to England but unfortunately the English weather didn't suit him. He also reported of mild racism towards him. 

After getting inspired by his book Orders of Infinity, he wrote a letter to famous English mathematician G.H. Hardy (Who later became his mentor) in 1913. After a visit to India, G.H Hardy brought Ramanujam with him to England but unfortunately the English weather didn't suit him very well. Also, being a devout Brahman, this mathematical super-hero had a tough time adjusting with the culture and cuisine. 
6. 22nd December is called National Mathematics Day in India because of Ramanujan's birth-anniversary!
7. After a funny incident, 1729 is called Hardy-Ramanujam number in his honor, and such numbers are called Taxicab numbers.
         After moving to England, Ramanujan had a lot of health disorders. A visit to hospital in a taxi resulted in one of the most celebrated anecdotes- Once, when G.H. Hardy went to the hospital to visit him, he remarked that he had ridden in a taxicab with the number 1729, adding "what a dull number to ride in to the hospital". To which Ramanujam immediately said "No, on the contrary it is a very interesting number! It is the smallest number expressible as the sum of two cubes in two different ways". He was a pure genius, isn't it?


Sunday, July 1, 2018

MODEL QUESTION PAPERS

Engineering Mathematics- Anna University Syllabus Reg. 2017

Common to 1st and 2nd semester for all departments:

MA8151 Engineering Mahematics - I
MA8251 Engineering Mahematics - II

For 3rd and 4th Semester Mathematics Subjects

Department of Mechanical Engineering 

MA8353 Transforms and Partial Dofferential Equations
MA8452 Statistics and Numerical Methods

Department of Civil Engineering 

MA8353 Transforms and Partial Dofferential Equations
MA8491 Numerical Methods

Department of Computer Science and Engineering 

MA8351 Discrete Mathematics
MA8402 Probability and Queueing Theory
MA8551 Algebra and Number Theory

Department of Information Technology

MA8351 Discrete Mathematics
MA8391 Probability and Statistics

Department of Electrical and Electronic Engineering

MA8353 Transforms and Partial Dofferential Equations
MA8491 Numerical Methods

Department of Electronic and Communication Engineering

MA8352 Linear Algebra and Partial Differential Equations
MA8451 Probability and Random Processes

Thursday, June 28, 2018

MA8491 NUMERICAL METHODS


OBJECTIVES : 
 To introduce the basic concepts of solving algebraic and transcendental equations.
 To introduce the numerical techniques of interpolation in various intervals in real life situations.
 To acquaint the student with understanding of numerical techniques of differentiation and
integration which plays an important role in engineering and technology disciplines.
 To acquaint the knowledge of various techniques and methods of solving ordinary differential
equations.
 To understand the knowledge of various techniques and methods of solving various types of
partial differential equations.

UNIT I SOLUTION OF EQUATIONS AND EIGENVALUE PROBLEMS 12
Solution of algebraic and transcendental equations - Fixed point iteration method – Newton Raphson
method - Solution of linear system of equations - Gauss elimination method – Pivoting - Gauss Jordan method – Iterative methods of Gauss Jacobi and Gauss Seidel - Eigenvalues of a matrix by Power method and Jacobi’s method for symmetric matrices.

UNIT II INTERPOLATION AND APPROXIMATION 12
Interpolation with unequal intervals - Lagrange's interpolation – Newton’s divided difference
interpolation – Cubic Splines - Difference operators and relations - Interpolation with equal intervals - Newton’s forward and backward difference formulae.

UNIT III NUMERICAL DIFFERENTIATION AND INTEGRATION 12
Approximation of derivatives using interpolation polynomials - Numerical integration using
Trapezoidal, Simpson’s 1/3 rule – Romberg’s Method - Two point and three point Gaussian
quadrature formulae – Evaluation of double integrals by Trapezoidal and Simpson’s 1/3 rules.

UNIT IV INITIAL VALUE PROBLEMS FOR ORDINARY DIFFERENTIAL EQUATIONS 12
Single step methods - Taylor’s series method - Euler’s method - Modified Euler’s method - Fourth
order Runge - Kutta method for solving first order equations - Multi step methods - Milne’s and
Adams - Bash forth predictor corrector methods for solving first order equations.

UNIT V BOUNDARY VALUE PROBLEMS IN ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS 12
Finite difference methods for solving second order two - point linear boundary value problems - Finite difference techniques for the solution of two dimensional Laplace’s and Poisson’s equations on
rectangular domain – One dimensional heat flow equation by explicit and implicit (Crank Nicholson)
methods – One dimensional wave equation by explicit method.

TOTAL : 60 PERIODS

OUTCOMES :
Upon successful completion of the course, students should be able to:
 Understand the basic concepts and techniques of solving algebraic and transcendental equations.
 Appreciate the numerical techniques of interpolation and error approximations in various
intervals in real life situations.
 Apply the numerical techniques of differentiation and integration for engineering problems.
 Understand the knowledge of various techniques and methods for solving first and second
order ordinary differential equations.
 Solve the partial and ordinary differential equations with initial and boundary conditions by
using certain techniques with engineering applications.

TEXTBOOKS :
1. Burden, R.L and Faires, J.D, "Numerical Analysis", 9th Edition, Cengage Learning, 2016.
2. Grewal, B.S., and Grewal, J.S., "Numerical Methods in Engineering and Science", Khanna
Publishers, 10th Edition, New Delhi, 2015.

REFERENCES :
1. Brian Bradie, "A Friendly Introduction to Numerical Analysis", Pearson Education, Asia, New
Delhi, 2007.
2. Gerald. C. F. and Wheatley. P. O., "Applied Numerical Analysis", Pearson Education, Asia, 6th
Edition, New Delhi, 2006.
3. Mathews, J.H. "Numerical Methods for Mathematics, Science and Engineering", 2nd Edition,
Prentice Hall, 1992.
4. Sankara Rao. K., "Numerical Methods for Scientists and Engineers", Prentice Hall of India Pvt.
Ltd, 3rd Edition, New Delhi, 2007.
5. Sastry, S.S, "Introductory Methods of Numerical Analysis", PHI Learning Pvt. Ltd, 5th Edition,
2015.

MA8251 ENGINEERING MATHEMATICS – II


OBJECTIVES
This course is designed to cover topics such as Matrix Algebra, Vector Calculus, Complex
Analysis and Laplace Transform. Matrix Algebra is one of the powerful tools to handle practical
problems arising in the field of engineering. Vector calculus can be widely used for modelling the
various laws of physics. The various methods of complex analysis and Laplace transforms can
be used for efficiently solving the problems that occur in various branches of engineering
disciplines.

UNIT I                     MATRICES                                                                                            12
Eigenvalues and Eigenvectors of a real matrix – Characteristic equation – Properties of Eigenvalues
and Eigenvectors – Cayley-Hamilton theorem – Diagonalization of matrices – Reduction of a
quadratic form to canonical form by orthogonal transformation – Nature of quadratic forms.

UNIT II                   VECTOR CALCULUS                                                                           12
Gradient and directional derivative – Divergence and curl - Vector identities – Irrotational and
Solenoidal vector fields – Line integral over a plane curve – Surface integral - Area of a curved
surface - Volume integral - Green’s, Gauss divergence and Stoke’s theorems – Verification and
application in evaluating line, surface and volume integrals.

UNIT III                  ANALYTIC FUNCTIONS                                                                     12
Analytic functions – Necessary and sufficient conditions for analyticity in Cartesian and polar
coordinates - Properties – Harmonic conjugates – Construction of analytic function - Conformal
mapping – Mapping by functions 1 2 z w  z  c, cz, , - Bilinear transformation.

UNIT IV                  COMPLEX INTEGRATION                                                                 12
Line integral - Cauchy’s integral theorem – Cauchy’s integral formula – Taylor’s and Laurent’s series – Singularities – Residues – Residue theorem – Application of residue theorem for evaluation of real
integrals – Use of circular contour and semicircular contour.

UNIT V                    LAPLACE TRANSFORMS                                                                  12
Existence conditions – Transforms of elementary functions – Transform of unit step function and unit
impulse function – Basic properties – Shifting theorems -Transforms of derivatives and integrals –
Initial and final value theorems – Inverse transforms – Convolution theorem – Transform of periodic
functions – Application to solution of linear second order ordinary differential equations with constant coefficients.

TOTAL: 60 PERIODS

OUTCOMES :
After successfully completing the course, the student will have a good understanding of the following topics and their applications: 
  • Eigenvalues and eigenvectors, diagonalization of a matrix, Symmetric matrices, Positive definite matrices and similar matrices. 
  • Gradient, divergence and curl of a vector point function and related identities.  Evaluation of line, surface and volume integrals using Gauss, Stokes and Green’s theorems and their verification. 
  • Analytic functions, conformal mapping and complex integration. 
  • Laplace transform and inverse transform of simple functions, properties, various relatedtheorems and application to differential equations with constant coefficients.


TEXT BOOKS :
1. Grewal B.S., “Higher Engineering Mathematics”, Khanna Publishers, New Delhi,
43rd Edition, 2014.
2. Kreyszig Erwin, "Advanced Engineering Mathematics ", John Wiley and Sons,
10th Edition, New Delhi, 2016.

REFERENCES :
1. Bali N., Goyal M. and Watkins C., “Advanced Engineering Mathematics”, Firewall
Media (An imprint of Lakshmi Publications Pvt., Ltd.,), New Delhi, 7th Edition, 2009.
2. Jain R.K. and Iyengar S.R.K., “ Advanced Engineering Mathematics ”, Narosa
Publications, New Delhi , 3rd Edition, 2007.
3. O’Neil, P.V. “Advanced Engineering Mathematics”, Cengage Learning India
Pvt., Ltd, New Delhi, 2007.
4. Sastry, S.S, “Engineering Mathematics", Vol. I & II, PHI Learning Pvt. Ltd,
4th Edition, New Delhi, 2014.
5. Wylie, R.C. and Barrett, L.C., “Advanced Engineering Mathematics “Tata McGraw Hill
Education Pvt. Ltd, 6th Edition, New Delhi, 2012.

MA8151 ENGINEERING MATHEMATICS - I



OBJECTIVES :
 The goal of this course is to achieve conceptual understanding and to retain the best traditions of
traditional calculus. The syllabus is designed to provide the basic tools of calculus mainly for the
purpose of modelling the engineering problems mathematically and obtaining solutions. This is a
foundation course which mainly deals with topics such as single variable and multivariable
calculus and plays an important role in the understanding of science, engineering, economics and
computer science, among other disciplines.

UNIT I DIFFERENTIAL CALCULUS                                                                                           12
Representation of functions - Limit of a function - Continuity - Derivatives - Differentiation rules -
Maxima and Minima of functions of one variable.

UNIT II FUNCTIONS OF SEVERAL VARIABLES                                                                      12
Partial differentiation – Homogeneous functions and Euler’s theorem – Total derivative – Change of
variables – Jacobians – Partial differentiation of implicit functions – Taylor’s series for functions of two variables – Maxima and minima of functions of two variables – Lagrange’s method of undetermined multipliers.

UNIT III INTEGRAL CALCULUS                                                                                                 12
Definite and Indefinite integrals - Substitution rule - Techniques of Integration - Integration by parts,
Trigonometric integrals, Trigonometric substitutions, Integration of rational functions by partial fraction, Integration of irrational functions - Improper integrals.

UNIT IV MULTIPLE INTEGRALS                                                                                                 12
Double integrals – Change of order of integration – Double integrals in polar coordinates – Area
enclosed by plane curves – Triple integrals – Volume of solids – Change of variables in double and
triple integrals.

UNIT V DIFFERENTIAL EQUATIONS                                                                                          12
Higher order linear differential equations with constant coefficients - Method of variation of parameters – Homogenous equation of Euler’s and Legendre’s type – System of simultaneous linear differential equations with constant coefficients - Method of undetermined coefficients.

TOTAL : 60 PERIODS

OUTCOMES :
After completing this course, students should demonstrate competency in the following skills:
 Use both the limit definition and rules of differentiation to differentiate functions.
 Apply differentiation to solve maxima and minima problems.
 Evaluate integrals both by using Riemann sums and by using the Fundamental Theorem of
Calculus.
 Apply integration to compute multiple integrals, area, volume, integrals in polar coordinates, in
addition to change of order and change of variables.
20
 Evaluate integrals using techniques of integration, such as substitution, partial fractions and
integration by parts.
 Determine convergence/divergence of improper integrals and evaluate convergent improper
integrals.
 Apply various techniques in solving differential equations.

TEXT BOOKS :
1. Grewal B.S., “Higher Engineering Mathematics”, Khanna Publishers, New Delhi, 43rd Edition,
2014.
2. James Stewart, "Calculus: Early Transcendentals", Cengage Learning, 7th Edition, New Delhi,
2015. [For Units I & III - Sections 1.1, 2.2, 2.3, 2.5, 2.7(Tangents problems only), 2.8, 3.1 to 3.6,
3.11, 4.1, 4.3, 5.1(Area problems only), 5.2, 5.3, 5.4 (excluding net change theorem), 5.5, 7.1 -
7.4 and 7.8].

REFERENCES :
1. Anton, H, Bivens, I and Davis, S, "Calculus", Wiley, 10th Edition, 2016.
2. Jain R.K. and Iyengar S.R.K., “Advanced Engineering Mathematics”, Narosa Publications, New
Delhi, 3rd Edition, 2007.
3. Narayanan, S. and Manicavachagom Pillai, T. K., “Calculus" Volume I and II,
S. Viswanathan Publishers Pvt. Ltd., Chennai, 2007.
4. Srimantha Pal and Bhunia, S.C, "Engineering Mathematics" Oxford University Press, 2015.
5. Weir, M.D and Joel Hass, "Thomas Calculus", 12th Edition, Pearson India, 2016.

MA 8353 TRANSFORMS AND PARTIAL DIFFERENTIAL EQUATIONS




UNIT I       PARTIAL DIFFERENTIAL EQUATIONS               `                           12

Formation of partial differential equations – Singular integrals – Solutions of standard types of first order partial differential equations – Lagrange’s linear equation – Linear partial differentialequations of second and higher order with constant coefficients of both homogeneous and nonhomogeneoustypes.

UNIT II      FOURIER SERIES                                                                                   12

Dirichlet’s conditions – General Fourier series – Odd and even functions – Half range sine series –Half range cosine series – Complex form of Fourier series – Parseval’s identity – Harmonicanalysis.

UNIT III    APPLICATIONS OF PARTIAL DIFFERENTIAL EQUATIONS      12

Classification of PDE – Method of separation of variables – Fourier Series Solutions of one dimensional wave equation – One dimensional equation of heat conduction – Steady state solution of two dimensional equation of heat conduction.

UNIT IV    FOURIER TRANSFORMS                                                                     12

Statement of Fourier integral theorem – Fourier transform pair – Fourier sine and
cosine transforms – Properties – Transforms of simple functions – Convolution theorem –Parseval’s identity.

UNIT V      Z – TRANSFORMS AND DIFFERENCE EQUATIONS                     12

Z-transforms – Elementary properties – Inverse Z-transform (using partial fraction and residues) –Initial and final value theorems – Convolution theorem – Formation of difference equations –Solution of difference equations using Z – transform.

TOTAL: 60 PERIODS

TEXT BOOKS:

1. Grewal B.S., “Higher Engineering Mathematics”, 43rd Edition, Khanna Publishers, New
Delhi, 2014.
2. Narayanan S., Manicavachagom Pillay.T.K and Ramanaiah.G “Advanced Mathematics for
Engineering Students”, Vol. II & III, S.Viswanathan Publishers Pvt. Ltd, Chennai, 1998.

REFERENCES :
1. Andrews, L.C and Shivamoggi, B, “Integral Transforms for Engineers” SPIE Press, 1999.
2. Bali. N.P and Manish Goyal, “A Textbook of Engineering Mathematics”, 9th Edition, LaxmiPublications Pvt. Ltd, 2014.
3. Erwin Kreyszig, “Advanced Engineering Mathematics “, 10th Edition, John Wiley, India,2016.
4. James, G., “Advanced Modern Engineering Mathematics”, 3rd Edition, Pearson Education,2007.
5. Ramana. B.V., “Higher Engineering Mathematics”, McGraw Hill Education Pvt. Ltd, NewDelhi, 2016.
6. Wylie, R.C. and Barrett, L.C., “Advanced Engineering Mathematics “Tata McGraw HillEducation Pvt. Ltd, 6th Edition, New Delhi, 2012