ABSTRACT ALGEBRA-AM101
Paper I
Course Outcomes
1. To combime polynomial by additions or subtractions.
2. To solve problems of simple inequalities.
3. Interpret basic absolute value expression.
4. To simplify algebraic expression,using the commutative,Associative and Distributive properties.
Unit-I
Automorphisms.
Conjugacy and G-sets.
Normal series Solvable groups.
Nilpotent groups.
Unit-II
Structure theorems of groups
Direct product
Finitely generated abelian group
Invariants of a finite abelian group
Sylow's theorems-Groups of orders p2, pq.
Unit-III
Ideals and homomorphisms.
Sum and direct sum of ideals.
Maximal and prime ideals.
Nilpotent and nil ideals.
Zorn'slemma.
Unit-IV
Unique factorization domains.
Principal ideal domains.
Euclidean domains.
Polynomial rings over UFD.
Rings of Fractions.
TextBook:
Basic Abstract Algebra byP.B.Bhattacharya,S.K.Jainand S.R.Nagpaul.
Reference:
[1] Topics in Algebra by I.N.Herstein.
[2] Elements of Modern Algebra by Gibert & Gilbert.
[3] Abstract Algebra byJe reyBergen.
[4] Basic Abstract Algebra by RobertBAsh.
Mathematical Analysis-AM102
Paper-II
Course Outcomes
1.Define and recognize the basic properties of the field of real numbers and Series of real numbers and convergence
2.The basic topological properties of R
3.The real functions and its limits, the continuity of real functions
4.The differentiability of real functions and its related theorems
Unit-I
Metric spaces.
Compact sets.
Perfect sets.
Connected sets.
Unit-II
Limits of functions.
Continuous functions.
Continuity and compactness, Continuity and connectedness.
Discontinuities.
Monotone functions.
Unit-III
Riemann - Steiltjes integral.
Definition and Existence of the Integral.
Properties of the integral.
Integration of vector valued functions.
Rectifiable curves.
Unit-IV
Sequences and series of functions.
Uniform convergence.
Uniform convergence and continuity.
Uniform convergence and integration.
Uniform convergence and differentiation.
Approximation of a continuous function by a sequence of polynomials.
Text Book:
Principles of Mathematical Analysis (3rd Edition(Chapters 2, 4, 6 ) By Walter Rudin, Mc Graw - Hill Internation Edition.
References:
[1] The Real Numbers by John Stillwel.
[2] Real Analysis by Barry Simon
[3] Mathematical Analysis Vol - I by D J H Garling.
[4] Measure and Integral by Richard L.Wheeden and Antoni Zygmund.
Ordinary and Partial Differential Equations-AM103
Paper - III
Course Outcomes
1.Apply the fundamental concepts of Ordinary Differential Equations and Partial Differential Equations and the basic numerical methods for their resolution.
2.Solve the problems choosing the most suitable method.
3.Understand the difficulty of solving problems analytically and the need to use numerical approximations for their resolution.
4.Use computational tools to solve problems and applications of Ordinary Differential Equations and Partial Differential Equations.
Unit-I
Existence and Uniqueness of solution of dy dx = f(x, y) and problems there on.
The method of successive approximations - Picard’s theorem.
Non - Linear PDE of order one.
Charpit’s method.
Cauchy’s method of Characteristics for solving non - linear partial differential equations.
Linear Partial Differential Equations with constant coefficients.
Unit-II
Partial Differential Equations of order two with variable coefficients.
Canonical form.
Classification of second order Partial Differential Equations.
separation of variables method of solving the one - dimensional Heat equation, Wave equation and Laplace equation.
Sturm - Liouville’s boundary value problem.
Unit-III
Power Series solution of O.D.E. – Ordinary and Singular points.
Series solution about an ordinary point.
Series solution about Singular point - Frobenius Method.
Lagendre Polynomials: Lengendre’s equation and its solution.
Lengendre Polynomial and its properties - Generating function - Orthogonal properties - Recurrance relations.
Laplace’s definite integrals for Pn(x).
Rodrigue’s formula.
Unit-IV
Bessels Functions: Bessel’s equation and its solution.
Bessel function of the first kind and its properties - Recurrence Relations - Generating function - Orthogonality properties.
Hermite Polynomials: Hermite’s equation and its solution.
Hermite polynomial and its properties - Generating function - Alternative expressions (Rodrigue’s formula) - Orthogonality properties.
Recurrence Relations.
Text Books:
[1] Ordinary and Partial Differential Equations, By M.D. Raisingania, S. Chand Company Ltd., New Delhi.
[2] Text book of Ordinary Differential Equation, By S.G.Deo, V. Lakshmi Kantham, V. Raghavendra, Tata Mc.Graw Hill Pub. Company Ltd.
[3] Elements of Partial Differential Equations, By Ian Sneddon, Mc.Graw - Hill International Edition.
Reference:
[1] Worldwide Differential equations by Robert McOwen .
[2] Differential Equations with Linear Algebra by Boelkins, Goldberg, Potter.
[3] Differential Equations By Paul Dawkins.
Mechanics-AM104
Paper - IV
Course Outcomes
1.Relative motion. Inertial and non inertial reference frames.
2.Parameters defining the motion of mechanical systems and their degrees of freedom.
3.Study of the interaction of forces between solids in mechanical systems.
4.Centre of mass and inertia tensor of mechanical systems.
Unit-I
Newton’s Law of Motion.
Historical Introduction.
Rectilinear Motion.
Uniform Acceleration Under a Constant Force.
Forces that Depend on Position.
The Concepts of Kinetic and Potential Energy.
Dynamics of systems of Particles
Introduction - Centre of Mass and Linear Momentum of a system .
Angular momentum and Kinetic Energy of a system.
Mechanics of Rigid bodies .
Planar motion.
Centre of mass of Rigid body
some theorem of Static equilibrium of a Rigid body .
Equilibrium in a uniform gravitational field.
Unit-II
Rotation of a Rigid body about a fixed axis.
Moment of Inertia.
calculation of moment of Inertia Perpendicular and Parallel axis theorem .
Physical pendulum.
A general theorem concerning Angular momentum.
Laminar Motion of a Rigid body.
Body rolling down an inclined plane (with and without slipping).
Unit-III
Motion of Rigid bodies in three dimension.
Angular momentum of Rigid body products of Inertia.
Principles axes.
Determination of principles axes.
Rotational Kinetic Energy of Rigid body.
Momentum of Inertia of a Rigid body about an arbitrary axis.
The momental ellipsoid.
Euler’s equation of motion of a Rigid body
Unit-IV
Lagrange Mechanics.
Generalized Coordinates.
Generalized forces.
Lagrange’s Equations and their applications.
Generalized momentum.
Ignorable coordinates.
amilton’s variational principle.
Hamilton function.
Hamilton’s Equations - Problems - Theorems.
Text Book: Analytical Mechanics by G.R.Fowles, CBS Publishing, 1986 Reference: [1] textbfAnalytical Mechanics by Farano,Marmi. [2] textbfAnalytical Mechanics by L N Hand, J D Finch. 14
INTEGRAL TRANSFORMS-AM105(B)
Paper - V
Course Outcomes
1.Calculate the Laplace transform of standard functions both from the definition and by using tables.
2.Select and use the appropriate shift theorems in finding Laplace and inverse Laplace transforms.
3.Select and combine the necessary Laplace transform techniques to solve second-order ordinary differential equations involving the Dirac delta (or unit impulse).
4.Calculate both real and complex forms of the Fourier series for standard periodic waveforms and convert from real-form Fourier series to complex-form and vice-versa.
Unit-I
Fourier Transforms.
Introduction.
Classes of functions.
Fourier Series and Fourier Integral Formula.
Fourier Transforms.
Fourier sine and cosine Transforms.
Linearity property.
Change of Scale property.
Shifting property.
The Modulation theorem.
Evaluation of integrals by means of inversion theorems.
Fourier Transform of some particular functions.
Convolution or Faltung of two integrable functions.
Convolution or Falting or Faltung Theorem for FT.
Parseval’s relations.
Fourier Transform of the derivative of a function.
Fourier Transform of some more useful functions.
Fourier Transforms of Rational Functions.
Other important examples concerning derivative of FT.
The solution of Integral Equations of Convolution Type.
Fourier Transform of Functions of several variables.
Application of Fourier Transform to Boundary Value Problems.
Unit II
Laplace Transforms.
Introduction.
Definitions.
Sufficient conditions for existence of Laplace Transform.
Linearity property of Laplace Transform.
Laplace transforms of some elementary functions.
First shift theorem.
Second shift theorem.
The change of scale property.
Examples - Laplace Transform of derivatives of a function.
Laplace Transform of Integral of a function.
Laplace Transform of tnf(t).
Laplace Transform of f(t)/t .
Laplace Transform of a periodic function.
The Initial Value Theorem and the Final.
Value Theorem of Laplace Transform.
Examples - Laplace Transform of some special functions.
The Convolution of two functions - Applications.
Unit III
Inverse Laplace Transforms & Applications.
Introduction.
Calculation of Laplace inversion of some elementary functions.
Method of expansion into partial fractions of the ratio of two.
The general evaluation technique of inverse Laplace transform.
Application of Laplace Transforms.
Finite Laplace Transforms.
Introduction - Definition of Finite Laplace Transform.
Finite Laplace Transform of elementary functions.
Operational Properties.
The Initial Value and the Final Value Theorem - Applications.
Unit-IV
The Mellin Transform.
Introduction.
Definition of Mellin Transform.
Mellin Transform of derivative of a function.
Mellin Transform of Integral of a function.
Mellin Inversion theorem.
Convolution theorem of Mellin Transform.
Illustrative solved Examples.
Solution of Integral equations.
Application to Summation of Series.
The Generalised Mellin Transform.
Convolution of generalised Mellin Transform.
Finite Mellin Transform.
The Z-Transform: Introduction.
Transform: Definition.
Some Operational Properties of Z-Transform.
Application of Z-Transforms.
Text Book: [1] An Introduction to Integral Transforms by Baidyanath Patra, CRC Press, Taylor Francis Group.
References: [1] Integral Transforms by A.R.Vasishta and R.K.Guptha.