UG TRB MATHEMATICS SYLLABUS

 

UG TRB MATHEMATICS SYLLABUS

UNIT-1 ALGEBRA and TRIGONOMETRY

Polynomial Equations — Imaginary and Irrational Roots Relation between Roots and Coef1iclents symmetric function of Roots in terms of coefficient- Transformation of equation — Reciprocal equation - Increase or Decrease the roots of given equation - Removal of terms — Descartes's rule of signs — Approxmate solution of roots of polynomial by Homer's Method Cardan's method of solution of cubic polynomial — Summation of series using Binomial — Exponentialand Logarithmic series.

Symmetric — Skew symmetilc, Hermitian — Skew Hermitian, Orthogonal Matrices, Unitary Matrices Eigen Values — Eigen Vectors — Cayley-HamiIton Theorem — Similar Matrices — Diagonalizalion of Matrices.

Prime Number, Composite Number, Decomposition of a Composite Number as a Product of primes uniquely — Divisor of a positive Integer — Euler Function. Congruence Modulo n, Highest power of prime number p Contained in n! — Application of Maxima and Minima — Prime and Composite numbers — Euler's function Φ(N) — Congruences — Fermat‘s, Wilson's and Lagrange's theorems.

Expansions of Power of sinnX, cosnX, tannx — Summation by C + i S method, Telescopic Summation - Expansion of sinx, cosx, tanx in lerms of x - Sum of Roots of Trigonometric Equation, Formation of Equation With Trigonometric Roots - Hyperbolic Functions — Relation Between Circular and Hyperbolic Function — Inverse Hyperbolic Function — Logarithm of a complex number — Principal Value and General Values,

UNIT II DIFFERENTIAL CALCULUS, INTEGRAL CALCULUS and ANALYTICAL GEOMETRY

 

n* derivatives —Trigonometrical Transformations — Leibnitz Theorem — Implicit functions - Partial Differentiation— Maxima / Minima of a function of Ma variables — Lagrangian multiplier method - Radius of curvature in Cartesian and Polar forms — Angle between radius vector and tangent — Slope of tangent of a polar curve — p-r equations — Center of Curvature — Evolutes, Envelopes —Asymptotes of Algebraic curves - Asymptotes by inspection — Intersection of a curve with asymptotes. 

Evaluation of Double and Triple integrals — Applications of Multiple lntegrals in finding volumes, surface areas of solids — Areas of curved surfaces — Jacobians — Transformation of fntegrals using Jacobians — Indefinite integrals - Beta and Gamma Functions and their properties — Evaluation of lntegrals using Beta and Gamma Functions.

 Pole and Polar — Conjugate points and Conjugate lines, Conjugate diameters - Polar Coordinates — General Polar Equation of a Straight line — General Polar Equation of a Conic

UNIT III DIFFERENTIAL EQUATIONS and LAPLACE TRANSFORMATIONS

 Ordinary Differential Equations - Homogeneous Equations - Exact equations - Integrating Factors - Linear equations - Reduction of order — Second order Linear differential equations — General soluton of homogeneous Equations — Homogeneous equation with constant coefficients — Method of undetermined coefficients - method of Variation of Parameters - System of first order equations — Linear systems - Homogeneous linear systems with constant coefficients.

Partial Differential Differential Equations - Formation of Partial Oifferential Differential Equations by eliminating arbitrary constants and arbitrary functions. Solving PDEs: Complele integral - Singular Integral - general integral - Lagrange's equation Pp+Qq=R - Charpit's mefhod and special types of first order equations. 

Laplace transform of elementary functions — Laplace transforms of special functions like unit step function. Dirac Delta function — Properties of Laplace Transformation and Laplace Transforms of derivatives and integrals — Evaluation of integrals using Laplace transform - Initial value theorem - Final value theorem — Laplace transform of periodic functions — Inverse Laplace transforms — Convolution theorem — Application of Laplace transformations in solving first and second order linear differential equations and simultaneous linear ordinary differential equations.

UNIT -IV   VECTOR CALCULUS and FOURIER SERIES, FOURIER TRANSFORMS

Vector Differentiation — Velocity and Acceleration — Vector valued functions and Scalar potentials — Gradient — Divergence — Curl — Directional Derivative — Unit normal to a surface — Laplacian double operator — Harmonic functions.

Vector Integration — Line integral — Conservative force field — Determining Scalar Potential from a conservative force field — Work done by a force — Surface Integral — Volume inlegral — Theorems of Gauss, Stokes, and Green.

 Fourier Series — Expansions of Periodic functions of period 2x - Expansion of even and odd functions — half range series — Evaluation of Infinite Series using Fourier Series expansions — Fourier Transforms — Infinite Fourier Transform — Fourier Sine and Cosine transforms — Simple properties of Fourier Transforms — Convolution Theorem — Parseval's identity.

UNIT -V ALGEBRAIC STRUCTURES

Groups — Subgroups, cyclic Groups and properties of cyclic groups, Lagrange's Theorem — Counting Principles — Normal subgroups, Quotient groups, Homomorphism, Automorphism, Cayley's theorem, Permutation groups — Rings — Some special classes of Rings — Integral domain, Homomorphism of rings — Ideal and Quotient rings — Prime ideal, ktaximum Ideals —the field and quotients of an integral domain — Euclidean rings — Algebra of Linear transformation, Characteristic roots, matrices, Canonical forms, Triangular Foims — Problems of converting Linear Transformation to Matrices and vice-versa — Vector Space — Definition and examples — Linear dependence — Independence, Sub spaces and Dual spaces — Inner product spaces.

UNIT-VI REAL ANALYSlS

Sets — Countable and Uncountabfe sets — Real Number system R — Functions — Real Valued functions, Equivalence and Countability — Infremum and Supremum of a subset of R — BoIzanO- Weierstrass Theorem — Sequences of real numbers — Convergent and Divergent Sequences — Monatane Sequences — Cauchy Sequences — Limit Superior and Limit Inferior of a sequence — Sub Sequences — Infinite series — Alternating Series — Conditional convergence and Absolute convergence — Tesls of Absolute convergence — Continuity and Uniform Continuity of a real valued function of a real variable — Limit of a function at a point — Coninuity and Differentiabllity of real valued functions — Rolle’s Theorem — Mean Value Theorems — Inverse function theorem, Taylor‘s Theorem with remainder forms — Pawer senes expansion — Riemann Integrability — Sequences and Series of Functions.

Metric spaces — Limits of a function at a point in metric spaces — functions continuous on a metric space — various reformulations of continuity of a function in a metric space - open sets — closed sets — discontinuous functions on the real line.

UNIT VII COMPLEX ANALYSIS

Algebra of Complex Numbers — Function of Complex Variable — Mappings, Limits — Theorems on Limits, confinuity, differentiability — CauEhy-Riemann Equations - Analytic Functions — Harmonic Function — Conformal mapping

— Mobius Transformations — Elementary Transformation — Bilinear Transformations — Cross ratio — Fixed points of biJinear transformations — Special Bilinear transformations.

Contours — Contour Integrals - Anti Deñvatives — Cauchy-Goursat Theorem- Power Series — Complex Integration

— Cauchy*s theorem, Morera's theorem, Cauchy's Integral Formula — Liouville's Theorem — Maximum Modulus Principle

— Schwarz's Lemma — Taylor's series — Laurent's sefies — Calculus of Residues — Residue Theorem — Evaluation of Integrals - Definite integrafs of Trigonometrlc functions — Argument principle and Rouche's Theorem.

UNIT VIII MECHANICS

Statics: Farces on a rigid body —Moment of a force — General motion of a rigid body — Equivalent system of forces

— Parallel Forces — Forces along the sides of Triangle Couples.

Resultant of several coplanar forces — Equation of line of action of the resultant — Equilibrium of rigid body under three Coplanar forces — Reduction of Coplanar fDrces into single force and couples — Laws of friction, angle of friction. Equilibrium of a body on a rough inclined plane acted on by several forces — Equilibrium of a uniform Homogeneous string — Catenary — Suspension bridge — Centre of Gravity ol uniform rigid bodies,

Dynamics: Velocity and AcceJeration — Coplanar motion — Rectilinear motion under constant forces — Acceleration and retardation thrust on a plane — Motion atong a Vertical line under gravity — Motion along an inclinerl plane — motion of connected particles — Newton's Laws of mofion.

Work, Energy and power — Work — Conservative field of force — Power —Rectilinear motion under varying force Simple Harmonic Motion (S,H.M) — S.H.M along a horizontal line — S.H.M along a Vertical line — Motion under gravity in a resisting medium.

Path of a projectile - Particle projected on an inclined plane — Analysis of forces acting on particles and rigid bodies on static equilibrium, equivalent systems of forces, friction, centroids and moments of inertia — Elastic Medium, Impact - Impulsive force — Impact of sphere — Impact of two smooth spheres — Impact of two spheres of two smooth sphere on a plane — oblique impact of two smooth spheres.

Circular motion — Conical Pendulum motion of a cyclist on circular path — Circular motion on a vertical plane relative rest in revolving cone — simple pendulum — Central Orbits — Conic as Centered Orbit — Moment of inertia

UNIT IX OPERATIONS RESEARCH

Linear Programming — Formulation — Graphical Solution - Simplex Method — Big —M method — Two phase method - Duality — Primal dual relation — dual simplex method — revised simplex method — Sensitivity analysis — Transportation Problem — Assignment Problem — Queuing Theory — Basic Concepts — Steady State analysis of M/M/1 and M/M/Systems with infinite and finite capacities.

 PERT-and CPM — Project network diagram — Critical path — PERT computations-Inventory Models- Basic Concept -EOQ Models — uniform Demand rate infinite and finite protection rate with no shonage — Classical newspaper boy problem with discrete demand — purchase inventory model with one price brake — Game theory - Two person Zero — Sum game with saddle point — without saddle point — Dominance — Solving 2xn or mx2 game by graphical method — Integer programming — Branch and bound method

UNIT—X STATISTICS/PROBABILITY

Measures of central tendency - Measures of Dispersion — Moments - Skewness and Xurtosis — Correlation — Rank Correlation — Regression — Regression line of x on y and y on x — Index Numbers — Consumer Price Index numbers — Conversion of chain base Index Number into fixed base index numbers - Curve Fitting — Principle of Least Squares - Fitting a straight line — Fitting a second degree parabola — Fitting of power curves - Theory of Attributes — Attributes — Consistency of Data — Independence and Associate of data.

Theory of Probability — Sample Space - Axioms of Probability - Probability function — Laws of Addition - Conditional Probability — Law of multiplication - Independent — Boole's Inequality - Bayes’ Theorem — Random Variables Distribution function — Discrete and continuous random variables — Probability density functions — Mathematical Expectation — Moment Generating Functions — Cumutates - Characteristic functions — Theoretical distributions —

Binomial, Poisson, Normal distributions — Properties and conditions of a normal curve — Test of significance of sample and large samples — Z-test - Student's t-test — F-test - Chi square and contingency coefficient.