BSC MATHEMATICS
Investing in Tomorrow's Leaders
Go University 2016/17/18/19
BSc. (Maths Hons.) - structure
1ST SEMESTER/1ST YEAR
Unit – I
Symmetric, Skew-symmetric, Hermitian and skew Hermitian matrices. Elementary Operations on matrices. Rank of a matrices. Inverse of a matrix. Linear dependence and independence of rows and columns of matrices. Row rank and column rank of a matrix. Eigenvalues, eigenvectors and the characteristic equation of a matrix. Minimal polynomial of a matrix. Cayley Hamilton theorem and its use in finding the inverse of a matrix.
Unit - II
Applications of matrices to a system of linear (both homogeneous and non– homogeneous) equations. Theorems on consistency of a system of linear equations. Unitary and Orthogonal Matrices, Bilinear and Quadratic forms.
Unit – III
Relations between the roots and coefficients of general polynomial equation in one variable. Solutions of polynomial equations having conditions on roots. Common roots and multiple roots. Transformation of equations.
Unit - IV
Nature of the roots of an equation Descarte‘s rule of signs. Solutions of cubic equations (Cardon‘s method). Bi-quadratic equations and their solutions.
Unit - I
Definition of the limit of a function. Basic properties of limits, Continuous functions and classification of discontinuities. Differentiability. Successive differentiation. Leibnitz theorem. Maclaurin and Taylor series expansions.
Unit - II
Asymptotes in Cartesian coordinates, intersection of curve and its asymptotes, asymptotes in polar coordinates. Curvature, radius of curvature for Cartesian curves, parametric curves, polar curves.
Newton‘s method. Radius of curvature for pedal curves. Tangential polar equations. Centre of curvature. Circle of curvature. Chord of curvature, evolutes. Tests for concavity and convexity. Points of inflexion. Multiple points. Cusps, nodes & conjugate points. Type of cusps.
Unit - III
Tracing of curves in Cartesian, parametric and polar co-ordinates. Reduction formulae. Rectification, intrinsic equations of curve.
Unit - IV
Quardrature (area) Sectorial area. Area bounded by closed curves. Volumes and surfaces of solids of revolution. Theorems of Pappu‘s and Guilden.
Unit - I
General equation of second degree. Tracing of conics. Tangent at any point to the conic, chord of contact, pole of line to the conic, director circle of conic. System of conics. Confocal conics. Polar equation of a conic, tangent and normal to the conic.
Unit - II
Sphere: Plane section of a sphere. Sphere through a given circle. Intersection of two spheres, radical plane of two spheres. Co-oxal system of spheres Cones. Right circular cone, enveloping cone and reciprocal cone. Cylinder: Right circular cylinder and enveloping cylinder.
Unit - III
Central Conicoids: Equation of tangent plane. Director sphere. Normal to the conicoids. Polar plane of a point. Enveloping cone of a coincoid. Enveloping cylinder of a coincoid.
Unit - IV
Paraboloids: Circular section, Plane sections of conicoids. Generating lines. Confocal conicoid. Reduction of second degree equations.
Unit - I
Sets, principle of inclusion and exclusion, relations, equivalence relations and partition, denumerable sets, partial order relations, Mathematical Induction, Pigeon Hole Principle and its applications.
Unit –II
Propositions, logical operations, logical equivalence, conditional propositions, Tautologies and contradictions. Quantifier, Predicates and Validity.
Unit - III
Permutations and combinations, probability, basic theory of Graphs and rings. Unit - IV Discrete numeric functions, Generating functions, recurrence relations with constant coefficients. Homogeneous solution, particular relations, total rotation, Solution of recurrence relation by the method generating function.
Unit - I
Introduction of Statistics, Basic knowledge of various types of data, Collection, classification and tabulation of data. Presentation of data: histograms, frequency polygon, frequency curve and ogives. Stem- and- Leaf and Box plots.
Unit - II
Measures of Central Tendency and Location: Mean, median, mode, geometric mean, harmonic mean, partition values. Measures of Dispersion: Absolute and relative measures of range, quartile deviation, mean deviation, standard deviation ( s ), coefficient of variation.
Unit - III
Moments, Skewness and Kurtosis: Moments about mean and about any point and derivation of their relationships, effect of change of origin and scale on moments, Sheppard‘s correction for moments (without derivation), Charlier‘s checks, Concepts of Skewness and Kurtosis.
Unit - IV
Theory of Attributes: Symbolic notation, dichotomy of data, class frequencies, order of class frequencies, consistency of data, independence and association of attributes, Yule‘s coefficient of association and coefficient of colligation. Correlation for Bivariate Data: Concept and types of correlation, Scatter diagram, Karl Pearson Coefficient (r) of correlation and rank correlation coefficient.
UNIT - I
Atomic Structure: Idea of de Broglie matter waves, Heisenberg uncertainty principle, atomic orbital‘s, quantum numbers, radial and angular wave functions and probability distribution curves, shapes of s, p, d orbitals. Aufbau and Pauli exclusion principles, Hund's multiplicity rule,
Electronic configurations of the elements, effective nuclear charge, Slater‘s rules.
UNIT - II
Gaseous States: Maxwell‘s distribution o f velocities and energies (derivation excluded)
Calculation of root mean square velocity, average velocity and most probable velocity. Collis ion diameter, collision number, collision frequency and mean free path. Deviation of Real gases from ideal behaviour. Derivation of Vander Waal‘s Equation of State, its application in the calculation of Boyle‘s temperature (compression factor). Explanation of behaviour of real gases using Vander Waal‘s equation. Critical Phenomenon: Critical temperature, Critical pressure, critical volume and their determination. PV isotherms of real gases, continuity of states, the isotherms of Vander
Waal‘s equation, relationship between critical constants and Vander Waal‘s constants. Critical compressibility factor. The Law of corresponding states. Liquefaction of gases.
UNIT - III
Structure and Bonding: Localized and delocalized chemical bond, resonance effect and its applications, Stereochemistry of Organic Compounds Concept of isomerism. Types of isomerism. Optical isomerism elements of symmetry, molecular chirality, enantiomers, , optical activity, , chiral and achiral molecules with two stereogenic cent res, diastereomers, Relative and absolute configuration, sequence rules, R & S systems of nomenclature. Geometric isomerism determination of configuration of geometric isomers. E & Z system of nomenclature.
UNIT - IV
Mechanism of Organic Reactions: Curved arrow notation, drawing electron movements with arrows, half-headed and double-headed arrows, homolytic and heterolytic bond breaking. Types of reagents electrophiles and nucleophiles. Types of organic reactions. Reactive intermediates carbocations, carbanions, free radicals
COMPUTER FUNDAMENTALS AND MS - OFFICE
UNIT - I
Mechanics: Mechanics of single and system of particles, conservation of laws of linear momentum, angular momentum and mechanical energy, Centre of mass and equation of motion, constrained motion, degrees of freedom. Generalized coordinates, displacement, velocity, acceleration, momentum, force and potential. Hamilton‘s variation principle, Lagrange‘s equation of motion from Hamilton‘s Principle.
UNIT - II
Properties of Matter: Elasticity, Hooke‘s law, Elastic constants and their relations, Poisson‘s ratio, torsion of cylinder and twisting couple. Bending of beam (bending moment and its magnitude).
Theory of Relativity: Reference systems, inertial frames, Galilean invariance and Conservation laws, Newtonian relativity principle, Michelson - Morley experiment: Search for ether. Lorentz transformations, length contraction, time dilation, variation of mass with velocity and mass energy equivalence.
UNIT - III
Electrostatic Field: Derivation of field E from potential as gradient, derivation of Laplace and Poisson equations. Electric flux, Gauss‘s Law and its application to spherical shell, uniformly charged infinite plane and uniformly charged straight wire, mechanical force of charged surface, Energy per unit volume.
UNIT - IV
Magnetism: Magnetic Induction, magnetic flux, solenoidal nature of Vector field of induction. Properties of (i) Ñ.B = 0 (ii) Ñ´B = m0 J . Electronic theory of dia and para magnetism
(Langevin‘s theory). Domain theory of ferromagnetism. Cycle of Magnetization - Hysteresis (Energy dissipation, Hysteresis loss and importance of Hysteresis curve).
Practical - Descriptive Statistics
Practicals
1.To learn to interpret the meaning of three graphical representations of sets of data: stem and leaf diagrams, frequency histograms, and relative frequency histograms, the concept of the
―center‖ of a data set.
2.To learn the meaning of each of three measures of the center of a data set—the mean, the median, and the mode - and how to compute each one , the concept of the variability of a data set.
3.To learn how to compute three measures of the variability of a data set: the range, the variance, and the standard deviation, the concept of the relative position of an element of a data set.
4.To learn the meaning of each of two measures, the percentile rank and the z-score, of the relative position of a measurement and how to compute each one ,the meaning of the three
quartiles associated to a data set an how to compute them.
Inorganic
Volumetric Analysis
Redox titrations: Determination of Fe2+, C2O4 ( using KMnO4 , K2Cr2O7)
Physical
To determine the specific reaction rate of the hydrolysis of methyl acetate/ethyl acetatecatalyzed by hydrogen ions at room temperature.
Organic
Preparation and purif ication through crystallization or distillation and ascertaining their purity through melting point or boiling point ( i) Iodo form from ethanol ( or acetone)
PRACTICAL/ COMPUTATIONAL WORK COMPUTER FUNDAMENTALS AND MS – OFFICE
SPECIAL NOTES
Do any seven experiments.
.The students are required to calculate the error involved in a particular experiment (percentage error).
1.Moment of Inertia of a fly-wheel
2.M.I. of an irregular body using a torsion pendulum.
3.Young’s modulus by bending of beam.
4.Modulus of rigidity by Maxwell’s needle.
5.Elastic constants by Searle’s method.
6.“g” by Bar pendulum.
7.To study the variation of magnetic field (B) with distance along the axis of circular coil carrying current and to find the radius of coil.
8.To find the frequency of AC mains by sonometer using electromagnet.
9.To study B-H curve and to determine the energy loss due to hysteresis.
COMMUNICATION SKILL AND PERSONALITY DEVELOPMENT
Unit-I
Basic English Grammar: Present and Past Tenses, Using Modals- Can, Could, Shall, Should, Will, May, Might, Used to, Dare. etc. Using Correct Prepositions, Conditional Sentences, Proper Usage of Determiners/Adjectives, Using A/ An/ The
Unit-II
Communication Skills: Basic Introduction, Asking Questions, Making Polite Enquiries/ Request, Narrating Incidents, Describing Future Plans/ Goals, Asking/ Telling Directions, Public Speaking, Presentation Skills
Unit-III
Personality Development: Positive Attitude, Time Management, Confidence Building, Body Language, Dining Etiquette, Telephone Etiquette, Dressing Sense,
Unit-IV
Vocabulary: Words used in daily communication, Commonly misused/confused words, Words in English, from other Languages, Vocabulary specific to various professions
2ND SEMESTER / 1ST YEAR
NUMBER THEORY AND TRIGONOMETRY
Unit - I
Divisibility, G.C.D.(greatest common divisors), L.C.M.(least common multiple) Primes, Fundamental Theorem of Arithemetic. Linear Congruences, Fermat‘s theorem. Wilson‘s theorem and its converse. Linear Diophanatine equations in two variables
Unit - II
Complete residue system and reduced residue system modulo m. Euler‘s ø function Euler‘s generalization of Fermat‘s theorem. Chinese Remainder Theorem. Quadratic residues. Legendre symbols. Lemma of Gauss; Gauss reciprocity law. Greatest integer function [x]. The number of divisors and the sum of divisors of a natural number n (The functions d(n) and (n)). Moebius function and Moebius inversion formula.
Unit – III
De Moivre‘s Theorem and its Applications. Expansion of trigonometrical functions. Direct circular and hyperbolic functions and their properties.
Unit – IV
Inverse circular and hyperbolic functions and their properties. Logarithm of a complex quantity. Gregory‘s series. Summation of Trigonometry series
ORDINARY DIFFERENTIAL EQUATIONS
Unit - 1
Preliminaries : Initial value problem and equivalent integral equation. -approximate solution, Cauchy-Euler construction of an -approximate solution, Equicontinuous family of functions, Ascoli-Arzela lemma, Cauchy-Peano existence theorem. Uniqueness of solutions, Lipschitz condition, Picard-Lindelof existence and uniqueness theorem for Dependence of solutions on initial conditions and parameters, Solution of initial-value problems by Picard method .
Unit - 2
Sturm-Liouville BVPs, Sturms separation and comparison theorems, Lagrange’s identity and Green’s formula for second order differential equations, Properties of eigenvalues and eigenfunctions, Pruffer transformation, Adjoint systems, Self-adjoint equations of second order. Linear systems, Matrix method for homogeneous first order system of linear differential equations, Fundamental set and fundamental matrix, Wronskian of a system, Method of variation of constants for a nonhomogeneous system with constant coefficients, nth order differential equation equivalent to a first order system.
Unit – 3
Nonlinear differential system, Plane autonomous systems and critical points, Classification of critical points – rotation points, foci, nodes, saddle points. Stability, Asymptotical stability and unstability of critical points.
Unit - 4
Almost linear systems, Liapunov function and Liapunov’s method to determine stability for nonlinear systems, Periodic solutions and Floquet theory for periodic systems, Limit cycles, Bendixson non-existence theorem, Poincare-Bendixson theorem (Statement only), Index of a critical point.
Unit - I
Scalar and vector product of three vectors, product of four vectors. Reciprocal vectors. Vector differentiation. Scalar Valued point functions, vector valued point functions, derivative along a curve, directional derivatives
Unit - II
Gradient of a scalar point function, geometrical interpretation of grad f , character of gradient as a point function. Divergence and curl of vector point function, characters of Div f and Curl f as point function, examples. Gradient, divergence and curl of sums and product and their related vector identities. Laplacian operator.
Unit - III
Orthogonal curvilinear coordinates Conditions for orthogonality fundamental triad of mutually orthogonal unit vectors. Gradient, Divergence, Curl and Laplacian operators in terms of orthogonal curvilinear coordinates, Cylindrical co-ordinates and Spherical coordinates.
Unit - IV
Vector integration; Line integral, Surface integral, Volume integral. Theorems of Gauss, Green & Stokes and problems based on these theorems.
Unit- I
Lattices and their properties, lattice as algebraic system, Bounded, Complement and distributive lattices.
Unit - II
Boolean algebra, definition and examples, properties, duality, distributive and complmented Calculus. Design and implementation of digital networks, switching circuits, Karnaugh map.
Unit - III
Graph, definition, exemplary types of graphs, paths and circuits. Eulearian and Hermitian circuits. Seven bridges machine, shortest path traveling salesman problems. Planar graph. Matrix of graph.
Unit – IV
Directed Graphs, Trees, Isomorphism of Trees, Representation of Algebraic Expressions by Binary Trees, Spanning Tree of a Graph, Shortest Path Problem, Minimal spanning Trees, Cut Sets, Tree Searching
REGRESSION ANALYSIS AND PROBABILITY
Unit - I
Linear Regression: Concept of regression, principle of least squares and fitting of straight line, derivation of two lines of regression, properties of regression coefficients, standard error of estimate obtained from regression line, correlation coefficient between observed and estimated values. Angle between two lines of regression. Difference between correlation and regression. Curvilinear Regression: Fitting of second degree parabola, power curve of the type Y = axbexponential curves of the types Y = abx and Y = aebx
Unit - II
Concepts in Probability: Random experiment, trial, sample point, sample space, operation of events, exhaustive, equally likely and independent events, Definition of probability—classical,relative frequency, statistical and axiomatic approach, Addition and multiplication laws of probability, Boole‘s inequality.
Unit - III
Bayes‘ theorem and its applications. Random Variable and Probability Functions: Definition and properties of random variables, discrete and continuous random variable, probability mass and density functions, distribution function.
Unit - IV
Concepts of bivariate random variable: joint, marginal and conditional distributions. Mathematical Expectation: Definition and its properties –moments, measures of location, dispersion, skewness and kurtosis.
UNIT - I
Periodic Properties: Atomic and ionic radii, ionization energy, electron affinity and electronegativity –definition, trends in periodic table (in s & p block elements).
S - Block Elements: Comparative study of the elements including, diagonal relationships and salient features of hydrides (methods of preparation excluded).
P - Block Elements: Emphasis on comparative study of properties of p-block elements (including diagonal relationship and excluding methods of preparation).
Boron family (13th group):- Diborane – properties and structure (as an example of electron – deficient compound and multicentre bonding), Borazene – chemical properties and structure.
Carbon Family (14th group): Allotropy of carbon, Catenation, p – d bonding (an idea), carbides, fluorocarbons– general methods of preparations, properties and uses.
UNIT - II
Kinetics: Rate of reaction, rate equation, factors influencing the rate of a reaction –concentration,temperature, pressure, solvent, light, catalyst. Order of a reaction, integrated rate expression for zero order, first order, Half life period of a reaction. Methods of determination of order of reaction, effect of temperature on the rate of reaction – Arrhenius equation.
Electrochemistry: Electrolytic conduction, factors affecting electrolytic conduction, specific, conductance, molar conductance, equivalent conductance and relation among them, their vartion with concentration. Arrhenius theory of ionization, Ostwald‘s Dilution Law. Debye- Huckel– Onsager‘s equation for strong electrolytes (elementary treatment only), Kohlarausch‘s Law, calculation of molar ionic conductance and effect of viscosity temperature & pressure on it.
Application of Kohlarausch‘s Law in calculation of conductance of weak electrolytes at infinite diloution. Applications of conductivity measurements: determination of degree of dissociation, determination of Ka of acids determination of solubility product of sparingly soluble salts, conductometric titrations. Definition of pH and pKa, Buffer solution, Buffer action, (elementary idea only).
UNIT - III
Alkenes: Nomenclature of alkenes, mechanisms of dehydration of alcohols and dehydrohalogenation of alkyl halides,. The Saytzeff rule, Chemical reactions of alkenes mechanisms involved in hydrogenation, electrophilic and free radical additions, Markownikoff‘s rule,
UNIT - IV
Dienes and Alkynes: Nomenclature and classification of dienes: isolated, conjugated and cumulated dienes.,.Chemical reactions 1,2 and 1,4 additions (Electrophilic & free radicalmechanism),Diels-Alder reaction, Nomenclature, structure and bonding in alkynes., acidity ofalkynes. Mechanism of electrophilic and nucleophilic addition reactions,
UNIT - I
Semiconductor Diodes: Energy bands in solids. Intrinsic and extrinsic semiconductor, Hall effect, P-Njunction diode and their V-I characteristics. Zener and avalanche breakdown. Resistance of a diode, Light Emitting diodes (LED). Photo conduction in semiconductors, photodiode, Solar Cell.
Diode Rectifiers: P-N junction half wave and full wave rectifier. Types of filter circuits. Zener diode as voltage regulator, simple regulated power supply.
UNIT - II
Transistors: Junction Transistors, Bipolar transistors, working of NPN and PNP transistors, Transistor connections (C-B, C-E, CC mode), constants of transistor. Transistor characteristic curves (excluding h parameter analysis), advantage of C-B configuration.
UNIT - III
Transistor Amplifiers: Different methods of Transistor biasing and stabilization. D.C. load line. Classification of amplifiers, Common-base, common-emitteer amplifiers. Resistance-capacitance (R-C)coupled amplifier (two stage; concept of band width, no derivation). Feed-back in amplifiers, advantage of negative feedback..
Oscillators: Oscillators, Principle of Oscillation, Classification of Oscillator. Condition for self sustained oscillation ; Hartley oscillator.
UNIT - IV
Lasers ; Main features of a laser : Directionality, high intensity, high degree of coherence, spatial and
temporal coherence, Einstein's coefficients and possibility of amplification, momentum transfer, life time of a level, kinetics of optical absorption. Laser pumping, Population inversions and laser action, He-Nelaser and Semi-conductor laser (Principle, Construction and Working). Applications of laser.
INORGANIC
Volumetric analysis: Complexometric titrations: Determination of Mg2+, Zn2+ by EDTA.
Paper Chromatography: Qualitative Analysis of the any one of the following Inorganic cations and anions by paper chromatography (Pb2+ , Cu2+ , Ca2+ , Ni2+ , Cl- , Br- , I- and PO4 - and NO3- ).
PHYSICAL
1.To determine the surface tension of a given liquid by drop number method.
2.To determine the viscosity of a given liquid.
ORGANIC
1.Preparation and purification through crystallization or distillation and ascertaining
their purity through melting point or boiling point i) m-Dinitrobenzne from nitrobenzene (use 1:2 conc. HNO3-H2SO4 mixture if fuming HNO3 is not available)
ii)Aspirin from salicylic acid.
PRACTICAL/ COMPUTATIONAL WORK PROGRAMMING IN VISUAL BASIC
Unit - I
Environment Introduction:
Biotic and A biotic environment, adverse effects of Environmental Pollution, Control Strategies, Various Acts and Regulation.
Unit - II
Water Pollution:
Water Quality Standards for potable water, Surface and underground water sources, Impurities in water and their removal, Denomination, Adverse effects of domestic waste water and industrial effluent to surface water sources, Eutrophication of lakes, Self purification of streams.
Air Pollution:
Sources of air contaminations, Adverse effects on human health, Measurement of air quality standards and their permissible limits, Measure to check air pollution, Greenhouse effect, Global warming, Acid rain, Ozone depletion.
Land Pollution:
Soil Conservation, Land Erosion, A forestation, Ecology Business of Species, Biodiversity, Population Dynamics, Energy flow, Ecosystems
Unit - III
Bio Medical Waste Management:
Introduction to Bio-Medical Waste, Types of Bio-Medical Waste, Collection of Bio-MedicalWaste, Treatment and safe disposal of Bio-Medical Waste.
Solid Waste Management:
Introduction to Solid Waste, Its collection and disposal, Recovery of resources, Sanitary land- filling, Vermin-composting, Hazardous waste management.
Unit - IV
Social Issues and the Environment:
Sustainable development and life style, Urban problems related to energy, Resettlement and rehabilitating of people, Environmental ethics, Consumerism and waste products ,Water Harvesting and Rural Sanitation-Water harvesting techniques, Different schemes of Rural Water Supply in Rajasthan, Rural Sanitation, Septic Tank, Collection and disposal of wastes, Bio-gas,Community Awareness and participation, Miscellaneous, Non-Conventional (Renewable) sources of energy, Solar energy, Wind energy, Bio-mass energy, Hydrogen energy.
3RD SEMESTER/2ND YEAR
Unit - I
Continuity, Sequential Continuity, properties of continuous functions, Uniform continuity, chain rule of differentiability. Mean value theorems; Rolle‘s Theorem and Lagrange‘s mean value theorem and their geometrical interpretations. Taylor‘s Theorem with various forms of remainders, Darboux intermediate value theorem for derivatives, Indeterminate forms.
Unit - II
Limit and continuity of real valued functions of two variables. Partial differentiation. Total Differentials; Composite functions & implicit functions. Change of variables. Homogenous functions & Euler‘s theorem on homogeneous functions. Taylor‘s theorem for functions of two variables.
Unit - III
Differentiability of real valued functions of two variables. Schwarz and Young‘s theorem. Implicit function theorem. Maxima, Minima and saddle points of two variables. Lagrange‘s method of multipliers.
Unit - IV
Curves: Tangents, Principal normals, Binormals, Serret-Frenet formulae. Locus of the centre of curvature, Spherical curvature, Locus of centre of Spherical curvature, Involutes, evolutes, Bertrand Curves. Surfaces: Tangent planes, one parameter family of surfaces, Envelopes.
PARTIAL DIFFERENTIAL EQUATIONS
Unit - I
Partial differential equations: Formation, order and degree, Linear and Non-Linear Partial differential equations of the first order: Complete solution, singular solution, General solution,
Solution of Lagrange‘s linear equations, Charpit‘s general method of solution. Compatible systems of first order equations, Jacobi‘s method.
Unit - II
Linear partial differential equations of second and higher orders, Linear and non-linearhomogeneous and Nonhomogeneous equations with constant coefficients, Partial differential equation with variable coefficients reducible to equations with constant coefficients, their complimentary functions and particular Integrals, Equations reducible to linear equations with constant coefficients.
Unit - III
Classification of linear partial differential equations of second order, Hyperbolic, parabolic and elliptic types, Reduction of second order linear partial differential equations to Canonical (Normal) forms and their solutions, Solution of linear hyperbolic equations, Monge‘s method for partial differential equations of second order.
Unit - IV
Cauchy‘s problem for second order partial differential equations, Characteristic equations and characteristic curves of second order partial differential equation, Method of separation of variables: Solution of Laplace‘s equation, Wave equation (one and two dimensions), Diffusion (Heat) equation (one and two dimension) in Cartesian Coordinate system.
Unit - I
Composition and resolution of forces. Parallel forces. Moments and Couples.
Unit - II Analytical conditions of equilibrium of coplanar forces. Friction. Centre of Gravity.
Unit - III Virtual work. Forces in three dimensions. Poinsots central axis.
Unit - IV Wrenches. Null lines and planes. Stable and unstable equilibrium.
Unit - I
One Parameter family of Surfaces: Envelope, Characteristics, edge of regression, Developable surfaces. Developables Associated with a Curve: Osculating developable, Polar developable, Rectifying developable.
Unit - II
Two- parameter Family of Surfaces: Envelope, Characteristics points, Curvilinear coordinates, First order magnitudes, Directions on a surface, The normal, Second order magnitudes, Derivatives of n.
Unit - III
Curves on a Surface: Principal directions and curvatures, First and second curvatures, Euler‘s theorems, Dupin‘s indicatrix, The surfaces z = f(x,y), Surface of revolution. Conjugate directions, Conjugate systems. Asymptotic lines, Curvature and torsion, Isometric parameters, Null lines, or minimal curves.
Unit - IV
Geodesics and Geodesic Parallels: Geodesics: Geodesic property, Equation of Geodesics, Surface of revolution, Torsion of Geodesic. Curves in Relation to Geodesics: Bonnet‘s theorem,
Joachimsthal‘s theorems, Vector curvature, Geodesic curvature, g , Other formulae for g ,
Bonnet‘s formula.
Unit - I
Generating Functions: Moment generating function and cumulant generating function along with their properties and uses. Tchebychev‘s inequality, Convergence in probability, Weak and strong laws of large numbers (Statements only).
Unit - II
Bernoulli, binomial, Poisson, geometric and hyper-geometric distributions with their properties.
Unit - III
Uniform, gamma, beta (first and second kinds) and exponential distributions with their properties.
Unit - IV
Normal distribution with its properties. Central Limit Theorem (Statement only) and its applications.
UNIT - I
Chemistry of d-Block Elements: Definition of transition elements, position in the periodic table, General characteristics & properties of d-block elements, Comparison of properties of 3d elements with 4d & 5d elements with reference only to ionic radii, oxidation state, magnetic and spectral properties.
Coordination Compounds: Werner's coordination theory, effective atomic number concept, chelates, nomenclature of coordination compounds, isomerism in coordination compounds, valence bond theory of transition metal complexes.
UNIT - II
Thermodynamics: Definition of thermodynamic terms: system, surrounding etc. Types of systems, intensive and extensive properties. St ate and path funct ions and their differentials. Thermodynamic process. Concept of heat and work. Zeroth Law of thermodynamics, First law of thermodynamics statement, definition of internal energy and enthalpy. Heat capacity, heat capacit ies at constant volume and pressure and their relationship. Calculation of w.q. dU & dH for the expansion of ideal gases under isothermal and adiabatic conditions for reversible process, Kirchoff`s equation. Second law of thermodynamics, need for the law, different statements of the law, Carnot ‘s cycles and its efficiency, Carnot‘s theorem, Thermodynamics scale of temperature. Concept of entropy – entropy as a stat e function, entropy as a function of P, V & T.
UNIT - III
Alcohols: Monohydric alcohols nomenclature, methods of formation by reduction of aldehydes, ketones, carboxylic acids and esters. Hydrogen bonding. Acidic nature. Dihydric alcohols — nomenclature, methods of formation, chemical react ions o f vicinal glycols,
Phenols: Nomenclature, structure and bonding. Preparation of phenols, Comparative acidic strengths of alcohols and phenols, resonance stabilization of phenoxide ion. Mechanisms of Fries rearrangement , Claisen rearrangement, and Schott en and Baumann react ions.
UNIT - IV
Ultraviolet (UV) absorption spectroscopy: Absorption laws (Beer-Lambert law), molar absorptivity, presentation and analysis of UV spectra, types of electronic transitions, effect of conjugation. Concept of chromophore and auxochrome. Bathochromic, hypsochromic, hyperchromic and hypochromic shift s.
DATABASE MANAGEMENT SYSTEM AND ORACLE
Unit - I
Basic Concepts: File systems vs DBMS, advantages and disadvantages of DBMS, objectives of a database. Database systems concepts and architecture. Data Modeling for a database: records and files, abstraction and data integration. Database Management System: Relational, Network, and Hierarchical. Relational Data Manipulations: Relational Algebra, Relational Calculus, SQL.
Unit - II
Relational Database Design: Functional dependencies, Finding keys; 1st to 3rd NFs, CNF, Lossess Join and Dependency preserving decomposition. Query Processing: General strategies for query processing, query optimization, query processor. Database security issues and recovery techniques.
Unit - III
Introduction to Oracle: Modules of Oracle, Invoking SQLPLUS, Data types, Data Constraints, Operators, Data manipulation: Create, Modify, Insert, Delete and Update; Searching, Matching and Oracle Functions. SQL*Forms: Form Construction, user-defined form, multiple-record form,Master-detail form. PL/SQL Blocks in SQL*Forms, PL/SQL syntax, Data types, PL/SQL functions, Error handling in PL/SQL, package functions, package procedures, Oracle transactions.
Unit - IV
SQL*ReportWriter: Selective dump report, Master-detail Report, Control-break Report, Test report. Database Triggers: Use & type of database Triggers, Database Triggers Vs SQL*Forms, Database Triggers Vs. Declarative Integrity Constraints, BEFORE vs AFTER Trigger Combinations, Creating a Trigger, Dropping a Trigger.
4TH SEMESTER / 2ND YEAR
Unit - 1
Boundedness of the set of real numbers; least upper bound, greatest lower bound of a set, neighborhoods, interior points, isolated points, limit points, open sets, closed set, interior of a set, closure of a set in real numbers and their properties. Bolzano-Weiestrass theorem, Open covers, Compact sets and Heine-Borel Theorem.
Unit – 1I
Sequence: Real Sequences and their convergence, Theorem on limits of sequence, Bounded and monotonic sequences, Cauchy‘s sequence, Cauchy general principle of convergence, Subsequences, Subsequential limits. Infinite series: Convergence and divergence of Infinite Series, Comparison Tests of positive terms Infinite series, Cauchy‘s general principle of Convergence of series, Convergence and divergence of geometric series, Hyper Harmonic series or p-series.
Unit – 1II
Infinite series: D-Alembert‘s ratio test, Raabe‘s test, Logarithmic test, de Morgan and Bertrand‘s test, Cauchy‘s Nth root test, Gauss Test, Cauchy‘s integral test, Cauchy‘s condensation test.
Unit – 1V
Alternating series, Leibnitz‘s test, absolute and conditional convergence, Arbitrary series: abel‘s lemma, Abel‘s test, Dirichlet‘s test, Insertion and removal of parenthesis, rearrangement of terms in a series, Dirichlet‘s theorem, Riemann‘s Re-arrangement theorem, Pringsheim‘s theorem
(statement only), Multiplication of series, Cauchy product of series, (definitions and examples only) Convergence and absolute convergence of infinite products.
SPECIAL FUNCTIONS AND INTEGRAL TRANSFORMS
Unit - I
Series solution of differential equations – Power series method, Definitions of Beta and Gamma functions. Bessel equation and its solution: Bessel functions and their properties- Convergence, recurrence, Relations and generating functions, Orthogonality of Bessel functions.
Unit - II
Legendre and Hermite differentials equations and their solutions: Legendre and Hermite functions and their properties-Recurrence Relations and generating functions. Orhogonality of Legendre and Hermite polynomials. Rodrigues‘Formula for Legendre & Hermite Polynomials, Laplace Integral Representation of Legendre polynomial.
Unit - III
Laplace Transforms – Existence theorem for Laplace transforms, Linearity of the Laplace transforms, Shifting theorems, Laplace transforms of derivatives and integrals, Differentiation and integration of Laplace transforms, Convolution theorem, Inverse Laplace transforms, convolution theorem, Inverse Laplace transforms of derivatives and integrals, solution of ordinary differential equations using Laplace transform.
Unit - IV
Fourier transforms: Linearity property, Shifting, Modulation, Convolution Theorem, Fourier Transform of Derivatives, Relations between Fourier transform and Laplace transform, Parseval‘s identity for Fourier transforms, solution of differential Equations using Fourier Transforms.
PROGRAMMING IN C AND NUMERICAL METHODS
Unit - I
Programmer‘s model of a computer, Algorithms, Flow charts, Data types, Operators and expressions, Input / outputs functions.
Unit - II
Decisions control structure: Decision statements, Logical and conditional statements, Implementation of Loops, Switch Statement & Case control structures. Functions, Preprocessors and Arrays.
Unit - III
Strings: Character Data Type, Standard String handling Functions, Arithmetic Operations on Characters. Structures: Definition, using Structures, use of Structures in Arrays and Arrays in Structures. Pointers: Pointers Data type, Pointers and Arrays, Pointers and Functions. Solution of Algebraic and Transcendental equations: Bisection method, Regula-Falsi method, Secant method,Newton-Raphson‘s method. Newton‘s iterative method for finding pth root of a number, Order of convergence of above methods.
Unit - IV
Simultaneous linear algebraic equations: Gauss-elimination method, Gauss-Jordan method,
Triangularization method (LU decomposition method). Crout‘s method, Cholesky Decomposition method. Iterative method, Jacobi‘s method, Gauss-Seidal‘s method, Relaxation method.
Unit - I Pressure equation. Condition of equilibrium. Lines of force. Homogeneous and heterogeneous fluids. Elastic fluids. Surface of equal pressure. Fluid at rest under action of gravity. Rotating fluids. Unit - II Fluid pressure on plane surfaces. Centre of pressure. Resultant pressure on curved surfaces. Equilibrium of floating bodies. Curves of buoyancy. Surface of buoyancy. Unit - III Stability of equilibrium of floating bodies. Metacentre. Work done in producing a displacement. Vessels containing liquid. Unit - IV Gas laws. Mixture of gases. Internal energy. Adiabatic expansion. Work done in compressing a gas. Isothermal atmosphere. Connective equilibrium
Unit - I
Parameter and statistic, sampling distribution and standard error of estimate. Point and interval estimation, Unbiasedness, Efficiency, Consistency and Sufficiency.
Unit - II
Method of maximum likelihood estimation. Null and alternative hypotheses, Simple and composite hypotheses, Critical region, Level of significance, One tailed and two tailed tests, Types of errors, Neyman- Pearson Lemma.
Unit - III
Testing and interval estimation of a single mean, single proportion, difference between two means and two proportions. Fisher‘s Z transformation.
Unit - IV
Definition of Chi-square statistic, Chi-square tests for goodness of fit and independence of attributes. Definition of Student‘s ‗t‘ and Snedcor‘s F-statistics. Testing for the mean and variance of univariate normal distributions, Testing of equality of two means and two variances of two univariate normal distributions. Related confidence intervals. Analysis of variance(ANOVA) forone-way and two-way classified data.
- See more at: http://www.sgtuniversity.ac.in/faculty-of-physical-sciences/pages/elementary-inference#sthash.IUrc4WLH.dpuf
Unit - I
Data structure and its essence, Data structure types. Linear and list structures: Arrays, stacks, queues and lists; Sequential and linked structures; Simple lists, circular lists, doubly linked lists. Inverted lists, threaded lists, Operations on all these structures and applications.
Unit - II
Arrays, Multidimensional arrays, sequential allocation, address calculations, sparse arrays. Tree structures: Trees, binary trees and binary search trees. Implementing binary trees, Tree traversal algorithms, threaded trees, trees in search algorithms, AVL Trees.
Unit - III
Graph data structure and their applications. Graph traversals, shortest paths, spanning trees and related algorithms. Family of B-Trees: B-tree, B*-Trees, B+ Trees.
Unit - IV
Sorting: Internal and External sorting. Various sorting algorithms, Time and Space complexity of algorithms. Searching techniques and Merging algorithms. Applications of sorting and searching in computer science.
5TH SEMESTER / 3RD YEAR
Unit – I
Riemann integral, Integrabililty of continuous and monotonic functions, The Fundamental theorem of integral calculus. Mean value theorems of integral calculus.
Unit – II
Improper integrals and their convergence, Comparison tests, Abel‘s and Dirichlet‘s tests, Frullani‘s integral, Integral as a function of a parameter. Continuity, Differentiability and integrability of an integral of a function of a parameter.
Unit – III
Definition and examples of metric spaces, neighborhoods, limit points, interior points, open and closed sets, closure and interior, boundary points, subspace of a metric space, equivalent metrics, Cauchy sequences, completeness, Cantor‘s intersection theorem, Baire‘s category theorem, contraction Principle
Unit – IV
Continuous functions, uniform continuity, compactness for metric spaces, sequential compactness, Bolzano-Weierstrass property, total boundedness, finite intersection property, continuity in relation with compactness, connectedness , components, continuity in relation with connectedness.
Unit – I
Definition of a group with example and simple properties of groups, Subgroups and Subgroup criteria, Generation of groups, cyclic groups, Cosets, Left and right cosets, Index of a sub-group
Coset decomposition, Largrage‘s theorem and its consequences, Normal subgroups, Quotient groups,
Unit – II
Homoomorphisms, isomophisms, automorphisms and inner automorphisms of a group. Automorphisms of cyclic groups, Permutations groups. Even and odd permutations. Alternating groups, Cayley‘s theorem, Center of a group and derived group of a group.
Unit – III
Introduction to rings, subrings, integral domains and fields, Characteristics of a ring. Ring homomorphisms, ideals (principle, prime and Maximal) and Quotient rings, Field of quotients of an integral domain.
Unit – IV
Euclidean rings, Polynomial rings, Polynomials over the rational field, The Eisenstein‘s criterion, Polynomial rings over commutative rings, Unique factorization domain, R unique factorization domain implies so is R[X1 , X2,……Xn]
- See more at: http://www.sgtuniversity.ac.in/faculty-of-physical-sciences/pages/groups-and-rings#sthash.4QKBqj0s.dpuf
Unit – I
Finite Differences operators and their relations. Finding the missing terms and effect of error in a difference tabular values, Interpolation with equal intervals: Newton‘s forward and Newton‘s backward interpolation formulae. Interpolation with unequal intervals: Newton‘s divided difference, Lagrange‘s Interpolation formulae, Hermite Formula.
Unit – II
Central Differences: Gauss forward and Gauss‘s backward interpolation formulae, Sterling, Bessel Formula. Probability distribution of random variables, Binomial distribution, Poisson‘s distribution, Normal distribution: Mean, Variance and Fitting.
Unit – III
Numerical Differentiation: Derivative of a function using interpolation formulae as studied in Sections –I & II. Eigen Value Problems: Power method, Jacobi‘s method, Given‘s method, House- Holder‘s method, QR method, Lanczos method.
Unit – IV
Numerical Integration: Newton-Cote‘s Quadrature formula, Trapezoidal rule, Simpson‘s one- third and three-eighth rule, Chebychev formula, Gauss Quadrature formula. Numerical solution of ordinary differential equations: Single step methods- Picard‘s method. Taylor‘s series method, Euler‘s method, Runge-Kutta Methods. Multiple step methods; Predictor-correctormethod, Modified Euler‘s method, Milne-Simpson‘s method.
Unit - I
Linear integral equations, Some basic identities, Initial-value problems reduced to Volterra integral equations, Method of successive approximation to solve Volterra integral equations of second kind, Iterated kernels and Neumann series for Volterra equation. Resolvent kernel as a series in , Laplace transform method for a difference kernel, Solution of a Volterra integral equation of the first kind.
Unit - II
Boundary value problems reduced to Fredholm integral equations, method of successive approximations to solve Fredholm equation of second kind, Iterated kernels and Neumann series for Fredholm equations, Resolvent kernel as a sum of series, Fredholm resolvent kernel as a ratio of two series. Fredholm equations with degenerate kernel, approximation of a kernel by a degenerate kernel, Fredholm Alternative.
Unit - III
Green‘s function. Use of method of variation of parameters to construction the Green‘s function for a nonhomogeneous linear second degree BVP, Basic four properties of the Green‘s function, Alternate procedure for construction of the Green‘s function by using its basic four properties. Method of series representation of the Green‘s function in terms of the solutions of the associated homogeneous BVP. Reduction of a BVP to a Fredholm integral equation with kernel as Green‘s function.
Unit - IV
Homogeneous Fredholm equations with symmetric kernels, Solution of Fredholm equations of the second kind with symmetric kernel, Method of Fredholm Resolvent Kernel, Method of Iterated Kernels, Fredholm Equations of the First Kind with Symmetric Kernels.
METHODS OF APPLIED MATHEMATICS
Unit - I
Solution of 3D Laplace, wave and heat equations in spherical polar co-ordinates and cylindrical polar co-ordinates by the method of separation of variables. Fourier series solution of the wave equation, transformation of boundary value problems.
Unit - II
Fourier series solution of the heat equation, steady-state temperature in plates, The heat and wave equations in unbounded domains, Fourier transform solution of boundary value problems. The heat equation in an infinite cylinder and in a solid sphere.
Unit - III
Hankel transform of elementary functions. Operational properties of the Hankel transform. Applications of Hankel transforms to PDE. Definition and basic properties of finite Fourier sine and cosine transforms, its applications to the solutions of BVP‘s and IVP‘s.
Unit - IV
Moments and products of inertia, Angular momentum of a rigid body, principal axes and principal moment of inertia of a rigid body, kinetic energy of a rigid body rotating about a fixed point, Momental ellipsoid and equimomental systems, coplanar mass distributions, general motion of a rigid body.
Unit - I
Definition, scope, methodology and applications of OR. Types of OR models. Concept of optimization, Linear Programming: Introduction, Formulation of a Linear Programming Problem (LPP), Requirements for an LPP, Advantages and limitations of LP. Graphical solution: Multiple, unbounded and infeasible solutions.
Unit - II
Principle of simplex method: standard form, basic solution, basic feasible solution. Computational Aspect of Simplex Method: Cases of unique feasible solution, no feasible solution, multiple solution and unbounded solution and degeneracy. Two Phase and Big- M methods.
Unit - III
Duality in LPP, primal-dual relationship. Transportation Problem: Methods for finding basic feasible solution of a transportation problem, Modified distribution method for finding the optimum solution, Unbalanced and degenerate transportation problems, transshipment problem, maximization in a transportation problem.
Unit - IV
Assignment Problem: Solution by Hungarian method, Unbalanced assignment problem, maximization in an assignment problem, Crew assignment and Travelling salesman problem. Game Theory: Two person zero sum game, Game with saddle points, the rule of dominance; Algebraic, graphical and linear programming methods for solving mixed strategy games.
6TH SEMESTER / 3RD YEAR
REAL AND COMPLEX ANALYSIS
Unit – I
Jacobians, Beta and Gama functions, Double and Triple integrals, Dirichlets integrals, change of order of integration in double integrals.
Unit – II
Fourier‘s series: Fourier expansion of piecewise monotonic functions, Properties of Fourier Co- efficients, Dirichlet‘s conditions, Parseval‘s identity for Fourier series, Fourier series for even and odd functions, Half range series, Change of Intervals.
Unit – III
Extended Complex Plane, Stereographic projection of complex numbers, continuity and differentiability of complex functions, Analytic functions, Cauchy-Riemann equations. Harmonic functions.
Unit – IV
Mappings by elementary functions: Translation, rotation, Magnification and Inversion. Conformal Mappings, Mobius transformations. Fixed pints, Cross ratio, Inverse Points and critical mappings.
Unit – I
Vector spaces, subspaces, Sum and Direct sum of subspaces, Linear span, Linearly Independent and dependent subsets of a vector space. Finitely generated vector space, Existence theorem for basis of a finitely generated vactor space, Finite dimensional vector spaces, Invariance of the number of elements of bases sets, Dimensions, Quotient space and its dimension.
unit – II
Homomorphism and isomorphism of vector spaces, Linear transformations and linear forms on vactor spaces, Vactor space of all the linear transformations Dual Spaces, Bidual spaces, annihilator of subspaces of finite dimentional vactor spaces, Null Space, Range space of a linear transformation, Rank and Nullity Theorem,
Unit – III
Algebra of Liner Transformation, Minimal Polynomial of a linear transformation, Singular andnon-singular linear transformations, Matrix of a linear Transformation, Change of basis, Eigen values and Eigen vectors of linear transformations.
Unit – IV
Inner product spaces, Cauchy-Schwarz inequality, Orthogonal vectors, Orthogonal complements,
Orthogonal sets and Basis, Bessel‘s inequality for finite dimensional vector spaces, Gram-Schmidt,Orthogonalization process, Adjoint of a linear transformation and its properties, Unitary linear transformations.
Unit – I
Velocity and acceleration along radial, transverse, tangential and normal directions. Relative velocity and acceleration. Simple harmonic motion. Elastic strings.
Unit – II
Mass, Momentum and Force. Newton‘s laws of motion. Work, Power and Energy. Definitions of Conservative forces and Impulsive forces.
Unit – III
Motion on smooth and rough plane curves. Projectile motion of a particle in a plane. Vector angular velocity.
Unit – IV
General motion of a rigid body. Central Orbits, Kepler laws of motion. Motion of a particle in three dimensions. Acceleration in terms of different co-ordinate systems.
Unit - I
Statements only of (Axiom of choice, Zorn‘s lemma, Well ordering theorem and Continnum hypothesis). Definition and examples of topological spaces, Neighbourhoods, Interior point and interior of a set , Closed set as a complement of an open set , Adherent point and limit point of a set, Closure of a set, Derived set, Properties of Closure operator, Boundary of a set , Dense subsets, Interior, Exterior and boundary operators. Base and subbase for a topology, Neighbourhood system of a point and its properties, Base for Neighbourhood system. Relative(Induced) topology, Alternative methods of defining a topology in terms of neighbourhood system and Kuratowski closure operator. Comparison of topologies on a set, Intersection and union of topologies on a set.
Unit - II
Continuous functions, Open and closed functions, Homeomorphism. Connectedness and its characterization, Connected subsets and their properties, Continuity and connectedness, Components, Locally connected spaces,
Unit - III
Compact spaces and subsets, Compactness in terms of finite intersection property,Continuity and compact sets, Basic properties of compactness, Closedness of compactsubset and a continuous map from a compact space into a Hausdorff and its consequence. Sequentially and countably compact sets, Local compactness and one point compatification.
Unit - IV
First countable, second countable and separable spaces, hereditary and topological property, Countability of a collection of disjoint open sets in separable and second countable spaces, Lindelof theorem. T0, T1, T2 (Hausdorff) separation axioms, their characterization and basic properties.
Unit - I
Kinematics - Eulerian and Lagrangian methods. Stream lines, path lines and streak lines. Velocity potential. Irrotational and rotational motions. Vortex lines. Equation of continuity. Boundary surfaces.
Unit - II
Acceleration at a point of a fluid. Components of acceleration in cylindrical and spherical polar co- ordiantes, Pressure at a point of a moving fluid. Euler‘s and Lagrange‘s equations of motion.
Bernoulli‘s equation. Impulsive motion. Stream function.
Unit - III
Acyclic and cyclic irrotation motions. Kinetic energy of irrotational flow. Kelvin‘s minimum energy theorem. Axially symmetric flows. Liquid streaming past a fixed sphere. Motion of a sphere through a liquid at rest at infinity. Equation of motion of a sphere. Three-dimensional sources, sinks, doublets and their images. Stoke‘s stream function.
Unit - IV
Irrotational motion in two-dimensions. Complex velocity potential. Milne-Thomson circle theorem.Two-dimensional sources, sinks, doublets and their images. Blasius theorem. Two- dimensional irrotation motion produced by motion of circular and coaxial cylinders in an infinite mass of liquid.
Unit - I
Inventory Control: introduction of inventory, factors affecting inventory, Inventory models, Deterministic models: Economic order quantity model when shortages are allowed/not allowed, price discounts model, multi-item inventory models.
Unit - II
Queuing Theory : Basic characteristics of queuing system, Birth-death equations, Steady state solution of Markovian queuing models with single and multiple servers (M/M/1 and M/M/c), with limited capacity (M/M/1/K and M/M/c/K).
Unit - III
Sequencing problems: Processing of n jobs through 2 machines, n jobs through 3 machines, 2 jobs through m machines, n jobs through m machines. Replacement problems: Replacement of items whose running cost increases with time, Replacement policies for the items that fail completely - Individual and the group replacement policies.
Unit - IV
PERT and CPM: Introduction of PERT and CPM, Earliest and latest times, Determination of critical path and various types of floats, Probablistic and cost considerations in project scheduling
MATHEMATICS BSc.
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