Syllabus of Mathematics BTech

8/2/2019 Syllabus of Mathematics BTech

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B. Tech Ist Semester Syllabus

Subject Mathematics-I Code: - M-101

First Half

I. Functions of Single Variable: (2 Lectures)

1. Successive Differentiation

1.1 Introduction

1.2. nth

derivative of some common functions1.3. Leibnitzs theorem and its application.

2. Expansion of function: (5 Lectures)

2.1 Introduction

2.2. Rolles Theorem and its geometrical interpretation

2.3 Lagranges Mean Value Theorem (MVT) and its geometrical interpretation

2.4. Taylors and Maclaurins theorem in finite form with Lagranges form and

Cauchys form of remainders

2.5. Taylors and Maclaurins theorem extended up to infinity

2.6. Cauchys MVT

2.7. Applications.

3. Indeterminate form: (2 Lectures)

3.1. Introduction

3.2 LHospitals rule (no proof) and its applications.

4. Radius of Curvature: (4 Lectures)

4.1 Introduction

4.2 Radius of curvature of a curve in different coordinate system (Cartesian,

explicit and implicit forms, parametric form, polar form, tangential polarform, pedal form)

4.3. Applications

4.4. Radius of curvature at the origin and its applications.

5. Asymptotes: (3 Lectures)

5.1. Introduction

5.2. Derivation of vertical, horizontal and inclined asymptotes for algebraic

curves

5.3. Asymptotes of polar curves

5.4. Applications.

6. Concavity, convexity and point of inflexion (2 Lectures)

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6.1. Introduction

6.2. Necessary and sufficient condition for convexity and concavity

6.3. Definition of point of inflexion

6.4. Necessary and sufficient condition for existence of point of inflection

6.5. Applications.

II. Functions of several variables:

7. Functions, Limit and Continuity (2 Lectures)

7.1. Introduction

7.2. Function of two variables, definitions of limit and continuity,

7.3. Applications.

8. Partial Differentiation (4 Lectures)

8.1. Introduction8.2. Definition of first and higher order partial derivatives

8.3. Homogeneous function, Eulers theorem and its converse

8.4. Partial derivatives of implicit function

8.5. Total differential coefficient and differentials

8.6. Exact differential

8.7. Partial derivative of a function of two functions

8.8. Applications.

9. Jacobian (2 Lectures)

Introduction,

Properties and applications.

10. Taylors and Maclaurins series for several variables (3 Lectures)

10.1. Statements and proofs,

10.2. Applications.

11. Maxima and Minima (3 Lectures)

11.1 Introduction11.2. Definition

11.3. Necessary and sufficient condition for maxima and minima (no proof)

11.4. Stationary points

11.5. Lagranges method of multipliers

11.6. Applications.

* Class test and revision (2 Lectures)

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Second Half

I. Two Dimensional coordinate geometry:

1. General Equation of the Second Degree (2 Lectures)

1.1. Introduction

1.2. Classification and nature of conics

1.3. Reduction of the general equation of second degree into normal form

1.4. Applications.

II. Three Dimensional Coordinate Geometry (2 Lectures)

2.1 Recapitulation of introduction to three- Dimensional Coordinate Geometry

2.2 Recapitulation of the plane

2.3 Recapitulation of the straight line

2.5 Recapitulation of the sphere.

3. The Cone (2 Lectures)

3.1. Introduction

3.2. Cone with its vertex at the origin3.3. General equation of a cone containing the axes

3.4. Equation of the cone with the origin as vertex and a given curve as base

3.5. Equation of a right circular cone

3.6. Condition for perpendicular generators

3.7. Intersection of cone by a plane through the vertex

3.8. Applications.

4. The Cylinder (1 Lecture)

4.1. Introduction

4.2. Equation of a cylinder

4.3. Equation of a right circular cylinder

4.4. Applications.

5. The Conicoids: (1 Lecture)

5.1. Intersection

5.2. Definitions, equation (with figures) and brief discussions of ellipsoid,

hyperboloid and paraboloid

5.3 Applications.

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III. Vector Calculus:

6. Basics of vector Calculus (4 Lectures)

6.1. Introduction,

6.2. Differentiation of vectors,

6.3. Scalar and vector point functions,

6.4. Vector operator Del,

6.5. Del applied to scalar point function-Gradient

6.6. Del applied to vector point functions-divergence and curl,

6.7. Physical interpretation of gradient, divergence and curl,

6.8. Del applied twice to point functions,

6.9. Del applied to products of point functions,

6.10. Applications.

IV. Determinants: (2 Lectures)

7.1. Recapitulation,

7.2. Jacobis Theorem,

7.3. Symmetric and skew Symmetric determinant,

7.4. Cramers rule,

7.5. Applications.


Books: (Text/References)

1. E. Kreyszig, Advanced Engineering Mathematics, 8th edition, John Wiley.

2. B. S. Grewal : Higher Engineering Mathematics,

3. Das and Mukherjee : Differential Calculus, U. N. Dhar

4. S.C.Malik and S. Arora : Mathematical Analysis

5. Maity and Ghosh : Vector Analysis6. Ghosh and Chakraborty : Analytical Geometry

7. Piskunov : Differential and Integral Calculus Vol.-I and Vol. II.

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B. Tech 2nd Semester Syllabus

Subject: Mathematics-II Code: - M-201

First Half

I. Integral Calculus

1. Recapitulation of Definite Integrals (1 Lecture)

1.1 Introduction1.2 Recapitulation of Integration as the limit of a sum

1.3 Recapitulation of Geometrical meaning of definite integral and its properties

1.4 Mean value theorem and fundamental theorem of integral calculus

1.5 Applications.

2. Reduction Formulae (2 Lectures)

2.1. Introduction

2.2. Reduction formulae involving two parameters

2.4 Applications.

3. Improper Integrals (2 Lectures)

3.1. Introduction

3.2. Definition and classification of improper integrals

3.3. Beta and gamma functions and their properties

3.4. Applications.

4. Application of definite integral to find the areas, lengths of plane curves, volumes and

surface areas of solids of revolution (4 Lectures)

4.1 Introduction

4.2 Areas in Cartesian and polar co-ordinates

4.3 Lengths determined from Cartesian polar and pedal equations

4.4 Intrinsic equation derived from Cartesian, polar and pedal equations

4.5 Solids of revolution in Cartesian and polar forms

4.8. Applications.

II. Ordinary Differential Equations:

5. Ordinary Differential Equations of first order (2 Lectures)

5.1. Recapitulation of equations of first order and first degree

5.2. Differential equations of first order but higher degree

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5.3. Clairauts equation

5.4. Applications.

6. Linear Differential Equations of higher order (5 Lectures)

6.1 Introduction

6.2 Equations with constant coefficients

6.3 Equations of second order-right hand member zero

6.4 Equations of second order-right hand member a function of x

6.5 Methods of finding particular integrals

6.6 Equation of nth order

6.7 Homogeneous linear equations,

6.9 Equation reducible to homogeneous linear form,

6.10 Applications.

III. Laplace transform (4 Lectures)

7.1. Introduction

7.2 Definition

7.3 Laplace transforms of some elementary functions

7.4 Properties of Laplace transform

7.5 Transforms of derivatives

7.6 Transforms of integrals

7.7 Multiplication by tn

7.8 Division by t

7.9 Applications

7.10 Inverse Laplace transforms and its properties

7.11 Methods of finding inverse transforms7.12 Convolution theorem

7.13 Applications to deferential equations.

IV. Differentiation under the sign of integration (1 Lectures)

8.1. Introduction

8.2. Leibnitzs rule

8.3. Applications.


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Second Half

1. Matrix Theory (8 Lectures)

1.1. Recapitulation of Matrix operations, adjoint and inverse

1.2. Symmetric and skew-symmetric matrices-their properties

1.3. Rank of a matrix

1.4. Elementary transformations

1.5. Elementary matrices, equivalent matrices and Echelon matrices

1.6. Normal form of a matrix

1.7. Rank by elementary transformation of echelon matrix1.8. Matrix inversion by elementary transformation

1.9. System of Linear equations

1.10. Consistent and inconsistent system of equations

1.11. Non-homogeneous and homogeneous system of equation

1.12. Solution of system of Equations by Matrix method

1.13. Applications.

2. Characteristic Equations and Quadratic Forms (5 Lectures)

2.1 Matrix polynomial

2.2. Characteristic equation and characteristic polynomial

2.3. Cayley-Hamilton Theorem

2.4. Eigen values and Eigen vectors

2.5 Diagonalisation of matrices

2.6 Orthogonal Diagonalisation

2.7 Quadratic form

2.8 Some definitions

2.9 Reduction of a quadratic form into its normal form

2.10 Reduction by orthogonal transformation

2.11 Value classes of quadratic form

2.12 Applications.

3. Vector Spaces (4 Lectures)

3.1. Introduction

3.2. Definition and examples of vector space

3.3. Properties of vector space

3.4. Vector sub-spaces

3.5. Linear dependence and independence of vectors

3.6. Theorems on linear dependence

3.7. Linear span3.8. Basis and dimension of a vector space

3.9. Theorems on basis and dimension

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3.10. Rank definition as number of linearly independent vectors

3.11. Applications.

4. Infinite Series (6 Lectures)

4.1. Introduction4.2. Sequences

4.3. Series, its definition and convergence

4.4. General properties of series

4.5. Series of positive terms

4.6. Necessary condition for convergence

4.7. Various tests for convergence-comparison test, DAlemberts ratio test, Cauchys

root test and Integral test

4.8. Alternating series-Leibnitzs rule

4.9. Absolute and conditional Convergence

4.10. Applications.




2. E. Kreyszig, Advanced Engineering Mathematics, 8th edition, John Wiley.3. S. K. Mapa : Higher Algebra, Asoka Prakasan

4. Maity and Ghosh : Differential Equations

5. Piaggio : Differential Equations

8. Das and Mukherjee : Integral Calculus

9. G. F. Simpsons, Ordinary Differential Equation, McGraw-Hill, 1972.

10. N. D. Raisinghania : Integral transforms [including Boundary Value Problems]

S.Chand [1988]

11. S. L. Ross : Differential Equations, John Willey and Sons

12. I.N. Sneddon,The use of Integral Transforms, McGraw-Hill, 1974.

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B. Tech 3rd Semester Syllabus

Subject: Mathematics-III Code: - M-301

First Half

I. Differential Equations:

1. Linear Differential Equations of the 2nd Order (4 Lectures)

Introduction

Complete solution of the 2nd order D.E. in terms of a known integral belonging to the

complimentary function

Rules for finding the Integral solution belonging to C.F.

Complete solution of the 2nd order D.E. by the method of variation of parameters

Linear dependence of solutions and related theorems on the Wronskian determinant

Applications.

2. Simultaneous Linear Differential Equations with Constant coefficients (4 Lectures)

Introduction

Method of Elimination for solving simultaneous Differential Equations

Simullincous Equation of the typeRdz

Qdy

Pdx == where P, Q and R are functions of x, y and z

and their methods of solution

Geometrical Interpretation of the equationR

dz

Q

dy

P

dx==

Applications.

II. Partial Differential Equations ( 4 Lectures)

3. Partial Differential Equations of the 1st order (3 Lectures)

Introduction

Formation of partial differential Equations

Lagranges Equation- its formation and Method of solution

Applications.

4. Homogeneous Linear partial Differential.Equations with Constant

Coefficients of the from F(D, D) z = f(x, y) (4 Lectures)

Introduction

Method of Finding the C.F.(Complimentary Function)

Method of Finding the P.I.(Particular Integral)

Applications.

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5. Reduction to the canonical form of a PDE (4 Lectures)

5.1. Characteristic differential equationand characteristic curves

5.2. Canonical Forms-Elliptic, Hyperbolic and Parabolic type of equations.

5.3. Applications.

6. Fourier series (4 Lectures)

6.1 Introduction

6.2 Periodic function

6.3 Fourier series and Eulers formulae

6.4 Dirichlets condition

6.5 Fourier series for discontinuous functions

6.6 Even and odd function-expansion of even odd periodic functions

6.7 Change of interval

6.8 Half-range series

6.9 Expansion of a non-periodic function in Fourier series

6.10 Fourier series in complex form

6.11 Applications.


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2nd Half

I. Multiple integrals and its applications

1. Multiple Integrals (5 Lectures)

1.1 Double Integrals

1.2 Evaluation of Double Integrals

1.3 Evaluation of Triple Integrals

1.4 Change of order of Integration

1.5 Change of variables

1.6 Area by double integration

1.7 Volume as Double Integrals

1.8 Volume as a triple Integral

1.9 Area of a curved surface

1.10 Applications.

II. Vector Calculus

2. Vector Integration (4 Lectures)

2.1 Introduction

2.2 Line Integrals

2.3 Circulation

2.4 Work done by a force

2.5 Surface Integral2.6 Greens theorem in the plane

2.7 Stokes theorem

2.8 Volume Integrals

2.9 Gauss Divergence theorem

2.10 Applications.

III. Complex Analysis

3. Functions of a complex variable (3 Lectures)

3.1 Introduction3.2 Functions

3.3 Limit

3.4 Continuity

3.5 Derivative of f(z)

3.6 Analytic Function

3.7 Necessary and sufficient conditions for f(z) to be analytic

3.8 Cauchy Riemann Equations in polar co-ordinates

3.9 Harmonic functions

3.10 Orthogonal systems

3.11 Application of Analytic functions to flow problems

3.12 Applications.

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4. Transformation or Mapping (2 Lectures)

4.1 Introduction

4.2 Conformal Transformation

4.3 Theorem4.4 Some standard Transformations-Translation, Rotation and Magnification,

Inversion, Bilinear Transformation

4.5 Applications.

5. Complex Integration (4 Lectures)

Introduction

Simply and Multiply connected regions

Cauchys Integral Theorem

Cauchys Integral Formula

Applications.

6. Series of Complex Terms (2 Lectures)

Introduction

Taylors series

Laurents series (only Statement)

Applications.

7. Singular points and Residues (2 Lectures)

7.1 Definitions7.2 Calculation of Residues

7.3 Residue Theorem

7.4 Applications.



1. B. S. Grewal : Higher Engineering Mathematics,2. E. Kreyszig, Advanced Engineering Mathematics, 8th edition, John Wiley.3. Piaggio : Differential Equations



6. I.N. Sneddon,The use of Integral Transforms, McGraw-Hill, 1974.7. I.N. Sneddon, Elements of Partial Differential Equations, McGraw-Hill, 1988.

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B. Tech 3rd Semester Syllabus(for CSE and IT)

Subject: Discrete Mathematics Code: - M-302

1st Half

I. Number System

1. Number System (2 Lectures)

1.1 Introduction

1.2 Natural Numbers

1.3 Integers

1.4 Radix r Representation of integers

1.5 Rational Numbers

1.6 Real and complex Numbers

1.7 Binary, Decimal and Hexadecimal representation of numbers and their conversion

1.8. Floating point Notation.

II. Abstract algebra

2. Group Theory (7 Lectures)

2.1 Introduction

2.2 Binary operation

2.3 Groupoid, Semi group2.4 Group

2.5 Elementary theorems on group

2.6 Subgroup

2.7 Finite group and its properties

2.8 Order of an element of a group and its properties

2.9 Alternating subgroup

2.10 Cyclic group

2.11 Permutation group and its properties

2.12 Cosets, Lagranges Theorem on finite group and Normal subgroups

2.13 Homomorphism and isomorphism of groups

2.14 Applications.

3. Rings and fields (5 Lectures)

3.1 Introduction

3.2 Ring

3.3 Elementary properties of ring

3.4 Ring with zero divisors

3.5 Integral domain

3.6 Skew field

3.7 Subring and Subfield

3.8 Applications.

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4. Elements of coding Theory (3 Lectures)

Introduction

Group codes

Parity check matrix and generator matrix

Hamming code.

5. Combinatories (6 Lectures)

5.1 Introduction

5.2 Sum rule and Product rule

5.3 Inclusion and exclusion principle

5.4 Permutation and combination

5.5 Pigeonhole principle

5.6 Binomial theorem

5.5 Multinomial coefficients

5.6 Recurrence relation5.7 Generating function.

GRAPH THEORY

1. Introduction to graph theory (4 Lectures)

1.1 What is a graph? Definition,

1.2 Basic terminologies

1.3 Directed and undirected graph

1.4 Types of graphs

1.5 Isomorphism and Subgraphs

1.6 Operations on graphs

1.7 Application

2. Walks, paths and Circuits (4 Lectures)

2.1 Introduction

2.2 Connected and disconnected graphs

2.3 Component

2.4 Walk, path and circuit

2.5 Euler path and Euler circuit

2.6 Hamiltonian path and Hamiltonian circuit

2.7 Matrix representation of directed and undirected graph

2.8 Applications.

3. Trees and Fundamental circuit (4 Lectures)

3.1 Introduction

3.2 Some properties of trees

3.3 Distance in graphs and trees

3.4 Spanning tree, chord,

3.5 BFS algorithm

3.6 Weighted graph3.7 Labeled graph

3.8 Minimal spanning tree

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3.9 Prims and Kruskals algorithm

3.10 Fundamental circuits

3.11 Application.

III. Cut-set and Cut-vertices (3 Lectures)

4.1 Introduction,4.2 Definition of cut-set and its properties,

4.3 Rank and nullity,

4.4 Fundamental circuits and fundamental cut-sets,

4.5 Connectivity and separability,

4.6 Definition of cut-vertices,

4.7 Cut-edge and bridge

4.8 Net-work flows,

4.9 Applications.

IV. Planer Graphs (4 Lectures)

5.1 Introduction,

5.2 Definition of planar and non-planar graphs,

5.3 Kuratowskis two graphs,

5.4 Homeomorphic graphs,

5.5 Geometric dual and Combinatorial dual,

5.6 Applications.

VI. Colouring, Matching, Covering and Partitioning (4 Lectures)

6.1 Introduction,

6.2 Chromatic number,

6.3 Bipartite graph,6.4 Chromatic partitioning,

6.5 Chromatic polynomial,

6.6 Matching,

6.7 Covering,

6.8 Four-colour problem and five colour theorem,

6.9 Applications.



1. Kolman and R.C. Busby, Discrete Mathematical Structures for Computer Science,

PHI, New Delhi, 1994.

2. C. L. Liu, Elements of Discrete Mathematics, McGraw Hill, 2/e, Singapore, 1985.

3. Narsingh Deo, Graph Theory with Applications to Engineering and Computer

Science, Prentice-Hall of India, 1974.

4. S. K. Mapa : Higher Algebra, Asoka Prakasan

5. Kneth Rosen : Discrete Mathematics, PHI

6. Ghosh and Chakraborty : Higher Algebra

7. John Fraleigh: Abstract Algebra

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B. Tech 4th Semester Syllabus(Only for CSE and IT)

Subject: Probability, statistics and Stochastic Processes

Subject Code: - M-403

First half: Probability

1. Introduction and Background (1 Lecture)

1.1 Preconception idea of set related to Probability1.2 Random Experiment

1.3 Sample space

1.4 Events [Simple event, compound event, equally likely event, favorable event,

Exhaustive events, complimentary events, Independent and Dependent events.]

2. The meaning of probability (2 Lectures)

2.1 Definition [Classical, Frequency, Axiomatic odds in favor of an event and odds

against an event]

2.2 General addition rule of probability for two events, the events and extension2.3 Conditional probability and Multiplication Rules

2.4 Stochastic independencies of the events and related theorems

2.5 Bayes theorem with proof

2.6 Applications related to Industry, Engineering and Management.

3. The concept of random variable (4 Lectures)

3.1 Random variable

3.2 Discrete and continuous random variables

3.3 Joint probability distribution

3.4 Joint distribution table and marginal distribution of random variable

3.5 Expectation of random variable and its concept in management and industry

3.6 Properties of Exportation

3.7 Variance and covariance of random variables

3.8 Applications.

4. Probability distributions of random variables (4 Lectures)

4.1 Discrete and continuous probability distribution

4.2 Probability mass function

4.3 Cumulative distribution function for discrete random variables and itsgraphical representation

4.4 Probability density function

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4.5 Cumulative probability distribution function for continuous random variable

and their properties and graphical representation

4.6 Relation between at least and at most values of the Discreet random valuable

4.7 Consultation of at most table for Discrete Distribution.

5. Discrete probability Distributions (4 Lectures)

5.1 Bernouli trial,

5.2 Binomial distribution

5.3 Poisson distribution,

5.4 Derivation of mean, variance and standard deviation of the distribution,

5.5 Applications.

6. Continuous probability Distribution (4 Lectures)

6.1 Deviation of mean, variance and standard deviation and their applications

6.2 Properties of normal distribution6.3 Standard normal variate and standard normal distribution

6.4 Graphical representation,

6.5 Computation of mean median and mode of Normal Distribution

6.6 Applications


2nd Half : Statistics

1. Introduction and Background (1 Lectures)

1.1 Introduction

1.2 Data [Primary and Secondary data, Collection of data]

1.3 Representation of data-grouped and ungrouped data

1.4 Frequency distribution of data

1.5 Applications.

2. Measures of Central tendency and Measures of Dispersion (3 Lectures)

2.1 Introduction2.2 Mean, median and mode

2.3 Relation between mean median and mode

2.4 Range, quartile deviation and semi-inter quartile range

2.5 Variance, standard deviation for both grouped and ungrouped data

2.6 Coefficient of variance

2.7 Applications.

3. Correlation and Regression (5 Lectures)

3.1 Introduction

3.2 Scattered diagram

3.3 Concept of correlation and its properties

3.4 Correlation Coefficient and Coefficient of determination

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3.5 Rank Correlation

3.6 Principle of least square and curve fitting

3.7 Linear regression

3.8 Applications.

4. Sampling Distribution (3 Lectures)

4.1 Introduction

4.2 Concept of random sample

4.3 Central tendency in statistic

4.4 Standard error of sampling distribution of statistic

4.5 Central limit theorem

4.6 Sampling distribution of mean

4.7 Applications.

5. Estimation(3 Lectures)

5.1 Introduction

5.2 Estimation of parameters

5.3 Point Estimation

5.4 Interval Estimation-confidence interval and confidence limit, Method of

Maximum Likelihood

5.5 Applications.

6. Testing of Hypothesis (4 Lectures)

6.1 Introduction

6.2 Its meaning and difference with estimation

6.3 Null hypothesis

6.4 Alternative Hypothesis

6.5 Type-I error

6.6 Type-II error

6.7 Large sample tests with normal distribution

6.8 Applications.

7. Stochastic Processes (5 Lectures)

7.1 Classification

7.2 Stationary random process

7.3 Autocorrelation

7.4 Cross-correlation

7.5 Markov chain

7.6 Poisson Process

7.7 Gaussian Process

7.8 Power spectral density

7.9 Linear system with random input7.10 Applications.

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1. Ronald E Walpole and Raymond H Myers : Probability and Statistics for Engineersand Scientists.

2. T. Veerarajan : Probability, Statistics and Random Process,

3. O.P Gupta and C. B. Gupta : Probability and Statistics.

4. I. J. Medhi, Stochastic Process, Wiley Eastern Limited, Second Edition, 1994.

5. S. M. Ross, Stochastic Process, Wiley Eastern Limited, Second Edition, 1996.

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Elective Paper

B. Tech 4

th

Semester SyllabusSubject: Numerical Methods Code: - M-430

1. Errors and convergence of approximation (4 Lectures)

1.1 Accuracy and Precision

1.2 Significant digits

1.3 Truncation error

1.4 Round off error

1.5 Relative error

1.6 Norms and Normalization

1.7 Conditioning and Ill-conditioning

1.8 Convergence analysis of approximation

2. Roots of algebraic and transcendental equations (10 Lectures)

2.1 Introduction

2.2 Bisection Method

2.3 Method of false position

2.4 Secant Method2.5 Newtons Raphson method and its extension to system of equations

2.6 Fixed point iteration method0

2.7 Convergence and Errors in various methods

2.8 Applications.

3. Solution of system of equations and Eigen value problems (10 Lectures)

3.1 Gauss elimination and Gauss Jordans direct methods

3.2 Gauss-Jacobin and Gauss-Seidel iterative, methods

3.3 Matrix decomposition method

3.4 Inverse of a matrix by Gauss-Jordan method

3.5 Eigen value of a matrix by power methods3.6 Convergence analysis and errors in various methods

3.7 Applications.

4. Interpolation and approximation (10 Lectures)

4.1 Forward, Backward, Centeral and Shift operators

4.2 Newtons forward and backward interpolation

4.3 Lagrange interpolation

4.4 Sterling and Bessels central difference interpolation

4.5 Newtons divided difference interpolation

4.6 Interpolation with a cubic spline and Hermite polynomial4.7 Errors in interpolation

4.8 Applications

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5. Numerical differentiation and integration (10 Lectures)

5.1 Derivatives from difference table, [Divided difference and finite difference]5.2 Trapezoidal and Simpsons 1/3 and 3/8 rules of numerical integration

5.3 Newtons Cotes quadrature formula

5.4 Rombergs method

5.5 Two and three point Gaussian quartered formulas

5.6 Double integrals using trapezoidal and Simpsons rules

5.7 Errors in various methods

5.8 Applications.

6. Solution of ordinary differential equations (10 Lectures)

6.1 Single step and multi-step methods

6.2 Taylor, Eulers and modified Eulers methods,

6.3 Second and Fourth order Runge Kutta methods for solving first and second order

equation

6.4 Milnes and Adams predictor and corrector methods

6.5 Finite difference solution of boundary value problem described by second order

ordinary differential equations

6.6 Errors of approximation and convergence analysis of all methods

6.7 Applications.

7. Solution of partial differential equations (8 Lectures)

7.1 Hyperbolic and parabolic partial differential equations.

7.2 One dimensional heat equation by implicit and explicit methods

7.3 Two and three dimensional heat equations

7.4 One dimensional wave equation and two-dimensional membrane equation

7.5 Laplace and Poisson equations in two and three dimensions

7.6 Applications.



a. M. K. jain S. R. K. Iyengar and R. K. Jain: Numerical Methods for Scientific and

Engineering Computation, Wiley Eastern Ltd. 1985.

C. F. Gerald and P. O.Wheatley :Applied numerical Analysis ,Addison Wesley, 1984.

S. Ali Mollah : Numerical Analysis, New Central

S. S. Sastry, Introductory Methods of Numerical Analysis,Scarbarough. Numerical methods

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Elective Paper

B. Tech 4th Semester SyllabusSubject: Probability, statistics and Stochastic Processes

Subject Code: - M-431

First half: Probability

1. Introduction and Background (1 Lecture)

1.5 Preconception idea of set related to Probability1.6 Random Experiment

1.7 Sample space

1.8 Events [Simple event, compound event, equally likely event, favorable event,

Exhaustive events, complimentary events, Independent and Dependent events.]

2. The meaning of probability (2 Lectures)

2.2 Definition [Classical, Frequency, Axiomatic odds in favor of an event and odds

against an event]

2.2 General addition rule of probability for two events, the events and extension2.3 Conditional probability and Multiplication Rules

2.4 Stochastic independencies of the events and related theorems

2.5 Bayes theorem with proof

2.6 Applications related to Industry, Engineering and Management.

3. The concept of random variable (4 Lectures)

3.9 Random variable

3.10 Discrete and continuous random variables

3.11 Joint probability distribution

3.12 Joint distribution table and marginal distribution of random variable3.13 Expectation of random variable and its concept in management and industry

3.14 Properties of Exportation

3.15 Variance and covariance of random variables

3.16 Applications.

4. Probability distributions of random variables (4 Lectures)

4.1 Discrete and continuous probability distribution

4.2 Probability mass function

4.3 Cumulative distribution function for discrete random variables and its

graphical representation4.4 Probability density function

4.5 Cumulative probability distribution function for continuous random variable

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and their properties and graphical representation

4.6 Relation between at least and at most values of the Discreet random valuable

4.7 Consultation of at most table for Discrete Distribution.

5. Discrete probability Distributions (4 Lectures)

5.1 Bernouli trial,

5.2 Binomial distribution

5.3 Poisson distribution,

5.4 Derivation of mean, variance and standard deviation of the distribution,

5.5 Applications.

6. Continuous probability Distribution (4 Lectures)

6.1 Derivation of mean, variance and standard deviation and their applications

6.2 Properties of normal distribution

6.3 Standard normal variate and standard normal distribution6.4 Graphical representation,

6.5 Computation of mean median and mode of Normal Distribution

6.6 Applications


2nd Half : Statistics

4. Introduction and Background (1 Lectures)

1.1 Introduction

1.2 Data [Primary and Secondary data, Collection of data]

1.3 Representation of data-grouped and ungrouped data

1.4 Frequency distribution of data

1.5 Applications.

5. Measures of Central tendency and Measures of Dispersion (3 Lectures)

2.1 Introduction

2.2 Mean, median and mode2.3 Relation between mean median and mode

2.4 Range, quartile deviation and semi-inter quartile range

2.5 Variance, standard deviation for both grouped and ungrouped data

2.6 Coefficient of variance

2.7 Applications.

6. Correlation and Regression (5 Lectures)

3.1 Introduction

3.2 Scattered diagram

3.3 Concept of correlation and its properties

3.4 Correlation Coefficient and Coefficient of determination

3.5 Rank Correlation

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3.6 Principle of least square and curve fitting

3.7 Linear regression

3.8 Applications.

4. Sampling Distribution (3 Lectures)

4.1 Introduction

4.2 Concept of random sample

4.3 Central tendency in statistic

4.4 Standard error of sampling distribution of statistic

4.5 Central limit theorem

4.6 Sampling distribution of mean

4.7 Applications.

5. Estimation (3 Lectures)

5.1 Introduction

5.2 Estimation of parameters

5.3 Point Estimation

5.4 Interval Estimation-confidence interval and confidence limit, Method of

Maximum Likelihood

5.5 Applications.

6. Testing of Hypothesis (4 Lectures)

6.1 Introduction

6.2 Its meaning and difference with estimation

6.3 Null hypothesis

6.4 Alternative Hypothesis

6.5 Type-I error

6.6 Type-II error

6.7 Large sample tests with normal distribution

6.8 Applications.

7. Stochastic Processes (5 Lectures)

7.11 Classification

7.12 Stationary random process

7.13 Autocorrelation

7.14 Cross-correlation

7.15 Markov chain

7.16 Poisson Process

7.17 Gaussian Process

7.18 Power spectral density

7.19 Linear system with random input

7.20 Applications.

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6. Ronald E Walpole and Raymond H Myers : Probability and Statistics for Engineers

and Scientists.7. T. Veerarajan : Probability, Statistics and Random Process,

8. O.P Gupta and C. B. Gupta : Probability and Statistics.

9. I. J. Medhi, Stochastic Process, Wiley Eastern Limited, Second Edition, 1994.

10. S. M. Ross, Stochastic Process, Wiley Eastern Limited, Second Edition, 1996.

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Elective Paper

B. Tech 4

th

Semester Syllabus

Subject: Higher Engineering Mathematical Methods

Code: - M-432

1. Series solution of Ordinary differential Equations: (16 Lectures)

1.1 Introduction

1.2 Seriessolution and its validity

1.3 Series solution about an ordinary point

1.4 Series solution about a regular singular point

1.5 Bessels equation and Bessel functions

1.6 Recurrence relations of Bessel functions

1.7 Generating function for Jn(x)

1.8 Equation reducible to Bessels equation

1.9 Orthogonality of Bessel functions

1.10 Legendres Equation and Legendre functions, Legendre polynomial1.11 Rodrigues formula

1.12 Generating function for Pn(x)

1.13 Recurrence relations for Pn(x)

1.14 Orthogonality of Legendre polynomial

1.15 Applications.

2. Difference Equations: (8 Lectures)

2.1 Introduction2.2 Definition and formation of difference equations

2.3 Linear difference equations

2.4 Rules for finding C. F. and P.I

2.5 Difference equations reducible to linear form

2.6 Simultaneous difference equations with constant coefficients

2.7 Applications.

3. Fourier Integral and Fourier Transform: (16 Lectures)

3.1 Introduction3.2 Definition of Integral transform

3.3 Fourier Integral Transform

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3.4 Fourier sine and cosine integrals

3.5 Complex form of Fourier integrals

3.6 Fourier Transforms and Inverse Fourier Transforms

3.7 Fourier sine and cosine transforms and their inversion

3.8 Finite Fourier sine and cosine transforms and their inversion

3.9 Properties of Fourier Transforms

3.10 Convolution

3.11 Fourier Transforms of the derivatives of a function

3.12 Persevals Identity for Fourier Transform

3.13 Relations between Fourier and Laplace transforms

3.14 Inverse Laplace transforms by method of residues

3.15 Application of transforms to boundary value problems

3.16 General applications.


4. Z-transform (8 Lectures)

4.1 Introduction

4.2 Some standard Z-transforms

4.3 Linearity property

4.4 Damping rule and some standard results

4.5 Shifting rule

4.6 Initial and Final value theorem

4.7 Convolution theorem

4.8 Evaluation of inverse transforms

4.9 Applications.

5. Partial Differential Equations and its Applications: (8 Lectures)

5.1 Introduction

5.2 Non-linear equation of second order

5.3 Separation of variables

5.4 Formulation and solution of wave equation

5.5 Formulation and solution of Laplace equation

5.6 One and two-dimensional heat flow equation and solution

5.7 Applications.




2. E. Kreyszig, Advanced Engineering Mathematics, 8th edition, John Wiley.3. Piaggio : Differential Equations



6. I.N. Sneddon,The use of Integral Transforms, McGraw-Hill, 1974.7. I.N. Sneddon, Elements of Partial Differential Equations, McGraw-Hill, 1988.

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Elective Paper

B. Tech 7th Semester Syllabus

Subject: Operations Research - I Code: - M-730

First Half

I. Linear programming (L.P.)

1. Operations Research- an overview and Formulation of L.P. Problem

(3 Lectures)

1.1 Introduction1.1. Mathematical Formulation

1.2. Application.

2. Linear Programming Problem-Graphical Solution (3 Lectures)

2.1. Introduction

2.2. Graphical solution method

2.3. Some exceptional cases

2.4. Application.

3. Linear Programming- simplex Method (8 Lectures)

3.1 Introduction

3.2 General form of Linear Programming Problem

3.3 Canonical and standard forms of L.P. Problem

3.4 Simplex method & its development

3.5 Simplex Algorithm

3.6 Use of Artificial Variables

3.7 The Big-M Method

3.8 The two-phase Method

3.9 Solution of simultaneous linear equations

3.10 Degeneracy in simplex Method

3.11 Applications.

4. Duality in Linear Programming (5 Lectures)

4.1 Introduction

4.2 General Primal-Dual Problem

4.3 Formulation of dual when primal is in canonical form

4.4 Formulation of dual when primal is in standard form4.5 Some theorems on Duality

4.6 Properties of primal and Dual optimal solutions

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4.7 Duality and simplex method

4.8 Applications.

5. Transportation Problem (6 Lectures)

5.1 Introduction

5.2 Definition and Matrix terminology

5.3 Mathematical Formulation

5.4 Solution of Transportation Problem

5.5 Degeneracy in Transportation Problem

5.6 Unbalanced Transportation Problem

5.7 Maximization of Transportation Problem

5.8 Least time Transportation Problem

5.9 Applications.

6. Assignment Problem (4 Lectures)

6.1 Introduction

6.2 Definition and Matrix terminology

6.3 Comparison with Transportation Problem

6.4 Mathematical Formulation of Assignment Problem

6.5 Solution of Assignment Problem

6.6 Variation of Assignment Problem

6.7 Traveling Salesman Problem

6.8 Applications.

7. Revised simplex Method (4 Lectures)

Introduction

Revised simplex algorithm

Advantages of the Revised simplex method, over simplex method

Applications.


SECOND HALF

I. THEORY OF GAMES

1. Introduction to basic concepts of Game theory (2 Lectures)

1.1 Introduction

1.2 Useful Technology

1.3 Maxmin and Minmax Principle

1.4 Two-person Zero-sum games with saddle point

1.5 Applications

2. Reduction of a Game problem to the 22 Matrix form (4 Lectures)

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2.1 Game problems with out saddle points

2.2 Pure strategy and mixed strategy

2.3 Solution of a 22 game problem without saddle point

2.4 Graphical method of solution for n2 and 2n game problem

2.5 Method of subgames for n2 and 2n game problem

2.6 Reduction rule of a game problem (Dominance rule)

2.7 Applications.

3. More about game problems with mixed strategy (3 Lectures)

3.1 Algebraic method of solution of game problem without saddle point

3.2 Reduction of a game problem to linear programming problem

3.3 Applications.

II. NETWORK TECHNIQUE

1. Basic steps in PERT and CPM (3 Lectures)

1.1 Introduction

1.2 Phases of project management

1.3 Construction of Network diagram

1.4 Numbering of the events (Fulkersons Rule)

1.5 Activity on node diagram

1.6 Measure of activity

1.7 Frequency distribution curve for PERT1.8 Applications.

2. PERT Computations (3 Lectures)

2.1 Introduction

2.2 Forward pass computation

2.3 Backward pass computation

2.4 Slack

2.5 Critical path

2.6 Probability of meeting the schedule dates

2.7 Applications.

3. CPM Computation (3 Lectures)

3.1. Difference between PERT and CPM

3.2. Some CPM Terms

3.3. Critical path

3.4.Float

3.5. Negative float and negative slack

3.6. Applications.

4. Cost Analysis, Contracting and Updating (2 Lectures)

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4.1. Project cost

4.2. Crashing

4.3. Updating (PERT and CPM)

4.4. Applications.

5. General Remarks PERT and CPM (3 Lectures)

5.1. PERT cost

5.2. Decision CPM

5.3. How the networks (PART/CPM) help management?

5.4. Difficulties in using network methods

5.5 Applications.



1. H. A. Taha, Operations Research An introduction, PHI2. J. K. Sharma : Fundamentals of Operations Research, Macmillan.

3. F.S. Hiller and G. J. Leiberman, Introduction to Operations Research (6thEdition), McGraw-Hill International Edition, 1995.

4. Hira and Gupta : Operations Research

5. Kanti Swarup, P. K. Gupta and Man Mohan, Operations Research- AnIntroduction, S. Chand & Company, New Delhi.

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Elective Paper


Subject: Mathematical Modeling and Calculus of Variations

Code: - M-731

1. Introduction: (4 Lectures)

1.1 What is Mathematical modeling?

1.2 Types of models

1.3 Five steps of modeling

1.4 Modeling as a major part of Mathematics and as a tool to solve real world

problems

1.5 Richardsons linear Arms race model etc. and methods of solution

1.6 Applications.

2. Dynamical systems: (9 Lectures)

2.1 Discrete and continuous single species models- Malthusian and logistic models

2.2 Nonlinear models of two interacting species of population, prey-predator, competition,symbiosis, comensal and amensal models

2.3 Local and global stability

2.4 Structural, asymptotic and robust stability of equilibrium of a system, Routh-Hurwitz

criteria

2.5 Behaviors of solutions near equilibriums - phase plane diagram, node, centre, focus, saddle

point and spiral

2.6 Liapunovs direct (second) theorem of stability

2.7 Applications.

3. Time delay and diffusion: (12 Lectures)

3.1 Discrete and distributed delays in single and two species models and their effects on the

populations in the long run

3.2 Bifurcation theory in prey-predator models of Lotka-Volterra and others

3.3 Advection of populations

3.4 Diffusion and dispersion, self and cross dispersion

3.5 Effect of advection and dispersion on stability

3.6 Applications.

4. Stochastic population models: (03 Lectures)

4.1 Use of probability generating functions in birth-death models4.2 Applications.

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5. Cell-growth: (05 Lectures)

5.1 Batch culture, Development of bacteria in a chemostat, Growth of Cancer, Enzyme

kinematics

5.2 One-chain, two-chain processes and cooperative systems

5.3 Applications.


6. Mathematical Bioeconomics: (12 Lectures)

Fisheries

6.1 Fish exploration, Beaverton-Holls model and maximization of profit for one species using

calculus of variation, Hamiltons maximum principle and dynamic programming

6.2 Optimization in many-species cases with and without interactions using aforesaid methods

6.3 Optimization of dynamically exhaustible resources as coal, oil, minerals etc.6.4 Applications.

Mining

6.5 Mathematical Models with and without monopoly exclusive/ inclusive of mining cost

6.6 Applications.

Forestry

6.7 Optimal harvesting of forest with various harvest functions

6.8 Applications.

Calculus of variations

7. Variation and its properties (4 Lectures)

7.1 Eulers equation and its solution

7.2 Brachistochrone problem

7.3 Curves of minimum arc of surface of revolution and similar problems

7.4 Applications

8 Geodesies (6 Lectures)

8.1 Geodesies in spherical polar and cylindrical coordinates

8.2 Functional dependent on higher order derivatives

8.3 Variational problems involving several unknown functions

8.4 Functional involving several independent variables-Ostrogradsky equation

8.5 Optimization under constraints and Lagrange multipliers

8.6 Applications.

8.7

9. Isoperimetric problems (5 Lectures)

9.1 Isoperimetric problems involving constraints as a functional

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9.2 Variational problems with moving boundaries

9.3 Transversality conditions

9.4 Applications.

10. Lagranges equations (7 Lectures)

10.1 Lagranges equations for dynamical systems and Hamiltons principle10.2 Sturm-Liovilles problem and variational methods

10.3 Raleighs principle

10.4 Direct methods of Ritz and Kantorovich methods

10.5 Applications.



1. M. K. jain S. R. K. Iyengar and R. K. Jain: Numerical Methods for Scientific and

Engineering Computation, Wiley Eastern Ltd. 1985.

. C. F. Gerald and P. O.Wheatley :Applied numerical Analysis ,Addison Wesley, 1984.

S. Ali Mollah : Numerical Analysis, New Central

S. S. Sastry, Introductory Methods of Numerical Analysis,


1. J. N. Kapur, Mathematical Modeling, Wiley Eastern.

2. D. N. Burghes, Mathematical Modeling in the Social Management and Life Science, Ellils

Horwood and John Wiley.

Elective Paper

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Subject: Operations Research - II Code: - M-830

First Half:

1. Dual simplex Method (4 Lectures)

1.1 Introduction

1.2 Dual simplex algorithm

1.3 Applications.

2. Integer Programming (6 Lectures)

2.1 Introduction

2.2 All-integer and mixed-integer programming problems2.3 Formulation of all-integer and mixed-Integer programming problem

2.4 Gomoris Fractional-cut method for all-integer and mixed-Integer programming problems

2.5 Branch and Bound Method for all-integer and mixed-Integer programming problems

2.6 Method of zero-one programming

2.7 Applications.

3. Goal Programming (4 Lectures)

3.1 Introduction

3.2 Graphical Method

3.3 Single goal problem3.4 Equal ranked goals

3.5 Priority ranked goals

3.6 Applications.

4. Dynamic Programming (5 Lectures)

4.1 Introduction

4.2 Deterministic Dynamic Programming

4.3 Probabilistic Dynamic Programming

4.4 Solution of Linear Programming using Dynamic Programming

4.5 Solution of Integer Programming using Dynamic Programming

4.6 Applications.

5. Non-linear Programming (5 Lectures)

5.1 Introduction

5.2 Unconstrained Optimization

5.3 Constrained Optimization for equality constraints - Lagrangean method

5.4 Constrained Optimization for inequality constraints Kuhn Tucker Conditions

5.5 Quadratic Programming

5.6 Convex Programming

5.7 Applications.

Class test and revision (3 Lectures)

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SECOND HALF.

1. Sequencing Models (4 Lectures)

1.1 Introduction

1.2 Classification of self problems

1.3 Processing of jobs through two machines

1.4 Processing of jobs through three machines

1.5 Processing of two jobs through two machines

1.6 Applications.

2. Replacement Methods (4 Lectures)

2.1 Introduction

2.2 Replacement policies for items whose efficiency deteriorates with time2.3 Replacement policies for items that fail completely

2.4 Applications.

3. Queuing Theory (8 Lectures)

3.1 Introduction

3.2 Characteristics of Queuing Systems

3.3 Steady state and transient state

3.4 Poisson birth and death process

3.5 Kendals notation for queuing models3.6 (M / M / 1) : ( / FCFS) Queuing model

3.7 (M / M / 1) : (N / FCFS) Queuing model

3.8 (M / M / L) : ( / FCFS) Queuing model

3.9 (M / M / L) : (N / FCFS) Queuing model

3.10 Applications.

4. Inventory Control (10 Lectures)

4.1 Introduction

4.2 Determination of Economic Order Quantity (EOQ)

4.3 EOQ model with shortage4.4 Determination of Economic Production Quantity (EPQ)

4.5 EPQ model with shortage

4.6 Multi-item Inventory and Multiple Constraints

4.7 Stochastic Inventory Models

4.7 Safety Stock and Buffer stock

4.8 Applications.

5. Decision Theory (7 Lectures)

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5.1 Introduction

5.2 Steps in decision theory approach

5.3 Decision making environment

5.4 Decision making under condition of certainty

5.5 Decision making under condition of uncertainty

5.6 Decision making under condition of risk

5.7 Maximum likelihood criterion

5.8 Expected value criterion for continuously distributed random variables

5.9 Variations of the expected value criterion

5.10 Applications.



1. H. A. Taha, Operations Research An introduction, PHI2. J. K. Sharma : Fundamentals of Operations Research, Macmillan.3. F.S. Hiller and G. J. Leiberman, Introduction to Operations Research (6th

Edition), McGraw-Hill International Edition, 1995.

4. Hira and Gupta : Operations Research

5. Kanti Swarup, P. K. Gupta and Man Mohan, Operations Research- AnIntroduction, S. Chand & Company, New Delhi.

Documents

Syllabus of Mathematics BTech