## Description

This book caters to the requirements of Post Graduate students of various Engineering Colleges affiliated to Anna University. This book has simple and lucid presentations with a range of solved examples which enables the students to self-study and understand the topics with ease. The book has a methodical approach towards problem solving and helps the students grasp the topics and solve the exercise problems with confidence. The answers for the exercise problems are given at the end of each chapter.

## Table of Content

**Chapter 1 LINEAR ALGEBRA AND ADVANCED MATRIX THEORY**

1.1 Vector Space

1.2 Inner Product

1.3 Norm

1.4 Generalised Eigenvector

1.5 Canonical Forms

1.5.1 Rational Normal Form or Companion Canonical Form or Natural

Normal Form

1.5.2 Jordan Normal Form or Classical Canonical Form

1.5.3 Smith’s Normal Form

Worked Examples 1(a)

Exercise 1(a)

1.6 QR-Factorisation

1.6.1 Revised Gram-Schmidt Method of Decomposing A

1.6.2 House Holders Method of Decomposing a Symmetric Matrix A

1.7 Least Square Method

1.8 Singular Value Decomposition of an (m × n) Rectangular Matrix A, when

m ≥ n

Worked Examples 1(b)

Exercise 1(b)

Answers

**Chapter 2 CALCULUS OF VARIATIONS**

2.1 Euler-Lagrange Equation for the Extremum of the Functional v[y(x)] between Fixed End Points

2.1.1 Variational Notation

2.1.2 Alternate Method of Deriving Euler’s Equation, using Variational Notation

2.1.3 First Integral of Euler’s Equation, when F is a Function of y

and y_ only, viz., F does contain x Explicitly

2.1.4 Necessary Conditions for the Extremum of a Functional of Two Functions

Between Fixed End Points

2.1.5 Necessary Condition for the Extremum of a Functional Depending on the First and Second Derivatives of the Function y

2.2 Necessary Condition for the Extremum of a Functional Depending on a Function of Two Independent Variables

Worked Examples 2(a)

Exercise 2(a)

2.3 Variational Problems Involving a Conditional Extremum

2.4 Variational Problems with Moving Boundaries/Variables End Points

2.4.1 Natural Boundary Conditions

2.5 Sufficient Conditions for the Extremum of the Functional

2.6 Direct Methods

2.6.1 Ritz Method

2.6.2 Kantorovich Method (or) Semi Direct Method

Worked Examples 2(b)

Exercise 2(b)

Answers

**Chapter 3 TENSOR ANALYSIS**

3.1 Summation Convention

3.2 Covariant and Contravariant Vectors

3.3 Second Order Tensors

3.3.1 The Kronecker Delta δji

3.3.2 Tensors of Higher Rank (Order)

3.4 Basic Operations of Tensors

3.4.1 Addition and Subtraction

3.4.2 Outer Multiplication

3.4.3 Contraction

3.4.4 Inner Multiplication

3.4.5 Quotient Law

3.5 Symmetric and Skew-Symmetric Tensors

3.5.1 Metric Tensor or Fundamental Tensor

3.5.2 Conjugate or Reciprocal Tensor of gij

Worked Examples 3(a)

Exercise 3(a)

3.6 Christoffel’s Symbols

3.6.1 Derivatives of the Fundamental Tensors

3.6.2 Transformation Law for Christoffel Symbol of the I Kind

3.6.3 Transformation Law for Christoffel Symbol of the II Kind

3.7 Covariant Differentiation

3.7.1 The Covariant Derivative Ai, j is a Covariant Tensor of Rank 2

3.7.2 The Covariant Derivative Ai, j is a Mixed Tensor of Rank 2

3.8 Gradient, Divergence and Curl in Tensor Notation

Worked Examples 3(b)

Exercise 3(b)

Answers

**Chapter 4 LAPLACE TRANSFORMS AND GENERALISED**

FOURIER SERIES

4.1 Laplace Transforms of the Bessel’s Functions J0(x) and J1(x)

4.2 Laplace Transform of Error Function

4.3 Complex Inverse Formula

Worked Examples 4(a)

Exercise 4(a)

4.4 Generalised Fourier Series

4.4.1 Orthogonal Functions

4.5 Sturm−Liouville Systems

Worked Examples 4(b)

Exercise 4(b)

Answers

**Chapter 5 ESTIMATION THEORY**

5.1 Interval Estimation

5.2 Point Estimation

5.2.1 Unbiased Estimator

5.2.2 Consistent Estimator

5.2.3 Efficient Estimator

5.2.4 Most Efficient Estimator

5.2.5 Sufficient Estimator

5.2.6 Neymann’s Factorisation Criterion for a Sufficient Estimator

Worked Examples 5(a)

Exercise 5(a)

5.3 Methods of Finding Estimators

5.3.1 Method of Maximum Likelihood

5.3.2 Method of Moments

Worked Examples 5(b)

Exercise 5(b)

Answers

**Chapter 6 MULTIPLE AND PARTIAL CORRELATIONS**

6.1 A Note on Yule’s Subscript Notation

6.2 Plane of Regression

6.3 Properties of Residuals

6.4 Coefficient of Multiple Correlation

6.4.1 Multiple Correlation Coefficient in terms of Simple Correlation Coefficients

6.5 Partial Correlation Coefficient in Terms of Simple Correlation Coefficients

Worked Examples 6

Exercise 6

Answers

**Chapter 7 MULTIVARIATE ANALYSIS**

7.1 Mean Vectors and Covariance Matrices

7.2 Partitioning the Covariance Matrix

7.3 Mean Vector and Covariance Matrix for Linear Combination of Random Variables

7.4 Multivariate Normal (MVN) Distribution

7.4.1 Importance of the MVN Distribution

7.4.2 Properties of a MVN Distribution

7.5 Principal Components

7.5.1 Population Principal Components

7.5.2 Principal Components got from Standardised Variables

Worked Examples 7

Exercise 7

Answers

**Chapter 8 LINEAR PROGRAMMING**

8.1 Linear Programming Problem (L.P.P)

8.2 Formulation of a Linear Programming Problem

8.2.1 Solution of a Linear Programming Problem

8.2.2 Step by Step Procedure of the Graphical Method

8.2.3 Important Special Cases

Worked Examples 8(a)

Exercise 8(a)

8.3 Simplex Method

8.3.1 Some Definitions and Notations

8.3.2 Slack Variable

8.3.3 Surplus Variable

8.3.4 Initial Basic Feasible Solution

8.3.5 Computational Procedure of the Simplex Method

Worked Examples 8(b)

Exercise 8(b)

8.4 Transportation Problem

8.4.1 Initial B.F.S. by North-West Corner Rule

8.4.2 Initial B.F.S. by the Lowest Cost Entry Method or Matrix Minima Method

8.4.3 Initial B.F.S. by Vogel’s Approximation Method (VAM) or Unit Cost Penalty Method

8.4.4 Step II Optimality Test – Computational Procedure

8.4.5 Step III

8.4.6 Unbalanced Transportation Problem

8.4.7 Alternative Optimal Solution of a Transportation Problem

8.4.8 Maximisation Case in Transportation Problem

Worked Examples 8(c)

Exercise 8(c)

8.5 Assignment Problem

8.5.1 Assignment Algorithm (Hungarian Method or Flood’s Technique)

Worked Examples 8(d)

Exercise 8(d)

Answers

**Chapter 9 DYNAMIC PROGRAMMING**

9.1 Bellman’s Principle of Optimality

Worked Examples 9

9.2 Solution of Linear Programming Problem as a Dynamic Programming Problem

Exercise 9

Answers

## About The Author

**T. Veerarajan** is Dean (Retd.), Department of Mathematics, Velammal College of Engineering and Technology, Madurai, Tamil Nadu. A Gold Medalist from Madras University, he has had a brilliant academic career all through. He has 50 years of teaching experience at undergraduate and postgraduate levels in various established engineering colleges in Tamil Nadu including Anna University, Chennai.

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