## Table of Contents

**Chapter 1 Matrices **

1.1 Introduction

1.2 Matrix – Definition

1.2.1 Order of a Matrix

1.2.2 General form of a Matrix

1.3 Types of Matrices

1.4 Characteristic Equation

*Examples 1.1 to 1.2 *

1.5 Eigen values and Eigenvectors

1.5.1 Working Rule for Finding Eigenvalues and Eigenvectors

1.5.2 Problems in Non-symmetric Matrix with Non-repeated Eigenvalues

*Examples 1.3 to 1.7 *

1.5.3 Problems in Non-symmetric Matrix with Repeated Eigenvalues

*Examples 1.8 to 1.10 *

1.5.4 Problems based on Symmetric Matrices with Non-repeated Eigenvalues

*Examples 1.11 to 1.13 *

1.5.5 Problems based on Symmetric Matrix with Repeated Eigenvalues

*Examples 1.14 to 1.15 *

*Exercise 1.1 *

1.6 Properties of Eigenvalues and Eigenvectors

*Examples 1.16 to 1.22 *

*Exercise 1.2 *

*Examples 1.23 to 1.30 *

*Exercise 1.3 *

*Examples 1.31 to 1.36 *

*Exercise 1.4 *

*Examples 1.37 to 1.50 *

*Exercise 1.5 *

**Chapter 2 Vector Calculus **

2.1 Gradient

*Examples 2.1 to 2.10 *

2.1.1 Directional Derivatives (DD)

2.1.2 Normal Derivative

*Examples 2.11 to 2.17 *

*Exercise 2.1 *

*Examples 2.18 to 2.23 *

*Exercise 2.2 *

2.2 Divergence and Curl

2.2.1 Vector Identities

*Examples 2.24 to 2.41 *

*Exercise 2.3 *

2.3 Line Integrals

2.3.1 Line Integral Over a Plane Curve

*Examples 2.42 to 2.46 *

2.3.2 Surface Integral –Area of a Curved Surface

2.4 Green’s Theorem (for Plane)

*Examples 2.47 to 2.58 *

*Exercise 2.4 *

*Examples 2.59 to 2.67 *

*Exercise 2.5 *

*Examples 2.68 to 2.74 *

*Exercise *

**Chapter 3 Analytic Function **

3.1 Complex Numbers

3.2 Necessary Condition for *f*(*z*) to be Analytic [Cauchy – Riemann Equations]

*Examples 3.1 to 3.17 *

3.3 Harmonic Functions

*Examples 3.18 to 3.22 *

3.4 Properties of Analytic Functions

*Examples 3.23 to 3.27 *

3.5 Construction of Analytic Function

*Examples 3.28 to 3.47 *

3.6 Some Standard Transformations

3.6.1 Translation

*Examples 3.48 to 3.52 *

3.6.2 Magnification and Rotation

*Examples 3.53 to 3.56 *

3.6.3 Transformation *w *=1/*z*

*Examples 3.57 to 3.65 *

3.6.4 Transformation *w *= *z*^{2}

*Examples 3.66 to 3.68 *

3.7 Bilinear Transformation

*Examples 3.69 to 3.82 *

**Chapter 4 Complex Integration **

4.1 Fundamentals of Complex Integrals

4.2 Cauchy’s Integral Theorem or Cauchy’s Fundamental Theorem

*Examples 4.1 to 4.5 *

4.3 Cauchy’s Integral Formula for Derivatives

*Examples 4.6 to 4.21 *

*Exercise 4.1 *

*Examples 4.22 to 4.30 *

*Exercise 4.2 *

*Examples 4.31 to 4.49 *

*Exercise 4.3 *

*Examples 4.50 to 4.55 *

*Exercise 4.4 *

*Examples 4.56 to 4.67 *

*Exercise 4.5 *

*Examples 4.68 to 4.75 *

*Exercise 4.6 *

*Examples 4.76 to 4.92 *

**Chapter 5 Laplace Transform **

5.1 Laplace Transform – Sufficient Condition for Existence

5.2 Laplace Transform of Elementary Functions

*Examples 5.1 to 5.5 *

*Exercise 5.1 *

5.3 Properties of Laplace Transforms

*Examples 5.6 to 5.9 *

*Exercise 5.2 *

5.4 Derivatives of Laplace Transform [Multiplication by *t*]

*Examples 5.10 to 5.11 *

*Exercise 5.3 *

5.5 Integration of Laplace Transform [Division by *t*]

*Examples 5.12 to 5.16 *

*Exercise 5.4 *

5.6 Transforms of Derivatives and Integrals

5.7 Laplace Transforms of Integrals

*Examples 5.17 to 5.20 *

*Exercise 5.5 *

*Examples 5.21 to 5.22 *

*Exercise 5.6 *

5.8 Evaluation of Integrals Using Laplace Transform

*Examples 5.23 to 5.24 *

*Exercise 5.7 *

5.9 Initial and Final Value Theorems

*Examples 5.25 to 5.32 *

*Exercise 5.6 *

5.10 Unit Step Function or Heaviside’s Unit Step Function

*Examples 5.33 to 5.34 *

5.11 Unit Impulse Function [Dirac Delta]

*Examples 5.35 to 5.36 *

5.12 Laplace Transform of Periodic Functions

*Examples 5.37 to 5.45 *

*Exercise 5.9*

5.13 Inverse Laplace Transform

*Examples 5.46 to 5.48 *

*Exercise 5.10 *

*Examples 5.49 to 5.56 *

*Exercise 5.11 *

5.14 Multiplication by *s *

*Examples 5.57 *

5.15 Division by *s *

*Examples 5.58 *

*Exercise 5.12 *

5.16 Inverse Laplace Transform of Derivatives

*Examples 5.59 to 5.61 *

*Exercise 5.13 *

5.17 Inverse Laplace Transform by Method of Partial Fraction

*Examples 5.62 to 5.70 *

5.18 Method of Convolution

*Examples 5.71 to 5.80 *

*Exercise 5.14 *

5.19 Application of Laplace Transform to the Solution of Linear Second Order

Ordinary Differential Equations with Constant Coefficients

*Examples 5.81 to 5.102 *

*Exercise 5.15 *

5.20 Solving Integral Equations

*Examples 5.103 to 5.105 *