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# Applied Mathematics

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CONTENTS

1. Integration

1.1 Introduction

Definition of Integration

Indefinite Integration or General Integral

Standard Formulae

1.2 Rules of Integration

Modification of Formulae for Linear Expression (nx + d)

Integration of Simple Rational Function

Integration by Trigonometric Transformation

1.3 Methods of Integration

Integration by Substitution

Important Deductions

Some Standard Substitutions

Integration by Parts and Partial Fraction

2. Definite Integration

2.1 Introduction

2.2 Definition

Rules of Definite Integrals

2.3 Properties of Definite Integrals

3. Application of Definite Integrals

3.1   INTRODUCTION

Tracing of Curves

Area between Two Curves

Volume of Revolution

Centre of Gravity25

Centroid of Plane Lamina

Mean Value OR Average Value

Root Mean Square Value (RMS value)

4. Differential Equation

4.1 INTRODUCTION

Definition

Formation of Differential Equation

4.2 Solution of first order first degree differential Equation

Variable Separable

Equation Reducible to Method of Variables Separable form by suitable substitutions

Homogeneous Differential Equations

Equations Reducible to Homogeneous Form

Linear Differential Equations

Exact differential Equation

4.3 BERNOULLI'S Equations

Non-linear differential equation

5. Application of Differential Equation

5.1 Application of Differential equations

Introduction

Applications of Differential Equations

Kirchhoff’s Laws

Application to LRC Circuits

6. Interpolation

6.1 Introduction

Lagrange’s Interpolation Formula

6.2 Forward and Backward Differences

Forward Differences

Backward Differences

6.3 Newton’s Forward Difference Interpolation Formula

6.4 Newton’s Backward Difference Interpolation Formula

6.5 Concept of Extrapolation

7. Numerical differentiation and integration

7.1   Newton's Forward and Backward Difference Formula for Differentiation

Newton's Forward Difference Formula for Differentiation

Newton's Backward Difference Formula for Differentiation

7.2 Numerical Integration

Trapezoidal Rule

Simpson's One-third Rule

8. Numerical Solution of Ordinary Differential Equation

8.1 Introduction

8.2 Runge-Kutta Method of Second Order

8.3 Runge-Kutta Method for Fourth Order

9. Laplace Transform

6.1 Introduction

6.2 Definition of Laplace Transform

6.3 Laplace Transform of Elementary Functions

Properties of Laplace Transform

First Shifting Property

Second Shifting Property

Multiplication by tn

Division by t Property

Use of Laplace transform in evaluation of certain integrals

6.4   Inverse Laplace Transform

Definition of Inverse Laplace transform

Inverse Laplace Transform of Standard Function

Properties of Inverse Laplace Transform

Inverse Laplace Transform by Method of Partial Fraction

6.5 Convolution Theorem

6.6 Laplace transform of Derivatives

6.7 Solution of Differential Equation using Laplace Transform (upto Second Order Equation)

10. Fourier Series

7.1 Introduction

7.2 Definition of Fourier Series (Euler’s Formula)

7.3 Series Expansion of continuous functions

Series Expansion of continuous Functions in (0, 2p)

Series expansion of continuous functions in (–p,p)

Series expansion of continuous functions in the interval (c, c + 2l)

Fourier Series expansion in (0, 2l)

Fourier Series Expansion in (–l, l)

7.4 Series Expansion of even and odd functions

Fourier Series Expansion of even and odd functions in (–p,p)

Fourier Series Expansion of Even and Odd Functions in (–l, l)

7.5 Half range Expansion

Half range cosine series

Half Range Sine Series

11. Numerical Solution of algebraic equation

8.1 Introduction

8.2 Bisection method

8.3 Regula Falsi method

8.4 Newton-Rapson method

12. Numerical solution of simultaneous equation

9.1 Introduction

9.2 Elimination method – Gauss elimination method

9.3 Gauss-Seidal method (Iterative Method)

9.4 Jacobi’s method

13. Probability

6.1 Introduction

Terms in Probability

Definition

Types of Event

6.2 Definition of Probability of an Event

Addition Theorem of Probability

14. Probability Distribution

7.1 Introduction

7.2 Binomial distribution

7.3 Poisson Distribution

Mean and Variance of Poisson’s Distribution

7.4 Normal distribution

15. Discrete Mathematics

9.1 Relational Algebra

9.2 Definition of a set

Methods for Describing Sets

9.3 Types of Sets

Operations on Set

Counting Principle

Principle of Inclusion - Exclusion

Cartesian Product

Relation

Function