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# GATE Electronics & Communication Vol-2- Engineering Mathematics

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PUBLISHED FOR GATE 2018

 Edition 8th Authors R K Kanodia & Ashish Murolia Publisher NODIA Pages 642 Binding Paper Back Language English

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SALIENT FEATURES

• Brief Theory

• Problem Solving Methodology

• Fundamental Concepts & Formulae Review

• Vast Question book with Full Solutions

• Multiple Choice Questions, Memory Based Questions and Numerical Types Questions

• Full width coverage of GATE Syllabus

• Well explained and error free solutions

CHAPTER 1 MATRIX ALGEBRA

1.1 INTRODUCTION

1.2 MULTIPLICATION OF MATRICES

1.3 TRANSPOSE OF A MATRIX

1.4 DETERMINANT OF A MATRIX

1.5 RANK OF MATRIX

1.6 ADJOINT OF A MATRIX

1.7 INVERSE OF A MATRIX

1.7.1 Elementary Transformations

1.7.2 Inverse of Matrix by Elementary Transformations

1.8 ECHELON FORM

1.9 NORMAL FORM

EXERCISE 1

SOLUTIONS 1

CHAPTER 2 SYSTEMS OF LINEAR EQUATIONS

2.1 INTRODUCTION

2.2 VECTOR

2.2.1 Equality of Vectors

2.2.2 Null Vector or Zero Vector

2.2.3 A Vector as a Linear Combination of a Set of Vectors

2.2.4 Linear Dependence and Independence of Vectors

2.3 SYSTEM OF LINEAR EQUATIONS

2.4 SOLUTION OF A SYSTEM OF LINEAR EQUATIONS

EXERCISE 2

SOLUTIONS 2

CHAPTER 3 EIGENVALUES AND EIGENVECTORS

3.1 INTRODUCTION

3.2 EIGENVALUES AND EIGEN VECTOR

3.3 DETERMINATION OF EIGENVALUES AND EIGENVECTORS

3.4 CAYLEY-HAMILTON THEOREM

3.4.1 Computation of the Inverse Using Cayley-Hamilton Theorem

3.5 REDUCTION OF A MATRIX TO DIAGONAL FORM

3.6 SIMILARITY OF MATRICES

EXERCISE 3

SOLUTIONS 3

CHAPTER 4 LIMIT, CONTINUITY AND DIFFERENTIABILITY

4.1 INTRODUCTION

4.2 LIMIT OF A FUNCTION

4.2.1 Left Handed Limit

4.2.2 Right Handed Limit

4.2.3 Existence of Limit at Point

4.2.4 Lâ€™ Hospitalâ€™s Rule

4.3 CONTINUITY OF A FUNCTION

4.3.1 Continuity in an Interval

4.4 DIFFERENTIABILITY

EXERCISE 4

SOLUTIONS 4

CHAPTER 5 MAXIMA AND MINIMA

5.1 INTRODUCTION

5.2 MONOTONOCITY

5.3 MAXIMA AND MINIMA

EXERCISE 5

SOLUTIONS 5

CHAPTER 6 MEAN VALUE THEOREM

6.1 INTRODUCTION

6.2 ROLLEâ€™S THEOREM

6.3 LAGRANGEâ€™S MEAN VALUE THEOREM

6.4 CAUCHYâ€™S MEAN VALUES THEOREM

EXERCISE 6

SOLUTIONS 6

CHAPTER 7 PARTIAL DERIVATIVES

7.1 INTRODUCTION

7.2 PARTIAL DERIVATIVES

7.2.1 Partial Derivatives of Higher Orders

7.3 TOTAL DIFFERENTIATION

7.4 CHANGE OF VARIABLES

7.5 DIFFERENTIATION OF IMPLICIT FUNCTION

7.6 EULERâ€™S THEOREM

EXERCISE 7

SOLUTIONS 7

CHAPTER 8 DEFINITE INTEGRAL

8.1 INTRODUCTION

8.2 DEFINITE INTEGRAL

8.3 IMPORTANT FORMULA FOR DEFINITE INTEGRAL

8.4 DOUBLE INTEGRAL

EXERCISE 8

SOLUTIONS 8

CHAPTER 9 DIRECTIONAL DERIVATIVES

9.1 INTRODUCTION

9.2 DIFFERENTIAL ELEMENTS IN COORDINATE SYSTEMS

9.3 DIFFERENTIAL CALCULUS

9.4 GRADIENT OF A SCALAR

9.5 DIVERGENCE OF A VECTOR

9.6 CURL OF A VECTOR

9.7 CHARACTERIZATION OF A VECTOR FIELD

9.8 LAPLACIAN OPERATOR

9.9 INTEGRAL THEOREMS

9.9.1 Divergence Theorem

9.9.2 Stokeâ€™s Theorem

9.9.3 Greenâ€™s Theorem

9.9.4 Helmholtzâ€™s Theorem

EXERCISE 9

SOLUTIONS 9

CHAPTER 10 FIRST ORDER DIFFERENTIAL EQUATIONS

10.1 INTRODUCTION

10.2 DIFFERENTIAL EQUATION

10.2.1 Ordinary Differential Equation

10.2.2 Order of a Differential Equation

10.2.3 Degree of a Differential Equation

10.3 DIFFERENTIAL EQUATION OF FIRST ORDER AND FIRST DEGREE

10.4 SOLUTION OF A DIFFERENTIAL EQUATION

10.5 VARIABLES SEPARABLE FORM

10.5.1 Equations Reducible to Variable Separable Form

10.6 HOMOGENEOUS EQUATIONS

10.6.1 Equations Reducible to Homogeneous Form

10.7 LINEAR DIFFERENTIAL EQUATION

10.7.1 Equations Reducible to Linear Form

10.8 BERNOULLIâ€™S EQUATION

10.9 EXACT DIFFERENTIAL EQUATION

10.9.1 Necessary and Sufficient Condition for Exactness

10.9.2 Solution of an Exact Differential Equation

10.9.3 Equations Reducible to Exact Form: Integrating Factors

10.9.4 Integrating Factors Obtained by Inspection

EXERCISE 10

SOLUTIONS 10

CHAPTER 11 HIGHER ORDER DIFFERENTIAL EQUATIONS

11.1 INTRODUCTION

11.2 LINEAR DIFFERENTIAL EQUATION

11.2.1 Operator

11.2.2 General Solution of Linear Differential Equation

11.3 DETERMINATION OF COMPLEMENTARY FUNCTION

11.4 PARTICULAR INTEGRAL

11.4.1 Determination of Particular Integral

11.5 HOMOGENEOUS LINEAR DIFFERENTIAL EQUATION

11.6 EULER EQUATION

EXERCISE 11

SOLUTIONS 11

CHAPTER 12 INITIAL AND BOUNDARY VALUE PROBLEMS

12.1 INTRODUCTION

12.2 INITIAL VALUE PROBLEMS

12.3 BOUNDARY-VALUE PROBLEM

EXERCISE 12

SOLUTIONS 12

CHAPTER 13 PARTIAL DIFFERENTIAL EQUATION

13.1 INTRODUCTION

13.2 PARTIAL DIFFERENTIAL EQUATION

13.2.1 Partial Derivatives of First Order

13.2.2 Partial Derivatives of Higher Order

13.3 HOMOGENEOUS FUNCTIONS

13.4 EULERâ€™S THEOREM

13.5 COMPOSITE FUNCTIONS

13.6 ERRORS AND APPROXIMATIONS

EXERCISE 13

SOLUTIONS 13

CHAPTER 14 ANALYTIC FUNCTIONS

14.1 INTRODUCTION

14.2 BASIC TERMINOLOGIES IN COMPLEX FUNCTION

14.3 FUNCTIONS OF COMPLEX VARIABLE

14.4 LIMIT OF A COMPLEX FUNCTION

14.5 CONTINUITY OF A COMPLEX FUNCTION

14.6 DIFFERENTIABILITY OF A COMPLEX FUNCTION

14.6.1 Cauchy-Riemann Equation: Necessary Condition for Differentiability of a Complex Function

14.6.2 Sufficient Condition for Differentiability of a Complex Function

14.7 ANALYTIC FUNCTION

14.7.1 Required Condition for a Function to be Analytic

14.8 HARMONIC FUNCTION

14.8.1 Methods for Determining Harmonic Conjugate

14.8.2 Milne-Thomson Method

14.8.3 Exact Differential Method

14.9 SINGULAR POINTS

EXERCISE 14

SOLUTIONS 14

CHAPTER 15 CAUCHYâ€™S INTEGRAL THEOREM

15.1 INTRODUCTION

15.2 LINE INTEGRAL OF A COMPLEX FUNCTION

15.2.1 Evaluation of the Line Integrals

15.3 CAUCHYâ€™S THEOREM

15.3.1 Cauchyâ€™s Theorem for Multiple Connected Region

15.4 CAUCHYâ€™S INTEGRAL FORMULA

15.4.1 Cauchyâ€™s Integral Formula for Derivatives

EXERCISE 15

SOLUTIONS 15

CHAPTER 16 TAYLORâ€™S AND LAURENTâ€™ SERIES

16.1 INTRODUCTION

16.2 TAYLORâ€™S SERIES

16.3 MACLAURINâ€™S SERIES

16.4 LAURENTâ€™S SERIES

16.5 RESIDUES

16.5.1 The Residue Theorem

16.5.2 Evaluation of Definite Integral

EXERCISE 16

SOLUTIONS 16

CHAPTER 17 PROBABILITY

17.1 INTRODUCTION

17.2 SAMPLE SPACE

17.3 EVENT

17.3.1 Algebra of Events

17.3.2 Types of Events

17.4 DEFINITION OF PROBABILITY

17.4.1 Classical Definition

17.4.2 Statistical Definition

17.4.3 Axiomatic Definition

17.5 PROPERTIES OF PROBABILITY

17.5.1 Addition Theorem for Probability

17.5.2 Conditional Probability

17.5.3 Multiplication Theorem for Probability

17.5.4 Odds for an Event

17.6 BAYEâ€™S THEOREM

EXERCISE 17

SOLUTIONS 17

CHAPTER 18 RANDOM VARIABLE

18.1 INTRODUCTION

18.2 RANDOM VARIABLE

18.2.1 Discrete Random Variable

18.2.2 Continuous Random Variable

18.3 EXPECTED VALUE

18.3.1 Expectation Theorems

18.4 MOMENTS OF RANDOM VARIABLES AND VARIANCE

18.4.1 Moments about the Origin

18.4.2 Central Moments

18.4.3 Variance

18.5 BINOMIAL DISTRIBUTION

18.5.1 Mean of the Binomial Distribution

18.5.2 Variance of the Binomial Distribution

18.5.3 Fitting of Binomial Distribution

18.6 POISSON DISTRIBUTION

18.6.1 Mean of Poisson Distribution

18.6.2 Variance of Poisson Distribution

18.6.3 Fitting of Poisson Distribution

18.7 NORMAL DISTRIBUTION

18.7.1 Mean and Variance of Normal Distribution

EXERCISE 18

SOLUTIONS 18

CHAPTER 19 STATISTICS

19.1 INTRODUCTION

19.2 MEAN

19.3 MEDIAN

19.4 MODE

19.5 MEAN DEVIATION

19.6 VARIANCE AND STANDARD DEVIATION

EXERCISE 19

SOLUTIONS 19

CHAPTER 20 CORRELATION AND REGRESSION ANALYSIS

20.1 INTRODUCTION

20.2 CORRELATION

20.3 MEASURE OF CORRELATION

20.3.1 Scatter or Dot Diagrams

20.3.2 Karl Pearsonâ€™s Coefficient of Correlation

20.3.3 Computation of Correlation Coefficient

20.4 RANK CORRELATION

20.5 REGRESSION

20.5.1 Lines of Regression

20.5.2 Angle Between Two Lines of Regression

EXERCISE 20

SOLUTIONS 20

CHAPTER 21 SOLUTIONS OF NON-LINEAR ALGEBRAIC EQUATIONS

21.1 INTRODUCTION

21.2 SUCCESSIVE BISECTION METHOD

21.3 FALSE POSITION METHOD (REGULA-FALSI METHOD)

21.4 NEWTON - RAPHSON METHOD (TANGENT METHOD)

EXERCISE 21

SOLUTIONS 21

CHAPTER 22 INTEGRATION BY TRAPEZOIDAL AND SIMPSONâ€™S RULE

22.1 INTRODUCTION

22.2 NUMERICAL DIFFERENTIATION

22.2.1 Numerical Differentiation Using Newtonâ€™s Forward Formula

22.2.2 Numerical Differentiation Using Newtonâ€™s Backward Formula

22.2.3 Numerical Differentiation Using Central Difference Formula

22.3 MAXIMA AND MINIMA OF A TABULATED FUNCTION

22.4 NUMERICAL INTEGRATION

22.4.1 Newton-Coteâ€™s Quadrature Formula

22.4.2 Trapezoidal Rule

22.4.3 Simpsonâ€™s One-Third Rule

22.4.4 Simpsonâ€™s Three-Eighth Rule

EXERCISE 22

SOLUTIONS 22

CHAPTER 23 SINGLE AND MULTI STEP METHODS FOR DIFFERENTIAL EQUATIONS

23.1 INTRODUCTION

23.2 PICARDâ€™S METHOD

23.3 EULERâ€™S METHOD

23.3.1 Modified Eulerâ€™s Method

23.4 RUNGE-KUTTA METHODS

23.4.1 Runge-Kutta First Order Method

23.4.2 Runge-Kutta Second Order Method

23.4.3 Runge-Kutta Third Order Method

23.4.4 Runge-Kutta Fourth Order Method

23.5 MILNEâ€™S PREDICTOR AND CORRECTOR METHOD

23.6 TAYLORâ€™S SERIES METHOD

EXERCISE 23

SOLUTIONS 23

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