**Engineering Mathematics** incorporates in one volume the material covered in the mathematics course of undergraduate programmes in engineering and technology.

In this revised edition, five new chapters on solutions of differential equations in series, beta and gamma functions, analytical geometry in three dimensions and complex analysis have been added in keeping with the current engineering curriculum. The existing chapters have been revised and several new worked-out examples have been included.

**Koneru Sarveswara Rao**obtained his Ph D in mathematics from the University of Saskatchewan, Canada. For about thirteen years he provided professional advice to scientists and engineers in computer-based mathematical problem solving using numerical methods. He taught at the department of mathematics, IIT Mumbai for sixteen years and guided several Ph D students.

*Foreword *

*Preface to Second Edition *

*Preface to First Edition *

**1 Sequences and Infinite Series **

1.1 Sequences

1.2 Infinite Series

1.3 Geometric Series

1.4 Test for Divergence

1.5 Comparison Test for Non-negative Series

1.6 Integral Test

1.7 Limit Form of the Comparison Test

1.8 Ratio Test

1.9 Root Test

1.10 Comparison of Ratios Test

1.11 Raabe’s Test

1.12 Logarithmic Test

1.13 Leibnitz’s Test for Alternating Series

1.14 Absolute and Conditional Convergence

1.15 Suggestions for Choosing an Appropriate Test

**2 Mean Value Theorems, Envelopes and Evolutes **

2.1 Introduction

2.2 Rolle’s Theorem

2.3 Taylor’s Series and Maclaurin’s Series

2.4 Envelopes

2.5 Radius of Curvature

**3 Ordinary Differential Equations of First Order **

3.1 Introduction

3.2 Forming a Differential Equation

3.3 Separable Equations

3.4 Equations Reducible to Separable Forms

3.5 Linear Equations

3.6 Bernoulli Equation

3.7 Exact Equations

3.8 Equations that can be made Exact by Integrating Factors

3.9 Application to Problems of Geometry

3.10 Orthogonal Trajectories

3.11 Newton’s Law of Cooling

3.12 Law of Natural Growth or Decay

**4 Linear Differential Equations of Second and Higher Order **

4.1 Introduction

4.2 Complementary Function for Equations with Constant Coefficients

4.3 Particular Integral

4.4 Particular Integral by the Method of Undetermined Coefficients

4.5 Particular Integral by the Method of Variation of Parameters

4.6 Equation with Variable Coefficients

4.7 Simultaneous Equations with Constant Coefficients

4.8 Second Solution by Reduction of Order Method

**5 Laplace Transforms **

5.1 Introduction

5.2 Laplace Transforms of some Elementary Functions

5.3 Transforms of Derivatives

5.4 Laplace Transform of the Integral of *f*(*t*)

5.5 Laplace Transform of *tf*(*t*)

5.6 Unit Step Function: *t*-Shifting

5.7 Convolution 122

5.8 Laplace Transform of Periodic Functions 124

5.9 Applications 126

**6 Solution of Differential Equations in Power Series 131**

6.1 Introduction 131

6.2 Power Series Solutions 132

6.3 Legendre Polynomials 135

6.4 Recursion Relation (Bonnet’s Relation) 138

6.5 Frobenius Method 139

6.6 Bessel Functions 144

6.7 Interlacing of Zeros of Bessel Functions of Integral Order 149

**7 Beta and Gamma Functions 153**

7.1 Introduction 153

7.2 Beta and Gamma Functions 153

7.3 Gamma Function

7.4 Relation between Beta and Gamma Functions 155

**8 Analytical Geometry in Three Dimensions 164**

8.1 Introduction 164

8.2 Distance Between Two Points and Related Results 164

8.3 Direction Cosines and Ratios of a Line 167

8.4 Straight Line 174

8.5 The Plane 178

8.6 Shortest Distance between Two Skew Lines 193

8.7 Right Circular Cone 195

8.8 Right Circular Cylinder 200

**9 Functions of Several Variables 205**

9.1 Introduction 205

9.2 Limit and Continuity of a Function 205

9.3 Partial Derivatives 206

9.4 Homogeneous Functions**–**Euler’s Theorem 212

9.5 Change of Variables: Chain Rule 216

9.6 Jacobians 220

9.7 Taylor’s Theorem for Functions of Two Variables 225

9.8 Maxima and Minima of Functions of Two Variables 226

**10 Curve Tracing and Some Properties of Polar Curves 233**

10.1 Introduction 233

10.2 Curves in Cartesian Coordinates: *f*(*x, y*) = 0 233

10.3 Curves in Parametric Form: *x *= *f*(*t*); *y *= *g*(*t*) 237

10.4 Polar Curves: *f*(*r, ?*) = 0 239

10.5 Properties of Polar Curves 242

**11 Lengths, Volumes, Surface Areas and Multiple Integrals 244**

11.1 Introduction 244

11.2 Length of a Plane Curve 244

11.3 Volume of Revolution 247

11.4 Surface Area of Revolution 250

11.5 Double Integrals 252

11.6 Triple Integrals 260

**12 Vector Calculus 263**

12.1 Scalar Fields and Vector Fields 263

12.2 Curvature and Torsion of a Curve in Space 264

12.3 Velocity and Acceleration of a Particle 267

12.4 Directional Derivative: Gradient of a Scalar Field 269

12.5 Divergence and Curl of a Vector Field 272

12.6 Line Integrals

12.7 Green’s Theorem in the Plane 277

12.8 Surface Integrals 281

12.9 Gauss Divergence Theorem 284

12.10Stoke’s Theorem 289

12.11Irrotational Fields and Potentials 292

**13 Matrices and Linear Systems 295**

13.1 Introduction 295

13.2 Sub-matrices and Partitions of a Matrix 296

13.3 Rank of a Matrix 297

13.4 Elementary Operations and Matrices 299

13.5 Normal Form 301

13.6 Inverse of a Matrix by Gauss**–**Jordan Method 303

13.7 Linear Independence of Vectors 304

13.8 Linear Systems: Properties of Solution 306

**14 Eigen Values and Eigen Vectors 317**

14.1 Linear Transformations 317

14.2 Eigen Values and Eigen Vectors 318

14.3 Some Properties of Eigen Values 322

14.4 Cayley**–**Hamilton Theorem 325

14.5 Similar Matrices 327

14.6 Diagonalisation of a Matrix 328

14.7 Quadratic Forms 337

14.8 A Canonical Form using the Normal Form of the Matrix 339

**15 Fourier Series 343**

15.1 Orthogonal Functions: General Introduction 343

15.2 Introduction to Trigonometric Fourier Series 344

15.3 Fourier Coefficients 345

15.4 Functions with any Period *T *354

15.5 Half Range Expansions 358

**16 Complex Analysis 362**

16.1 Complex Numbers and Functions 362

16.2 Analytic Functions and Cauchy**–**Riemann Equations 370

16.3 Laplace Equation, Harmonic Functions and Conjugate Functions 372

16.4 Conformal Mapping 380

16.5 Complex Integration 390

16.6 Cauchy’s Integral Theorem 393

16.7 Cauchy’s Integral Formula 398

16.8 Power Series, Taylor’s Series 406

16.9 Laurent’s Series

16.10Singularities and Zeros 415

16.11Integration using Residues 418

16.12Evaluation of Real Integrals 424

**17 Partial Differential Equations 438**

17.1 Introduction 438

17.2 Formation of Partial Differential Equations 439

17.3 Solution of Partial Differential Equations 442

17.4 Lagrange’s Equations 443

17.5 Solutions of Some Standard Types of Equations 448

17.6 General Method of Finding Solutions: Charpit’sMethod 452

17.7 Particular Integrals from Complete Integrals 453

17.8 Homogeneous Linear Equations with Constant Coefficents 458

**18 Applications of Partial Differential Equations 466**

18.1 Introduction 466

18.2 One-dimensional Heat Equation 467

18.3 One-dimensionalWave Equation 477

18.4 Two-dimensional Laplace Equation 485

18.5 Laplace Equation in Polar Coordinates 493

**19 Fourier and ***Z***-transforms 498**

19.1 Introduction 498

19.2 *Z*-transform 498

19.3 Some Properties of a *Z*-transform 500

19.4 Inverse *Z*-transforms 504

19.5 Solution of Difference Equations 510

19.6 Fourier Transforms 511

19.7 Solution of Differential Equations using Fourier Transforms 522

**20 Probability 530**

20.1 Introduction 530

20.2 Algebra of Sets 530

20.3 Random Experiments, Sample Spaces, Outcomes and Events 531

20.4 Probability 532

20.5 Conditional Probability 533

20.6 Bayes’ Theorem 534

**21 Random Variables and Probability Distributions 545**

21.1 Density and Distribution Function 545

21.2 Continuous Random Variable and its Distribution 546

21.3 Expectation and Variance 550

21.4 Chebyshev’s Inequality 553

21.5 Binomial Distribution

21.6 Poisson Distribution 555

21.7 Normal Distribution for Continuous Variable 557

**22 Joint Distributions 564**

22.1 Discrete Variables 564

22.2 Expectation, Variance and Covariance of Joint Distributions 566

22.3 Conditional Distribution 568

22.4 Distribution of the Sum of Two Random Variables 574

22.5 Functions of Random Variables 578

**23 Sampling Distributions 584**

23.1 Random Samples from Populations 584

23.2 Sampling Distribution of the Mean when the Variance is Known 585

23.3 Sampling Distribution of the Mean when the Population Variance is Unknown 588

23.4 Sampling Distribution of Difference of TwoMeans 588

23.5 Sampling Distribution of a Single Proportion 591

23.6 Sampling Distribution of the Difference of Two Proportions 593

23.7 Sampling Distribution for Several Proportions: *?*2 Distribution 594

23.8 Sampling Distribution of the Variance with a Known Population Variance 596

23.9 Sampling Distribution of the Ratio of Two Sample Variances 596

23.10Contingency Tables: *?*2 Distribution 599

23.11Testing the Goodness of Fit of a Distribution to Observed Data 600

**24 Statistical Estimation and Inference 603**

24.1 Introduction 603

24.2 Estimation of Population Parameters 603

24.3 Interval Estimates: Confidence Intervals 607

24.4 Testing of Hypotheses 608

24.5 Operating Characteristic Curves 620

**25 Curve Fitting, Regression and Correlation 624**

25.1 Introduction 624

25.2 Regression Line 625

25.3 Residual Sum of Squares and Correlation Coefficient 627

25.4 Polynomial Regression 629

25.5 Multiple Regression 631

**26 Numerical Methods 634**

26.1 Introduction 634

26.2 Solution of Non-linear Equations 634

26.3 Solution of a Linear System of Equations 640

26.4 Interpolation 646

26.5 Numerical Differentiation 656

26.6 Numerical Integration 657

26.7 Solution of Ordinary Differential Equations (ODE) 662

**27 Epilogue 670**

27.1 Introduction 670

27.2 Summation of Series 670

27.3 Modelling using Second Order Equation 671

27.4 Fourier Transform in Signal Processing 674

27.5 Problem Solving in Real Life 674

*Appendix *689

*Reference *696

*Index *697