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.
Sarveswara Rao Koneru 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, θ) = 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: χ2 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: χ2 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