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