Preface xiii
Foreword xvii
About the Authors xix
1 Preliminary Notions 1
1.1 Introduction 1
1.2 Versions of Mathematica 1
1.3 Getting Started 2
1.4 Simple Calculations 2
1.4.1 Arithmetic Operations 2
1.4.2 Approximate Numerical Results 3
1.4.3 Algebraic Calculations 3
1.4.4 Defining Variables 4
1.4.5 Using the Previous Results 5
1.4.6 Suppressing the Output 6
1.4.7 Sequences of Operations 6
1.5 Built-in Functions 7
1.6 Additional Features 9
1.6.1 Arbitrary-Precision Calculations 9
1.6.2 Value for Symbols 10
1.6.3 Defining Naming and Evaluating Functions 10
1.6.4 Composition of Functions 11
1.6.5 Conditional Assignment 12
1.6.6 Warnings and Messages 13
1.6.7 Interrupting Calculations 13
1.6.8 Using Symbols to Tag Objects 13
2 Basic Mathematical Operations 15
2.1 Introduction 15
2.2 Basic Algebraic Operations 15
2.3 Basic Trigonometric Operations 20
2.4 Basic Operations with Complex Numbers 21
3 Lists and Tables 25
3.1 Introduction 25
3.2 Lists 25
3.3 Arrays 26
3.4 Tables 26
3.5 Extracting the Elements from the Arrays/Tables 29
4 Two-Dimensional Graphics 31
4.1 Introduction 31
4.2 Plotting Functions of a Single Variable 31
4.3 Additional Commands 34
4.4 Plot Styles 44
4.5 Probability Distribution 58
4.5.1 Binomial Distribution 58
4.5.2 Poisson Distribution 58
4.5.3 Normal or Gaussian Distribution 59
4.6 Some More Useful Commands 61
5 Parametric, Polar, Contour, Density, and List Plots 65
5.1 Introduction 65
5.2 Parametric Plotting 65
5.3 Polar Plots 72
5.3.1 Polar Plots of Circles 72
5.3.2 Polar Plots of Ellipse, Parabola, and Hyperbola 72
5.4 Implicit Plot 80
5.5 Contour Plots 81
5.6 Density Plot 85
5.7 ListPlot and ListLinePlot 85
5.8 LogPlot, LogLogPlot, ErrorListPlot 88
5.9 Least Square Fit 89
5.10 Plotting of Complex Numbers 92
6 Three-Dimensional Graphics 97
6.1 Introduction 97
6.2 Plotting Function of Two Variables 97
6.3 Parametric Plots 101
6.4 3D Plots in Cylindrical and Spherical Coordinates 102
6.5 ContourPlot3D 105
6.6 ListContourPlot3D 108
6.7 ListSurfacePlot3D 110
6.8 Surface of Revolution 112
6.9 Conicoids 114
7 Matrices 123
7.1 Introduction 123
7.2 Properties of Matrices 123
7.2.1 Matrix Multiplication 123
7.3 Types of Matrices 123
7.4 The Rank of the Matrix 124
7.5 Special Matrices 124
7.6 Creation of a Matrix and Matrix Operations 125
7.6.1 Extraction of the Submatrices or the Elements of the Matrices 126
7.7 Properties of the Special Matrices 133
7.8 Direct Sum of Matrices 137
7.9 Direct Product of Matrices 137
7.10 Examples from Group Theory 138
7.10.1 SO(3) Group 138
7.10.2 SU(n)Group 139
7.10.3 SU(2) Group 140
7.10.4 SU(3) Group 141
8 Solving Algebraic and Transcendental Equations 143
8.1 Introduction 143
8.2 Solving System of Linear Equations 143
8.2.1 Number of Equations Equal to Number of Unknowns 144
8.2.2 Number of Equations Less than the Number of Unknowns 146
8.2.3 Number of Equations More than Number of Unknowns 146
8.3 Nonlinear Algebraic Equations 147
8.4 Solving Complex Equations 149
8.5 Solving Transcendental Equations 153
9 Eigenvalues and Eigenvectors of a Matrix 161
9.1 Introduction 161
9.2 Eigenvalues and Eigenvectors 161
9.2.1 Distinct Eigenvalues Having Independent Eigenvectors 162
9.2.2 Multiple Eigenvalues Having Independent Eigenvectors 163
9.2.3 Multiple Eigenvalues Not Having Independent Eigenvectors 165
9.3 Cayley–Hamilton Theorem 166
9.4 Diagonalization of a Matrix 167
9.4.1 Gram–Schmidt Orthogonalization Method 167
9.4.2 Diagonalizability of a Matrix 169
9.4.3 Case of a Non-diagonalizable Matrix 170
9.5 Some More Properties of the Special Matrices 172
9.6 Power of a Matrix 173
9.6.1 Roots of a Matrix 174
9.6.2 Exponential of a Matrix 174
9.6.3 Logarithm of a Matrix 174
9.6.4 Matrix Power Series 174
9.7 Power of a Matrix by Diagonalization 174
9.8 Bilinear, Quadratic, and Hermitian Forms 177
9.9 Principal Axes Transformation 178
10 Differential Calculus 183
10.1 Introduction 183
10.2 Limits 183
10.2.1 Evaluation of the Limits Using L’Hospital’s Rule 184
10.2.2 Application of L’Hospital’s Rule for the “Indeterminate Form” ∞ 185 ∞
10.2.3 Evaluation of the Limit Using Taylor’s Theorem of Mean 186
10.3 Differentiation 188
10.3.1 Computation of Partial Derivatives 191
10.3.2 Total Derivative 193
10.4 Derivatives of Functions in Parametric Forms 195
10.4.1 Chain Rule for a Function of Two Independent Variables 196
10.4.2 Chain Rule for a Function of Three Independent Variables 196
10.5 Rolle’s Theorem 198
10.6 Mean Value Theorem 198
10.7 Series 200
10.8 Maxima and Minima 209
10.8.1 First Derivative Test 210
10.8.2 Second Derivative Test 211
10.8.3 Maximum and Minimum Values of a Function in a Closed Interval 213
10.8.4 Maxima and Minima of Two Variables 218
10.9 Differential Equations 222
10.9.1 Simple Harmonic Oscillator 225
10.9.2 LCR Circuit – Discharging of a Condenser Through an LR Circuit 227
11 Integral Calculus 235
11.1 Introduction 235
11.1.1 Indefinite Integral 235
11.1.2 Definite Integral 235
11.1.3 Numerical Value of the Integral 235
11.1.4 Assumptions While Evaluating the Integral 236
11.1.5 Multiple Integrals 236
11.1.6 Triple Integral 236
11.2 Evaluation of Indefinite Integrals 236
11.3 Evaluation of Definite Integrals 238
11.3.1 Numerical Value of the Integral 238
11.3.2 Options for Integration 239
11.4 Two and Three-Dimensional Integrals 240
11.5 Evaluation of the Integral in Polar Coordinates 242
11.6 Evaluation of Special Integrals 242
11.7 Orthogonal Polynomials 248
11.8 Area Between Curves 252
11.9 Application of Green’s Theorem in a Plane 256
11.10 Area of Surfaces of Revolution 257
12 Dirac Delta Function 263
12.1 Introduction 263
12.2 The Limiting Form of the Dirac Delta Function 263
12.3 Integral Representation of the Dirac Delta Function 265
12.4 Some Important Properties of the Dirac Delta Function 267
12.5 The Three-Dimensional Dirac Delta Function 270
13 Fourier Transforms 273
13.1 Introduction 273
13.2 Fourier Transforms 273
13.3 Scaling Property 280
13.4 Shifting Property 280
13.5 Fourier Sine and Cosine Transforms 281
13.6 Fourier Transform of the Derivative 282
13.7 Inverse Fourier Transform 282
13.8 Convolution 283
13.9 Convolution Theorem for Fourier Transforms 291
13.10 Parseval’s Theorem 293
14 Laplace Transforms 295
14.1 Introduction 295
14.2 Some Simple Examples 296
14.3 Properties of the Laplace Transforms 297
14.3.1 Linearity 297
14.3.2 Shifting Property 297
14.3.3 Scaling Property 297
14.4 Laplace Transform of the Derivative 298
14.5 Laplace Transform of Certain Special Functions 299
14.6 The Laplace Transform of Error and Complementary Error Functions 300
14.7 The Evaluation of a Certain Class of Definite Integrals Using Laplace Transforms 300
14.8 The Inverse Laplace Transform 302
14.8.1 Inverse Laplace Transform of Standard Functions 303
14.8.2 Shifting Properties 303
14.8.3 Inverse Laplace Transforms of Derivatives 305
14.9 Solving the Differential Equation by Laplace Transform 306
14.10 Convolution Theorem 307
14.11 Graphical Treatment of the Convolution 308
15 Vectors 315
15.1 Introduction 315
15.2 Properties 315
15.3 Vector Differentiation 319
15.4 Directional Derivative 320
15.5 Unit Vector Normal to the Surface 320
15.6 Gradient, Divergence, and Curl in the Cartesian Coordinate System 320
15.6.1 Gradient 320
15.6.2 Divergence 321
15.6.3 Curl 321
15.6.4 Laplacian Operator (∇ 2) 321
15.6.5 Examples 322
15.7 Expressing the Gradient, Divergence, and Curl in Other Coordinate Systems 326
15.7.1 Spherical Coordinate System 326
15.7.2 Cylindrical Coordinate System 330
15.8 Vector Plots 337
16 Linear Vector Spaces and Quantum Mechanics 343
16.1 Introduction 343
16.2 Linear Independence, Basis, and Dimension 343
16.3 Dimension of the Vector Space 343
16.4 Basis of the Vector Space 343
16.5 Completeness 344
16.6 Scalar Product in a Linear Vector Space 344
16.7 Norm of the Vector 344
16.8 Orthonormal Basis 344
16.9 Linear Independence of Functions 348
16.10 Hilbert Space 349
16.11 Completeness in Functional Space 350
16.12 The Dirac Ket and Bra Notation 351
16.12.1 The Scalar Product of Kets and Bras 351
16.12.2 Schwartz Inequality 352
16.12.3 The Orthonormal States 352
16.12.4 Basis 352
16.12.5 Probability Density 352
16.13 The Hermitian and Skew-Hermitian Operators in Dirac Ket and Bra Notation 352
16.14 Expectation Values 353
16.15 Matrix Representation of the Linear Operator 359
17 Application of Mathematica to Quantum Mechanics 361
17.1 Introduction 361
17.2 A Particle in a One-Dimensional Box 361
17.3 A Particle in a Two-Dimensional Box 365
17.4 The Hydrogen Atom Problem 368
17.4.1 The Orthonormal Property of the Hydrogen Atom Wave Functions 371
17.5 The One-Dimensional Linear Harmonic Oscillator Atom Problem 373
17.6 Three-Dimensional Harmonic Oscillator 377
17.7 Miscellaneous Problems 382
References 385
Index 387
