Details

Mathematica for Physicists and Engineers


Mathematica for Physicists and Engineers


1. Aufl.

von: K. B. Vijaya Kumar, Antony P. Monteiro

75,99 €

Verlag: Wiley-VCH
Format: PDF
Veröffentl.: 26.05.2023
ISBN/EAN: 9783527843237
Sprache: englisch
Anzahl Seiten: 416

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Beschreibungen

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

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