| Description |
1 online resource (xx, 820 pages) |
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1 online resource (1 CD-ROM (4 3/4 in.) |
| Contents |
Machine generated contents note: 1.1. Algebra -- 1.1.1. Systems of Equations -- 1.1.2.Completing the Square -- 1.1.3.Common Denominator -- 1.1.4. Partial Fractions Decomposition -- 1.1.5. Inverse Functions -- 1.1.6. Exponential and Logarithmic Equations -- 1.1.7. Logarithms of Powers, Products and Ratios -- 1.1.8. Radioactive Decay -- 1.1.9. Transcendental Equations -- 1.1.10. Even and Odd Functions -- 1.1.11. Examples in Maple -- 1.2. Trigonometry -- 1.2.1. Polar Coordinates -- 1.2.2.Common Identities -- 1.2.3. Law of Cosines -- 1.2.4. Systems of Equations -- 1.2.5. Transcendental Equations -- 1.3.Complex Numbers -- 1.3.1.Complex Roots -- 1.3.2.Complex Arithmetic -- 1.3.3.Complex Conjugate -- 1.3.4. Euler's Formula -- 1.3.5.Complex Plane -- 1.3.6. Polar Form of Complex Numbers -- 1.3.7. Powers of Complex Numbers -- 1.3.8. Hyperbolic Functions -- 1.4. Elements of Calculus -- 1.4.1. Derivatives -- 1.4.2. Prime and Dot Notation -- 1.4.3. Chain Rule for Derivatives |
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Note continued: 1.4.4. Product Rule for Derivatives -- 1.4.5. Quotient Rule for Derivatives -- 1.4.6. Indefinite Integrals -- 1.4.7. Definite Integrals -- 1.4.8.Common Integrals and Derivatives -- 1.4.9. Derivatives of Trigonometric and Hyperbolic Functions -- 1.4.10. Euler's Formula -- 1.4.11. Integrals of Trigonometric and Hyperbolic Functions -- 1.4.12. Improper Integrals -- 1.4.13. Integrals of Even and Odd Functions -- 1.5. MATLAB Examples -- 1.5.1. Functional Calculator -- 1.6. Exercises -- 2.1. Vectors and Scalars in Physics -- 2.1.1. Vector Addition and Unit Vectors -- 2.1.2. Scalar Product of Vectors -- 2.1.3. Vector Cross Product -- 2.1.4. Triple Vector Products -- 2.1.5. The Position Vector -- 2.1.6. Expressing Vectors in Different Coordinate Systems -- 2.2. Matrices in Physics -- 2.2.1. Matrix Dimension -- 2.2.2. Matrix Addition and Subtraction -- 2.2.3. Matrix Integration and Differentiation -- 2.2.4. Matrix Multiplication and Commutation -- 2.2.5. Direct Product |
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Note continued: 2.2.6. Identity Matrix -- 2.2.7. Transpose of a Matrix -- 2.2.8. Symmetric and Antisymmetric Matrices -- 2.2.9. Diagonal Matrix -- 2.2.10. Tridiagonal Matrix -- 2.2.11. Orthogonal Matrices -- 2.2.12.Complex Conjugate of a Matrix -- 2.2.13. Matrix Adjoint (Hermitian Conjugate) -- 2.2.14. Unitary Matrix -- 2.2.15. Partitioned Matrix -- 2.2.16. Matrix Trace -- 2.2.17. Matrix Exponentiation -- 2.3. Matrix Determinant and Inverse -- 2.3.1. Matrix Inverse -- 2.3.2. Singular Matrices -- 2.3.3. Systems of Equations -- 2.4. Eigenvalues and Eigenvectors -- 2.4.1. Matrix Diagonalization -- 2.5. Rotation Matrices -- 2.5.1. Rotations in Two Dimensions -- 2.5.2. Rotations in Three Dimensions -- 2.5.3. Infinitesimal Rotations -- 2.6. MATLAB Examples -- 2.7. Exercises -- 3.1. Single-Variable Calculus -- 3.1.1. Critical Points -- 3.1.2. Integration with Substitution -- 3.1.3. Work-Energy Theorem -- 3.1.4. Integration by Parts -- 3.1.5. Integration with Partial Fractions |
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Note continued: 3.1.6. Integration by Trig Substitution -- 3.1.7. Differentiating Across the Integral Sign -- 3.1.8. Integrals of Logarithmic Functions -- 3.2. Multivariable Calculus -- 3.2.1. Partial Derivatives -- 3.2.2. Critical Points -- 3.2.3. Double Integrals -- 3.2.4. Triple Integrals -- 3.2.5. Orthogonal Coordinate Systems -- 3.2.6. Cartesian Coordinates -- 3.2.7. Cylindrical Coordinates -- 3.2.8. Spherical Coordinates -- 3.2.9. Line, Volume, and Surface Elements -- 3.3. Gaussian Integrals -- 3.3.1. Error Functions -- 3.4. Series and Approximations -- 3.4.1. Geometric Series -- 3.4.2. Taylor Series -- 3.4.3. Maclaurin Series -- 3.4.4. Index Labels -- 3.4.5. Convergence of Series -- 3.4.6. Ratio Test -- 3.4.7. Integral Test -- 3.4.8. Binomial Theorem -- 3.4.9. Binomial Approximations -- 3.5. Special Integrals -- 3.5.1. Integral Functions -- 3.5.2. Elliptic Integrals -- 3.5.3. Gamma Functions -- 3.5.4. Riemann Zeta Function -- 3.5.5. Writing Integrals in Dimensionless Form |
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Note continued: 3.5.6. Black-Body Radiation -- 3.6. MATLAB Examples -- 3.7. Exercises -- 4.1. Vector and Scalar Fields -- 4.1.1. Scalar Fields -- 4.1.2. Vector Fields -- 4.1.3. Field Lines -- 4.2. Gradient of Scalar Fields -- 4.2.1. Gradient in Cartesian Coordinates -- 4.2.2. Unit Normal -- 4.2.3. Gradient in Curvilinear Coordinates -- 4.2.4. Cylindrical Coordinates -- 4.2.5. Spherical Coordinates -- 4.2.6. Scalar Field from the Gradient -- 4.3. Divergence of Vector Fields -- 4.3.1. Flux through a Surface -- 4.3.2. Divergence of a Vector Field -- 4.3.3. Gradient in Curvilinear Coordinates -- 4.3.4. Cylindrical Coordinates -- 4.3.5. Spherical Coordinates -- 4.4. Curl of Vector Fields -- 4.4.1. Line Integral -- 4.4.2. Curl of a Vector Field -- 4.4.3. Curl in Cartesian Coordinates -- 4.4.4. Curl in Curvilinear Coordinates -- 4.4.5. Cylindrical Coordinates -- 4.4.6. Spherical Coordinates -- 4.4.7. Vector Potential -- 4.5. Laplacian of Scalar and Vector Fields |
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Note continued: 4.5.1. Laplacian in Curvilinear Coordinates -- 4.5.2. Cylindrical Coordinates -- 4.5.3. Spherical Coordinates -- 4.5.4. The Vector Laplacian -- 4.6. Vector Identities -- 4.6.1. First Derivatives -- 4.6.2. First Derivatives of Products -- 4.6.3. Second Derivatives -- 4.6.4. Vector Laplacian -- 4.7. Integral Theorems -- 4.7.1. Gradient Theorem -- 4.7.2. Divergence Theorem -- 4.7.3. Cartesian Coordinates -- 4.7.4. Cylindrical Coordinates -- 4.7.5. Stokes's Curl Theorem -- 4.7.6. Navier-Stokes Equation -- 4.8. MATLAB Examples -- 4.9. Exercises -- 5.1. Classification of Differential Equations -- 5.1.1. Order -- 5.1.2. Degree -- 5.1.3. Solution by Direct Integration -- 5.1.4. Exact Differential Equations -- 5.1.5. Sturm-Liouville Form -- 5.2. First Order Differential Equations -- 5.2.1. Homogeneous Equations -- 5.2.2. Inhomogeneous Equations -- 5.3. Linear, Homogeneous with Constant Coefficients -- 5.3.1. Damped Harmonic Oscillator -- 5.3.2. Undamped Motion |
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Note continued: 5.3.3. Overdamped Motion -- 5.3.4. Underdamped Motion -- 5.3.5. Critically Damped Oscillator -- 5.3.6. Higher Order Differential Equations -- 5.4. Linear Independence -- 5.4.1. Wronskian Determinant -- 5.5. Inhomogeneous with Constant Coefficients -- 5.6. Power Series Solutions to Differential Solutions -- 5.6.1. Standard Form -- 5.6.2. Airy's Differential Equation -- 5.6.3. Hermite's Differential Equation -- 5.6.4. Singular Points -- 5.6.5. Bessel's Differential Equation -- 5.6.6. Legendre's Differential Equation -- 5.7. Systems of Differential Equations -- 5.7.1. Homogeneous Systems -- 5.7.2. Inhomogeneous Systems -- 5.7.3. Solution Vectors -- 5.7.4. Test for Linear Independence -- 5.7.5. General Solution of Homogeneous Systems -- 5.7.7. Charged Particle in Electric and Magnetic Fields -- 5.8. Phase Space -- 5.8.1. Phase Plots -- 5.8.2. Noncrossing Property -- 5.8.3. Autonomous Systems -- 5.8.4. Phase Space Volume -- 5.9. Nonlinear Differential Equations |
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Note continued: 5.9.1. Predator-Prey System -- 5.9.2. Fixed Points -- 5.9.3. Linearization -- 5.9.4. Simple Pendulum -- 5.9.5. Numerical Solution -- 5.10. MATLAB Examples -- 5.11. Exercises -- 6.1. Dirac Delta Function -- 6.1.1. Representations of the Delta Function -- 6.1.2. Delta Function in Higher Dimensions -- 6.1.3. Delta Function in Spherical Coordinates -- 6.1.4. Poisson's Equation -- 6.1.5. Differential Form of Gauss's Law -- 6.1.6. Heaviside Step Function -- 6.2. Orthogonal Functions -- 6.2.1. Expansions in Orthogonal Functions -- 6.2.2.Completeness Relation -- 6.3. Legendre Polynomials -- 6.3.1. Associated Legendre Polynomials -- 6.3.2. Rodrigues' Formulas -- 6.3.3. Generating Functions -- 6.3.4. Orthogonality Relations -- 6.3.5. Spherical Harmonics -- 6.4. Laguerre Polynomials -- 6.4.1. Rodrigues' Formula -- 6.4.2. Generating Function -- 6.4.3. Orthogonality Relations -- 6.5. Hermite Polynomials -- 6.5.1. Rodrigues' Formula -- 6.5.2. Generating Function |
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Note continued: 6.5.4. Orthogonality -- 6.6. Bessel Functions -- 6.6.1. Modified Bessel Functions -- 6.6.2. Generating Function -- 6.6.3. Spherical Bessel Functions -- 6.6.4. Rayleigh Formulas -- 6.6.5. Generating Functions -- 6.6.6. Useful Relations -- 6.7. MATLAB Examples -- 6.8. Exercises -- 7.1. Fourier Series -- 7.1.1. Fourier Cosine Series -- 7.1.2. Fourier Sine Series -- 7.1.3. Fourier Exponential Series -- 7.2. Fourier Transforms -- 7.2.1. Power Spectrum -- 7.2.2. Spatial Transforms -- 7.3. Laplace Transforms -- 7.3.1. Properties of the Laplace Transform -- 7.3.2. Inverse Laplace Transform -- 7.3.3. Properties of Inverse Laplace Transforms -- 7.3.4. Table of Laplace Transforms -- 7.3.5. Solving Differential Equations -- 7.4. MATLAB Examples -- 7.5. Exercises -- 8.1. Types of Partial Differential Equations -- 8.1.1. First Order PDEs -- 8.1.2. Second Order PDEs -- 8.1.3. Laplace's Equation -- 8.1.4. Poisson's Equation -- 8.1.5. Diffusion Equation -- 8.1.6. Wave Equation |
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Note continued: 8.1.7. Helmholtz Equation -- 8.1.8. Klein-Gordon Equation -- 8.2. The Heat Equation -- 8.2.1. Transient Heat Flow -- 8.2.2. Steady State Heat Flow -- 8.2.3. Laplace Transform Solution -- 8.3. Separation of Variables -- 8.3.1. The Heat Equation -- 8.3.2. Laplace's Equation in Cartesian Coordinates -- 8.3.3. Laplace's Equation in Cylindrical Coordinates -- 8.3.4. Wave Equation -- 8.3.5. Helmholtz Equation in Cylindrical Coordinates -- 8.3.6. Helmholtz Equation in Spherical Coordinates -- 8.4. MATLAB Examples -- 8.5. Exercises -- 9.1. Cauchy-Riemann Equations -- 9.1. Laplace's Equation -- 9.2. Integral Theorems -- 9.2.1. Cauchy's Integral Theorem -- 9.2.2. Cauchy's Integral Formula -- 9.2.3. Laurent Series Expansion -- 9.2.4. Types of Singularities -- 9.2.5. Residues -- 9.2.6. Residue Theorem -- 9.2.7. Improper Integrals -- 9.2.8. Fourier Transform Integrals -- 9.3. Conformal Mapping -- 9.3.1. Poisson's Integral Formulas |
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Note continued: 9.3.2. Schwarz-Christoffel Transformation -- 9.3.3. Conformal Mapping -- 9.3.4. Mappings on the Riemann Sphere -- 9.4. MATLAB Examples -- 9.5. Exercises -- 10.1. Velocity-Dependent Resistive Forces -- 10.1.1. Drag Force Proportional to the Velocity -- 10.1.2. Drag Force on a Falling Body -- 10.2. Variable Mass Dynamics -- 10.2.1. Rocket Motion -- 10.3. Lagrangian Dynamics -- 10.3.1. Calculus of Variations -- 10.3.2. Lagrange's Equations of Motion -- 10.3.3. Lagrange's Equations with Constraints -- 10.4. Hamiltonian Mechanics -- 10.4.1. Legendre Transformation -- 10.4.2. Hamilton's Equations of Motion -- 10.4.3. Poisson Brackets -- 10.5. Orbital and Periodic Motion -- 10.5.1. Kepler Problem -- 10.5.2. Periodic Motion -- 10.5.3. Small Oscillations -- 10.6. Chaotic Dynamics -- 10.6.1. Strange Attractors -- 10.6.2. Lorenz Model -- 10.6.3. Jerk Systems -- 10.6.4. Time Delay Coordinates -- 10.6.5. Lyapunov Exponents -- 10.6.6. Poincare Sections -- 10.7. Fractals |
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Note continued: 10.7.1. Cantor Set -- 10.7.2. Koch Snowflake -- 10.7.3. Mandelbrot Set -- 10.7.4. Fractal Dimension -- 10.7.5. Chaotic Maps -- 10.8. MATLAB Examples -- 10.9. Exercises -- 11.1. Electrostatics in 1D -- 11.1.1. Integral and Differential Forms of Gauss's Law -- 11.1.2. Laplace's Equation in 1D -- 11.1.3. Poisson's Equation in 1D -- 11.2. Laplace's Equation in Cartesian Coordinates -- 11.2.1.3D Cartesian Coordinates -- 11.2.2. Method of Images -- 11.3. Laplace's Equation in Cylindrical Coordinates -- 11.3.1. Potentials with Planar Symmetry -- 11.3.2. Potentials in 3D Cylindrical Coordinates -- 11.4. Laplace's Equation in Spherical Coordinates -- 11.4.1. Axially Symmetric Potentials -- 11.4.2.3D Spherical Coordinates -- 11.5. Multipole Expansion of Potential -- 11.5.1. Axially Symmetric Potentials -- 11.5.2. Off-Axis Trick -- 11.5.3. Asymmetric Potentials -- 11.6. Electricity and Magnetism -- 11.6.1.Comparison of Electrostatics and Magnetostatics |
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Note continued: 11.6.2. Electrostatic Examples -- 11.6.3. Magnetostatic Examples -- 11.6.4. Static Electric and Magnetic Fields in Matter -- 11.6.5. Examples: Electrostatic Fields in Matter -- 11.6.6. Examples: Magnetic Fields in Matter -- 11.7. Scalar Electric and Magnetic Potentials -- 11.8. Time-Dependent Fields -- 11.8.1. The Ampere-Maxwell Equation -- 11.8.2. Maxwell's Equations -- 11.8.3. Self-Inductance -- 11.8.4. Mutual Inductance -- 11.8.5. Maxwell's Wave Equations -- 11.8.6. Maxwell's Equations in Matter -- 11.8.7. Time Harmonic Maxwell's Equations -- 11.8.8. Magnetic Monopoles -- 11.9. Radiation -- 11.9.1. Poynting Vector -- 11.9.2. Inhomogeneous Wave Equations -- 11.9.3. Gauge Transformation -- 11.9.4. Radiation Potential Formulation -- 11.9.5. The Hertz Dipole Antenna -- 11.10. MATLAB Examples -- 11.11. Exercises -- 12.1. Schrodinger Equation -- 12.1.1. Time-Dependent Schrodinger Equation -- 12.1.2. Time-Independent Schrodinger Equation |
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Note continued: 12.1.3. Operators, Expectation Values and Uncertainty -- 12.1.4. Probability Current Density -- 12.2. Bound States I -- 12.2.1. Particle in a Box -- 12.2.2. Semi-Infinite Square Well -- 12.2.3. Square Well with a Step -- 12.3. Bound States II -- 12.3.1. Delta Function Potential -- 12.3.2. Quantum Bouncer -- 12.3.3. Harmonic Oscillator -- 12.3.4. Operator Notation -- 12.3.5. Excited States of the Harmonic Oscillator -- 12.4. Schrodinger Equation in Higher Dimensions -- 12.4.1. Particle in a 3D Box -- 12.4.2. Schrodinger Equation in Spherical Coordinates -- 12.4.3. Radial Equation -- 12.4.4. Hydrogen Radial Wavefunctions -- 12.5. Approximation Methods -- 12.5.1. WKB Approximation -- 12.5.2. Time-Independent Perturbation Theory -- 12.5.3. Degenerate Perturbation Theory -- 12.5.4. Stark Effect -- 12.6. MATLAB Examples -- 12.7. Exercises -- 13.1. Microcanonical Ensemble -- 13.1.1. Number of Microstates and the Entropy -- 13.2. Canonical Ensemble |
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Note continued: 13.2.1. Boltzmann Factor and Partition Function -- 13.2.2. Average Energy -- 13.2.3. Free Energy and Entropy -- 13.2.4. Specific Heat -- 13.2.5. Rigid Rotator -- 13.2.6. Harmonic Oscillator -- 13.2.7.Composite Systems -- 13.2.8. Stretching a Rubber Band -- 13.3. Continuous Energy Distributions -- 13.3.1. Partition Function and Average Energy -- 13.3.2. Particle in a Box -- 13.3.3. Maxwell-Boltzmann Distribution -- 13.3.4. Relativistic Gas -- 13.4. Grand Canonical Ensemble -- 13.4.1. Gibbs Factor -- 13.4.2. Average Energy and Particle Number -- 13.4.3. Single Species -- 13.4.4. Grand Potential -- 13.4.5.Comparison of Canonical and Grand Canonical Ensembles -- 13.4.6. Bose-Einstein Statistics -- 13.4.7. Black-Body Radiation -- 13.4.8. Debye Theory of Specific Heat -- 13.4.9. Fermi-Dirac Statistics -- 13.5. MATLAB Examples -- 13.6. Exercises -- 14.1. Kinematics -- 14.1.1. Postulates of Special Relativity -- 14.1.2. Time Dilatation -- 14.1.3. Length Contraction |
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Note continued: 14.1.4. Relativistic Doppler Effect -- 14.1.5. Galilean Transformation -- 14.1.6. Lorentz Transformations -- 14.1.7. Relativistic Addition of Velocities -- 14.1.8. Velocity Addition Approximation -- 14.1.9.4-Vector Notation -- 14.2. Energy and Momentum -- 14.2.1. Newton's Second Law -- 14.2.2. Mass Energy and Kinetic Energy -- 14.2.3. Low Velocity Approximation -- 14.2.4. Energy Momentum Relation -- 14.2.5.Completely Inelastic Collisions -- 14.2.6. Particle Decay -- 14.2.7. Energy Units -- 14.3. Electromagnetics in Relativity -- 14.3.1. Relativistic Transformation of Fields -- 14.3.2. Covariant Formulation of Maxwell's Equations -- 14.3.3. Homogeneous Maxwell Equations -- 14.3.4. Lorentz Force Equation -- 14.4. Relativistic Lagrangian Formulation -- 14.4.1. Lagrangian of a Free Particle -- 14.4.2. Relativistic 1D Harmonic Oscillator -- 14.4.3. Charged Particle in Electric and Magnetic Fields -- 14.5. MATLAB Examples -- 14.6. Exercises |
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Note continued: 15.1. The Equivalence Principle -- 15.1.1. Classical Approximation to Gravitational Redshift -- 15.1.2. Photon Emitted from a Spherical Star -- 15.1.3. Gravitational Time Dilation -- 15.1.4.Comparison of Time Dilation Factors -- 15.2. Tensor Calculus -- 15.2.1. Tensor Notation -- 15.2.2. Line Element and Spacetime Interval -- 15.2.3. Raising and Lowering Indices -- 15.2.4. Metric Tensor in Spherical Coordinates -- 15.2.5. Dot Product -- 15.2.6. Cross Product -- 15.2.7. Transformation Properties of Tensors -- 15.2.8. Quotient Rule for Tensors -- 15.2.9. Covariant Derivatives -- 15.3. Einstein's Equations -- 15.3.1. Geodesic Equations of Motion -- 15.3.2. Alternative Lagrangian -- 15.3.3. Riemann Curvature Tensor -- 15.3.4. Ricci Tensor -- 15.3.5. Ricci Scalar -- 15.3.6. Einstein Tensor -- 15.3.7. Einstein's Field Equations -- 15.3.8. Friedman Cosmology -- 15.3.9. Killing Vectors -- 15.4. MATLAB Examples -- 15.5. Exercises -- 16.1. Early Models -- 16.1.1.de Broglie waves |
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Note continued: 16.1.2. Klein-Gordon Equation -- 16.1.3. Probability Current Density -- 16.1.4. Lagrangian Formulation of the Klein-Gordon Equation -- 16.2. Dirac Equation -- 16.2.1. Derivation of a First Order Equation -- 16.2.2. Probability Current -- 16.2.3. Gamma Matrices -- 16.2.4. Positive and Negative Energies -- 16.2.5. Lagrangian Formulation of the Dirac Equation -- 16.3. Solutions to the Dirac Equation -- 16.3.1. Plane Wave Solutions -- 16.3.2. Nonplane Wave Solutions -- 16.3.3. Nonrelativistic Limit -- 16.3.4. Dirac Equation in an Electromagnetic Field -- 16.4. MATLAB Examples -- 16.5. Exercises |
| Summary |
This book may be used by students and professionals in physics and engineering that have completed first-year calculus and physics. An introductory chapter reviews algebra, trigonometry, units and complex numbers that are frequently used in physics. Examples using MATLAB and Maple for symbolic and numerical calculations in physics with a variety of plotting features are included in all 16 chapters. The book applies many of mathematical concepts in chapter 1-9 to fundamental physics topics in mechanics, electromagnetics, quantum mechanics and relativity in chapters 10-16. Companion files are included with MATLAB and Maple worksheets and files, and all of the figures from the text |
| Bibliography |
Includes bibliographical references and index |
| Notes |
Print version record |
| Subject |
Mathematical physics.
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SCIENCE / Physics / General.
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Mathematical physics
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| Form |
Electronic book
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| ISBN |
9781523120314 |
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1523120312 |
