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Mathematical methods for physics : using MATLAB & Maple /

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 n...

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Detalles Bibliográficos
Clasificación:	Libro Electrónico
Autor principal:	Claycomb, J. R. (Autor)
Formato:	Electrónico eBook
Idioma:	Inglés
Publicado:	Dulles, Virginia : Mercury Learning & Information, [2018]
Temas:	Mathematical physics. Physique mathématique. Mathematical physics
Acceso en línea:	Texto completo

Tabla de Contenidos:

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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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.

Mathematical methods for physics : using MATLAB & Maple /

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