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# Herman R.L. A course in mathematical methods for physicists

• Файл формата djvu
• размером 13,05 МБ
• Добавлен пользователем
• Отредактирован Boca Raton: CRC Press, 2013. - 788p.
This is a book on mathematical methods seen in physics and aimed at undergraduate students who have completed a year-long introductory course in physics. The intent of the course is to introduce students to many of the mathematical techniques useful in their undergraduate science education long before they are exposed to more focused topics in physics.
Contents:

Introduction
What Is Mathematical Physics?
An Overview of the Course
Tips for Students
Acknowledgments
Introduction
Trigonometric Functions
Hyperbolic Functions
Derivatives
Integrals
Geometric Series
Power Series
Binomial Expansion
What I Need from My Intro Physics Class?
Technology and Tables
Appendix: Dimensional Analysis
Problems
Free Fall
First-Order Differential Equations
Separable Equations
Linear First-Order Equations
Terminal Velocity
Mass-Spring Systems
Simple Pendulum
Second-Order Linear Differential Equations
Constant Coefficient Equations
LRC Circuits
Special Cases
Damped Oscillations
Method of Undetermined Coefficients
Periodically Forced Oscillations
Method of Variation of Parameters
Initial Value Green's Functions
Cauchy-Euler Equations
Numerical Solutions of ODEs
Euler's Method
Higher-Order Taylor Methods
Runge-Kutta Methods
Nonlinear Pendulum
Extreme Sky Diving
Flight of Sports Balls
Falling Raindrops
Two-Body Problem
Expanding Universe
Coefficient of Drag
Coupled Oscillators
Planar Systems
Equilibrium Solutions and Nearby Behaviors
Polar Representation of Spirals
Problems
Finite Dimensional Vector Spaces
Linear Transformations
Active and Passive Rotations
Rotation Matrices
Matrix Representations
Matrix Inverses and Determinants
Cramer's Rule
An Introduction to Coupled Systems
Eigenvalue Problems
Rotations of Conics
Matrix Formulation of Planar Systems
Solving Constant Coefficient Systems in 2D
Examples of the Matrix Method
Applications
Mass-Spring Systems
Circuits
Mixture Problems
Chemical Kinetics *
Love Affairs
Epidemics
Appendix: Diagonalization and Linear Systems
Problems
Introduction
Logistic Equation
Riccati Equation
Autonomous First-Order Equations
Bifurcations for First-Order Equations
Nonlinear Pendulum
Period of the Nonlinear Pendulum
Stability of Fixed Points in Nonlinear Systems
Nonlinear Population Models
Limit Cycles
Nonautonomous Nonlinear Systems
Exact Solutions Using Elliptic Functions
Problems
Harmonics and Vibrations
Boundary Value Problems
Partial Differential Equations
1D Heat Equation
1D Wave Equation
Introduction to Fourier Series
Fourier Trigonometric Series
Fourier Series over Other Intervals
Fourier Series on [a, b]
Sine and Cosine Series
Solution of the Heat Equation
Finite Length Strings
Gibbs Phenomenon
Green's Functions for D Partial Differential Equations
Heat Equation
Wave Equation
Derivation of Wave Equation for String
Derivation of D Heat Equation
Problems
Non-Sinusoidal Harmonics and Special Functions
Function Spaces
Classical Orthogonal Polynomials
Fourier-Legendre Series
Properties of Legendre Polynomials
Generating Functions: Generating Function for Legendre Polynomials
Differential Equation for Legendre Polynomials
Fourier-Legendre Series
Gamma Function
Fourier-Bessel Series
Sturm-Liouville Eigenvalue Problems
Sturm-Liouville Operators
Properties of Sturm-Liouville Eigenvalue Problems
Lagrange's and Green's Identities
Orthogonality and Reality
Rayleigh Quotient
Eigenfunction Expansion Method
Boundary Value Green's Function
Properties of Green's Functions
Differential Equation for the Green's Function
Series Representations of Green's Functions
Nonhomogeneous Heat Equation
Appendix: Least Squares Approximation
Appendix: Fredholm Alternative Theorem
Problems
Complex Representations of Waves
Complex Numbers
Complex Valued Functions
Complex Domain Coloring
Complex Differentiation
Complex Path Integrals
Cauchy's Theorem
Analytic Functions and Cauchy's Integral Formula
Laurent Series
Singularities and Residue Theorem
Infinite Integrals
Integration over Multivalued Functions
Appendix: Jordan's Lemma
Problems
Introduction
Example : Linearized KdV Equation
Example : Free Particle Wave Function
Transform Schemes
Complex Exponential Fourier Series
Exponential Fourier Transform
Dirac Delta Function
Properties of the Fourier Transform
Fourier Transform Examples
Convolution Operation
Convolution Theorem for Fourier Transforms
Application to Signal Analysis
Parseval's Equality
Laplace Transform
Properties and Examples of Laplace Transforms
Series Summation Using Laplace Transforms
Solution of ODEs Using Laplace Transforms
SteP and Impulse Functions
Convolution Theorem
Inverse Laplace Transform
Fourier Transform and the Heat Equation
Laplace's Equation on the Half Plane
Heat Equation on Infinite Interval, Revisited
Nonhomogeneous Heat Equation
Problems
Vector Analysis and EM Waves
A Review of Vector Products
Differentiation and Integration of Vectors
Integral Theorems
Vector Identities
Kepler Problem
Maxwell's Equations
Electromagnetic Wave Equation
Potential Functions and Helmholtz's Theorem
Curvilinear Coordinates
Tensors
Problems
Stationary and Extreme Values of Functions
Functions of One Variable
Functions of Several Variables
Linear Regression
Lagrange Multipliers and Constraints
Introduction
Variational Problems
Euler Equation
Isoperimetic Problems
Hamilton's Principle
Geodesics
Problems
Problems in Higher Dimensions
Vibrations of Rectangular Membranes
Vibrations of a Kettle Drum
Laplace's Equation in 3D
Poisson Integral Formula
Three-Dimensional Cake Baking
Laplace's Equation and Spherical Symmetry
Spherical Harmonics
Schrodinger Equation in Spherical Coordinates
Solution of the 2D Poisson Equation
Green's Functions for the 3D Poisson Equation
Introduction
Laplace's Equation: V2i/ =0
Homogeneous Time-Dependent Equations
Problems
Review of Sequences and Infinite Series
Convergence of Sequences
Limit Theorems
Infinite Series
Convergence Tests
Sequences of Functions
Infinite Series of Functions
Special Series Expansions
Order of Sequences and Functions
Problems
Bibliography
Index
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