CRC Press, 2014. - 765p. ISBN -13: 978-1-4665-8468-6 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 Adjoint Operators 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 Div, Grad, Curl 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 Inhomogeneous Steady-State Equation 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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