QUALITATIVE ANALYSIS OF PHYSICAL PROBLEMS

By MARY SANCHEZ

CONTENTS PART I PRELIMINARIES COMPLEX NUMBERS QUOTIENTS OF COMPLEX NUMBERS ROOTS OF COMPLEX NUMBERS SEQUENCES AND EULER’S CONSTANT POWER SERIES AND RADIUS OF CONVERGENCE MINKOWSKI SPACETIME RINDLER OBSERVERS THE LOGARITHM AND WINDING NUMBER BRANCH CUTS FOR LOG Z BRANCH CUTS FOR Z EXERCISES REFERENCE COMPLEX FUNCTION THEORY ANALYTIC FUNCTIONS CAUCHY’S INTEGRAL FORMULA EVALUATION OF A REAL INTEGRAL RESIDUE THEOREM MORERA’S THEOREM LIOUVILLE’S THEOREM POISSON KERNEL FLUX AND CIRCULATION EXAMPLES OF POTENTIAL FLOWS REFERENCES VECTORS AND LINEAR ALGEBRA INTRODUCTION INNER AND OUTER PRODUCTS ANGULAR MOMENTUM VECTOR ROTATIONS ANGULAR MOMENTUM AND MACH’S PRINCIPLE ENERGY AND TORQUE CORIOLIS FORCES SPINNING TOP ELEMENTARY TRANSFORMATIONS IN THE PLANE RE?ECTION MATRIX ROTATION MATRIX MATRIX ALGEBRA EIGENVALUE PROBLEMS EIGENVALUES OF RВÞ EIGENVALUES OF A REAL-SYMMETRIC MATRIX HERMITIAN MATRICES UNITARY MATRICES AND INVARIANTS HERMITIAN STRUCTURE OF MINKOWSKI SPACETIME EIGENVECTORS OF HERMITIAN MATRICES QR FACTORIZATION EXAMPLES OF IMAGE AND NULL SPACE DIMENSIONS OF IMAGE AND NULL SPACE QR FACTORIZATION BY GRAM-SCHMIDT LINEAR PARTIAL DIFFERENTIAL EQUATIONS HYPERBOLIC EQUATIONS INHOMOGENEOUS WAVE EQUATION (DUHAMEL) DIFFUSION EQUATION PHOTON DIFFUSION IN THE SUN ELLIPTIC EQUATIONS CHARACTERISTICS OF HYPERBOLIC SYSTEMS WEYL EQUATION PART II METHODS OF APPROXIMATION PROJECTIONS AND MINIMAL DISTANCES VECTORS AND DISTANCES PROJECTIONS OF VECTORS SNELL’S LAW AND FERMAT’S PRINCIPLE FITTING DATA BY LEAST SQUARES GAUSS-LEGENDRE QUADRATURE SPECTRAL METHODS AND SIGNAL ANALYSIS BASIS FUNCTIONS EXPANSION IN LEGENDRE POLYNOMIALS SYMBOLIC COMPUTATION IN MAXIMA FOURIER EXPANSION THE FOURIER TRANSFORM THE SINC FUNCTION PLANCHEREL’S THEOREM FOURIER SERIES BY SAMPLING ~F Ð!Þ DISCRETE FOURIER TRANSFORM (DFT) FOURIER SERIES CHEBYSHEV POLYNOMIALS WEIERSTRASS APPROXIMATION THEOREM DETECTOR SIGNALS IN THE PRESENCE OF NOISE CONVOLUTION AND CROSS-CORRELATION SIGNAL DETECTION BY FFT USING MAXIMA GPU-BUTTER?Y FILTER IN ( F ; _F ) ROOT FINDING SOLVING FOR ?? P AND CONVERGENCE IN NEWTON’S METHOD CONTRACTION MAPPING NEWTON’S METHOD IN TWO DIMENSIONS BASINS OF ATTRACTION ROOT FINDING IN HIGHER DIMENSIONS FINITE DIFFERENCING: DIFFERENTIATION AND INTEGRATION VECTOR FIELDS ESTIMATING VELOCITY ESTIMATING ACCELERATION GRADIENT OPERATOR INTEGRATION BY FINITE SUMMATION NUMERICAL INTEGRATION OF ODE’S EXAMPLES OF ORDINARY DIFFERENTIAL EQUATIONS OSMOSIS MIGRATION TIME OF THE MOON COSMOLOGICAL EXPANSION PERTURBATION THEORY, SCALING AND TURBULENCE ROOTS OF A CUBIC EQUATION DAMPED PENDULUM ORBITAL MOTION PERIHELION PRECESSION INERTIAL AND VISCOUS FLUID MOTION LARGE AND SMALL REYNOLDS NUMBERS VORTICITY EQUATION JEANS INSTABILITY IN LINEARIZED EULER’S EQUATIONS KOLMOGOROV SCALING OF HOMOGENEOUS TURBULENCE PART III SELECTED TOPICS THERMODYNAMICS OF N-BODY SYSTEMS THE ACTION PRINCIPLE MOMENTUM IN EULER-LAGRANGE EQUATIONS LEGENDRE TRANSFORMATION HAMILTONIAN FORMULATION GLOBULAR CLUSTERS COEF?CIENTS OF RELAXATION ACCRETION FLOWS ONTO BLACK HOLES BONDI ACCRETION HOYLE-LYTTLETON ACCRETION ACCRETION DISKS GRAVITATIONAL WAVE EMISSION MASS TRANSFER IN BINARIES RINDLER OBSERVERS IN ASTROPHYSICS AND COSMOLOGY THE MOVING MIRROR PROBLEM IMPLICATIONS FOR DARK MATTER