ESSENTIAL MATHEMATICAL METHODS FOR PHYSICISTS
CONTENTS VECTOR ANALYSIS ELEMENTARY APPROACH VECTORS AND VECTOR SPACE SUMMARY SCALAR OR DOT PRODUCT FREE MOTION AND OTHER ORBITS VECTOR OR CROSS PRODUCT TRIPLE SCALAR PRODUCT AND TRIPLE VECTOR PRODUCT TRIPLE SCALAR PRODUCT TRIPLE VECTOR PRODUCT GRADIENT, ? PARTIAL DERIVATIVES GRADIENT AS A VECTOR OPERATOR A GEOMETRICAL INTERPRETATION DIVERGENCE, ? A PHYSICAL INTERPRETATION CURL, ?× SUCCESSIVE APPLICATIONS OF ? VECTOR INTEGRATION LINE INTEGRALS CONTENTS SURFACE INTEGRALS VOLUME INTEGRALS INTEGRAL DE?NITIONS OF GRADIENT, DIVERGENCE, AND CURL GAUSS’S THEOREM GREEN’S THEOREM STOKES’S THEOREM POTENTIAL THEORY SCALAR POTENTIAL GAUSS’S LAW AND POISSON’S EQUATION GAUSS’S LAW POISSON’S EQUATION DIRAC DELTA FUNCTION ADDITIONAL READING VECTOR ANALYSIS IN CURVED COORDINATES AND TENSORS SPECIAL COORDINATE SYSTEMS RECTANGULAR CARTESIAN COORDINATES INTEGRALS IN CARTESIAN COORDINATES CIRCULAR CYLINDER COORDINATES INTEGRALS IN CYLINDRICAL COORDINATES GRADIENT DIVERGENCE CURL ORTHOGONAL COORDINATES DIFFERENTIAL VECTOR OPERATORS SPHERICAL POLAR COORDINATES INTEGRALS IN SPHERICAL POLAR COORDINATES TENSOR ANALYSIS ROTATION OF COORDINATE AXES INVARIANCE OF THE SCALAR PRODUCT UNDER ROTATIONS COVARIANCE OF CROSS PRODUCT COVARIANCE OF GRADIENT CONTENTS DE?NITION OF TENSORS OF RANK TWO ADDITION AND SUBTRACTION OF TENSORS SUMMATION CONVENTION SYMMETRY–ANTISYMMETRY SPINORS CONTRACTION AND DIRECT PRODUCT CONTRACTION DIRECT PRODUCT QUOTIENT RULE DUAL TENSORS LEVI–CIVITA SYMBOL DETERMINANTS AND MATRICES DETERMINANTS LINEAR EQUATIONS: EXAMPLES HOMOGENEOUS LINEAR EQUATIONS INHOMOGENEOUS LINEAR EQUATIONS LAPLACIAN DEVELOPMENT BY MINORS ANTISYMMETRY MATRICES BASIC DE?NITIONS, EQUALITY, AND RANK MATRIX MULTIPLICATION, INNER PRODUCT DIRAC BRA-KET, TRANSPOSITION MULTIPLICATION (BY A SCALAR) ADDITION PRODUCT THEOREM DIAGONAL MATRICES TRACE MATRIX INVERSION ORTHOGONAL MATRICES DIRECTION COSINES APPLICATIONS TO VECTORS ORTHOGONALITY CONDITIONS: TWO-DIMENSIONAL CASE CONTENTS EULER ANGLES SYMMETRY PROPERTIES AND SIMILARITY TRANSFORMATIONS RELATION TO TENSORS HERMITIAN MATRICES AND UNITARY MATRICES DE?NITIONS PAULI MATRICES DIAGONALIZATION OF MATRICES MOMENT OF INERTIA MATRIX EIGENVECTORS AND EIGENVALUES HERMITIAN MATRICES ANTI-HERMITIAN MATRICES NORMAL MODES OF VIBRATION ILL-CONDITIONED SYSTEMS FUNCTIONS OF MATRICES GROUP THEORY INTRODUCTION TO GROUP THEORY DE?NITION OF GROUP HOMOMORPHISM AND ISOMORPHISM MATRIX REPRESENTATIONS: REDUCIBLE AND IRREDUCIBLE GENERATORS OF CONTINUOUS GROUPS ROTATION GROUPS SO(2) AND SO(3) ROTATION OF FUNCTIONS AND ORBITAL ANGULAR MOMENTUM SPECIAL UNITARY GROUP SU(2) ORBITAL ANGULAR MOMENTUM LADDER OPERATOR APPROACH HOMOGENEOUS LORENTZ GROUP VECTOR ANALYSIS IN MINKOWSKI SPACE–TIME INFINITE SERIES FUNDAMENTAL CONCEPTS ADDITION AND SUBTRACTION OF SERIES CONVERGENCE TESTS COMPARISON TEST CAUCHY ROOT TEST D’ALEMBERT OR CAUCHY RATIO TEST CAUCHY OR MACLAURIN INTEGRAL TEST ALTERNATING SERIES LEIBNIZ CRITERION ABSOLUTE AND CONDITIONAL CONVERGENCE ALGEBRA OF SERIES MULTIPLICATION OF SERIES SERIES OF FUNCTIONS UNIFORM CONVERGENCE WEIERSTRASS M (MAJORANT) TEST ABEL’S TEST TAYLOR’S EXPANSION MACLAURIN THEOREM BINOMIAL THEOREM TAYLOR EXPANSION—MORE THAN ONE VARIABLE POWER SERIES CONVERGENCE UNIFORM AND ABSOLUTE CONVERGENCE CONTINUITY DIFFERENTIATION AND INTEGRATION UNIQUENESS THEOREM INVERSION OF POWER SERIES ELLIPTIC INTEGRALS SERIES EXPANSION LIMITING VALUES BERNOULLI NUMBERS AND THE EULER–MACLAURIN FORMULA BERNOULLI FUNCTIONS EULER–MACLAURIN INTEGRATION FORMULA IMPROVEMENT OF CONVERGENCE IMPROVEMENT OF CONVERGENCE BY RATIONAL APPROXIMATIONS ASYMPTOTIC SERIES ERROR FUNCTION FUNCTIONS OF A COMPLEX VARIABLE I COMPLEX ALGEBRA COMPLEX CONJUGATION FUNCTIONS OF A COMPLEX VARIABLE CAUCHY–RIEMANN CONDITIONS ANALYTIC FUNCTIONS CAUCHY’S INTEGRAL THEOREM CONTOUR INTEGRALS STOKES’S THEOREM PROOF OF CAUCHY’S INTEGRAL THEOREM MULTIPLY CONNECTED REGIONS CAUCHY’S INTEGRAL FORMULA DERIVATIVES MORERA’S THEOREM LAURENT EXPANSION TAYLOR EXPANSION SCHWARZ RE?ECTION PRINCIPLE ANALYTIC CONTINUATION LAURENT SERIES MAPPING TRANSLATION ROTATION INVERSION BRANCH POINTS AND MULTIVALENT FUNCTIONS CONFORMAL MAPPING FUNCTIONS OF A COMPLEX VARIABLE II SINGULARITIES POLES BRANCH POINTS CALCULUS OF RESIDUES RESIDUE THEOREM EVALUATION OF DE?NITE INTEGRALS CAUCHY PRINCIPAL VALUE POLE EXPANSION OF MEROMORPHIC FUNCTIONS PRODUCT EXPANSION OF ENTIRE FUNCTIONS METHOD OF STEEPEST DESCENTS ANALYTIC LANDSCAPE SADDLE POINT METHOD DIFFERENTIAL EQUATIONS INTRODUCTION FIRST-ORDER ODES SEPARABLE VARIABLES EXACT DIFFERENTIAL EQUATIONS LINEAR FIRST-ORDER ODES ODES OF SPECIAL TYPE SECOND-ORDER ODES INHOMOGENEOUS LINEAR ODES AND PARTICULAR SOLUTIONS INHOMOGENEOUS EULER ODE INHOMOGENEOUS ODE WITH CONSTANT COEF?CIENTS LINEAR INDEPENDENCE OF SOLUTIONS SINGULAR POINTS SERIES SOLUTIONS—FROBENIUS’S METHOD SYMMETRY OF ODE AND SOLUTIONS LIMITATIONS OF SERIES APPROACH—BESSEL’S EQUATION REGULAR AND IRREGULAR SINGULARITIES FUCHS’S THEOREM SUMMARY A SECOND SOLUTION SERIES FORM OF THE SECOND SOLUTION NUMERICAL SOLUTIONS FIRST-ORDER DIFFERENTIAL EQUATIONS TAYLOR SERIES SOLUTION RUNGE–KUTTA METHOD PREDICTOR–CORRECTOR METHODS INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS SEPARATION OF VARIABLES CARTESIAN COORDINATES CONTENTS CIRCULAR CYLINDRICAL COORDINATES STURM–LIOUVILLE THEORY—ORTHOGONAL FUNCTIONS SELF-ADJOINT ODES EIGENFUNCTIONS AND EIGENVALUES BOUNDARY CONDITIONS HERMITIAN OPERATORS HERMITIAN OPERATORS IN QUANTUM MECHANICS REAL EIGENVALUES ORTHOGONAL EIGENFUNCTIONS EXPANSION IN ORTHOGONAL EIGENFUNCTIONS DEGENERACY GRAM–SCHMIDT ORTHOGONALIZATION ORTHOGONAL POLYNOMIALS COMPLETENESS OF EIGENFUNCTIONS BESSEL’S INEQUALITY SCHWARZ INEQUALITY SUMMARY OF VECTOR SPACES—COMPLETENESS EXPANSION (FOURIER) COEF?CIENTS THE GAMMA FUNCTION (FACTORIAL FUNCTION) DE?NITIONS AND SIMPLE PROPERTIES IN?NITE LIMIT (EULER) DE?NITE INTEGRAL (EULER) IN?NITE PRODUCT (WEIERSTRASS) FACTORIAL NOTATION DOUBLE FACTORIAL NOTATION INTEGRAL REPRESENTATION DIGAMMA AND POLYGAMMA FUNCTIONS DIGAMMA FUNCTION POLYGAMMA FUNCTION CONTENTS MACLAURIN EXPANSION, COMPUTATION SERIES SUMMATION STIRLING’S SERIES DERIVATION FROM EULER–MACLAURIN INTEGRATION FORMULA NUMERICAL COMPUTATION THE INCOMPLETE GAMMA FUNCTIONS AND RELATED FUNCTIONS EXPONENTIAL INTEGRAL ERROR INTEGRALS LEGENDRE POLYNOMIALS AND SPHERICAL HARMONICS PHYSICAL BASIS: ELECTROSTATICS GENERATING FUNCTION LINEAR ELECTRIC MULTIPOLES VECTOR EXPANSION RECURRENCE RELATIONS AND SPECIAL PROPERTIES RECURRENCE RELATIONS DIFFERENTIAL EQUATIONS UPPER AND LOWER BOUNDS FOR PN (COS ?) ORTHOGONALITY EXPANSION OF FUNCTIONS, LEGENDRE SERIES ALTERNATE DE?NITIONS OF LEGENDRE POLYNOMIALS RODRIGUES’S FORMULA ASSOCIATED LEGENDRE FUNCTIONS SPHERICAL HARMONICS BESSEL FUNCTIONS BESSEL FUNCTIONS OF THE FIRST KIND, J?(X) GENERATING FUNCTION FOR INTEGRAL ORDER APPLICATIONS OF RECURRENCE RELATIONS BESSEL’S DIFFERENTIAL EQUATION CONTENTS INTEGRAL REPRESENTATIONS NORMALIZATION BESSEL SERIES BESSEL FUNCTIONS OF NONINTEGRAL ORDER NEUMANN FUNCTIONS, BESSEL FUNCTIONS OF THE SECOND KIND DE?NITION AND SERIES FORM OTHER FORMS WRONSKIAN FORMULAS ASYMPTOTIC EXPANSIONS EXPANSION OF AN INTEGRAL REPRESENTATION NUMERICAL EVALUATION SPHERICAL BESSEL FUNCTIONS HERMITE AND LAGUERRE POLYNOMIALS HERMITE POLYNOMIALS QUANTUM MECHANICAL SIMPLE HARMONIC OSCILLATOR RAISING AND LOWERING OPERATORS RECURRENCE RELATIONS AND GENERATING FUNCTION ALTERNATE REPRESENTATIONS LAGUERRE FUNCTIONS DIFFERENTIAL EQUATION—LAGUERRE POLYNOMIALS ASSOCIATED LAGUERRE POLYNOMIALS FOURIER SERIES GENERAL PROPERTIES COMPLETENESS BEHAVIOR OF DISCONTINUITIES ADVANTAGES AND USES OF FOURIER SERIES PERIODIC FUNCTIONS CHANGE OF INTERVAL COMPLEX FOURIER SERIES ABEL’S THEOREM PROPERTIES OF FOURIER SERIES INTEGRATION DIFFERENTIATION INTEGRAL TRANSFORMS INTRODUCTION AND DE?NITIONS LINEARITY FOURIER TRANSFORM LAPLACE TRANSFORM DEVELOPMENT OF THE INVERSE FOURIER TRANSFORM INVERSE FOURIER TRANSFORM—EXPONENTIAL FORM DIRAC DELTA FUNCTION DERIVATION FOURIER TRANSFORMS—INVERSION THEOREM EXPONENTIAL TRANSFORM COSINE TRANSFORM SINE TRANSFORM FOURIER TRANSFORM OF DERIVATIVES CONVOLUTION THEOREM PARSEVAL’S RELATION MOMENTUM REPRESENTATION LAPLACE TRANSFORMS DE?NITION INVERSE TRANSFORM PARTIAL FRACTION EXPANSION NUMERICAL INVERSION LAPLACE TRANSFORM OF DERIVATIVES 15 10 OTHER PROPERTIES SUBSTITUTION RLC ANALOG CONTENTS TRANSLATION DERIVATIVE OF A TRANSFORM INTEGRATION OF TRANSFORMS LIMITS OF INTEGRATION—UNIT STEP FUNCTION 15 11 CONVOLUTION OR FALTUNGS THEOREM 15 12 INVERSE LAPLACE TRANSFORM BROMWICH INTEGRAL SUMMARY: INVERSION OF LAPLACE TRANSFORM PARTIAL DIFFERENTIAL EQUATIONS EXAMPLES OF PARTIAL DIFFERENTIAL EQUATIONS AND BOUNDARY CONDITIONS HEAT FLOW OR DIFFUSION PDE ALTERNATE SOLUTIONS INHOMOGENEOUS PDE—GREEN’S FUNCTION PROBABILITY DE?NITIONS, SIMPLE PROPERTIES COUNTING OF PERMUTATIONS AND COMBINATIONS RANDOM VARIABLES BINOMIAL DISTRIBUTION POISSON DISTRIBUTION GAUSS’S NORMAL DISTRIBUTION STATISTICS ERROR PROPAGATION FITTING CURVES TO DATA THE ?2 DISTRIBUTION THE STUDENT T DISTRIBUTION CALCULUS OF VARIATIONS USES OF THE CALCULUS OF VARIATIONS A DEPENDENT AND AN INDEPENDENT VARIABLE CONCEPT OF VARIATION SEVERAL DEPENDENT VARIABLES TRANSVERSALITY CONDITION HAMILTON’S PRINCIPLE SEVERAL INDEPENDENT VARIABLES SEVERAL DEPENDENT AND INDEPENDENT VARIABLES RELATION TO PHYSICS LAGRANGIAN MULTIPLIERS: VARIATION WITH CONSTRAINTS VARIATION WITH CONSTRAINTS LAGRANGIAN EQUATIONS RAYLEIGH–RITZ VARIATIONAL TECHNIQUE GROUND-STATE EIGENFUNCTION NONLINEAR METHODS AND CHAOS THE LOGISTIC MAP SENSITIVITY TO INITIAL CONDITIONS AND PARAMETERS LYAPUNOV EXPONENTS FRACTALS NONLINEAR DIFFERENTIAL EQUATIONS BERNOULLI AND RICCATI EQUATIONS FIXED AND MOVABLE SINGULARITIES, SPECIAL SOLUTIONS AUTONOMOUS DIFFERENTIAL EQUATIONS LOCAL AND GLOBAL BEHAVIOR IN HIGHER DIMENSIONS