HIGHER MATHEMATICS FOR CHEMICAL STUDENTS
CONTENTS VARIABLES AND FUNCTIONS INTRODUCTION FUNCTIONS CLASSI?CATION AND PROPERTIES OF FUNCTIONS EXPONENTIAL AND LOGARITHMIC FUNCTIONS APPLICATIONS OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS COMPLEX NUMBERS CIRCULAR TRIGONOMETRIC FUNCTIONS HYPERBOLIC FUNCTIONS PROBLEMS LIMITS, DERIVATIVES AND SERIES DE?NITION OF A LIMIT CONTINUITY THE DERIVATIVE HIGHER DERIVATIVES IMPLICIT AND PARAMETRIC RELATIONS THE EXTREMA OF A FUNCTION AND ITS CRITICAL POINTS THE DIFFERENTIAL THE MEAN-VALUE THEOREM AND L’HOSPITAL’S RULE TAYLOR’S SERIES BINOMIAL EXPANSION TESTS OF SERIES CONVERGENCE FUNCTIONS OF SEVERAL VARIABLES EXACT DIFFERENTIALS INTEGRATION THE INDE?NITE INTEGRAL INTEGRATION FORMULAS METHODS OF INTEGRATION INTEGRATION BY SUBSTITUTION INTEGRATION BY PARTS INTEGRATION OF PARTIAL FRACTIONS DE?NITE INTEGRALS DE?NITION PLANE AREA LINE INTEGRALS FIDO AND HIS MASTER THE GAUSSIAN AND ITS MOMENTS INTEGRATING FACTORS TABLES OF INTEGRALS VECTOR ANALYSIS VECTOR ADDITION SCALAR PRODUCT VECTOR PRODUCT TRIPLE PRODUCTS RECIPROCAL BASES DIFFERENTIATION OF VECTORS SCALAR AND VECTOR ?ELDS THE GRADIENT THE DIVERGENCE THE CURL OR ROTATION THE LAPLACIAN MAXWELL’S EQUATIONS CURVILINEAR COORDINATES ORDINARY DIFFERENTIAL EQUATIONS FIRST-ORDER DIFFERENTIAL EQUATIONS SECOND-ORDER DIFFERENTIAL EQUATIONS SERIES SOLUTION THE CLASSICAL HARMONIC OSCILLATOR THE DAMPED OSCILLATOR THE DIFFERENTIAL OPERATOR HARMONIC OSCILLATOR INHOMOGENEOUS EQUATIONS FORCED VIBRATIONS APPLICATIONS IN QUANTUM MECHANICS THE PARTICLE IN A BOX SYMMETRIC BOX RECTANGULAR BARRIER: THE TUNNEL EFFECT THE HARMONIC OSCILLATOR IN QUANTUM MECHANICS SPECIAL FUNCTIONS HERMITE POLYNOMIALS ASSOCIATED LEGENDRE POLYNOMIALS THE ASSOCIATED LAGUERRE POLYNOMIALS THE GAMMA FUNCTION BESSEL FUNCTIONS MATHIEU FUNCTIONS THE HYPERGEOMETRIC FUNCTIONS PARTIAL DIFFERENTIAL EQUATIONS THE VIBRATING STRING THE WAVE EQUATION SEPARATION OF VARIABLES BOUNDARY CONDITIONS INITIAL CONDITIONS THE THREE-DIMENSIONAL HARMONIC OSCILLATOR QUANTUM-MECHANICAL APPLICATIONS DEGENERACY THE TWO-BODY PROBLEM CLASSICAL MECHANICS QUANTUM MECHANICS CENTRAL FORCES SPHERICAL COORDINATES SPHERICAL HARMONICS THE DIATOMIC MOLECULE THE RIGID ROTATOR THE VIBRATING ROTATOR CENTRIFUGAL FORCES THE HYDROGEN ATOM ENERGY WAVEFUNCTIONS AND THE PROBABILITY DENSITY BINARY COLLISIONS CONSERVATION OF ANGULAR MOMENTUM CONSERVATION OF ENERGY INTERACTION POTENTIAL: LJ (6-12) ANGLE OF DE?ECTION QUANTUM MECHANICAL DESCRIPTION: THE PHASE SHIFT OPERATORS AND MATRICES THE ALGEBRA OF OPERATORS HERMITIAN OPERATORS AND THEIR EIGENVALUES MATRICES THE DETERMINANT PROPERTIES OF DETERMINANTS JACOBIANS VECTORS AND MATRICES LINEAR EQUATIONS PARTITIONING OF MATRICES MATRIX FORMULATION OF THE EIGENVALUE PROBLEM COUPLED OSCILLATORS GEOMETRIC OPERATIONS THE MATRIX METHOD IN QUANTUM MECHANICS THE HARMONIC OSCILLATOR GROUP THEORY DE?NITION OF A GROUP EXAMPLES PERMUTATIONS CONJUGATE ELEMENTS AND CLASSES MOLECULAR SYMMETRY THE CHARACTER IRREDUCIBLE REPRESENTATIONS CHARACTER TABLES REDUCTION OF A REPRESENTATION: THE “MAGIC FORMULA” THE DIRECT PRODUCT REPRESENTATION SYMMETRY-ADAPTED FUNCTIONS: PROJECTION OPERATORS HYBRIDIZATION OF ATOMIC ORBITALS CRYSTAL SYMMETRY MOLECULAR MECHANICS KINETIC ENERGY MOLECULAR ROTATION EULER’S ANGLES CLASSI?CATION OF ROTATORS ANGULAR MOMENTA THE SYMMETRIC TOP IN QUANTUM MECHANICS VIBRATIONAL ENERGY INTERNAL COORDINATES: THE G MATRIX POTENTIAL ENERGY NORMAL COORDINATES SECULAR DETERMINANT AN EXAMPLE: THE WATER MOLECULE SYMMETRY COORDINATES APPLICATION TO MOLECULAR VIBRATIONS FORM OF NORMAL MODES NONRIGID MOLECULES MOLECULAR INVERSION INTERNAL ROTATION MOLECULAR CONFORMATION: THE MOLECULAR MECHANICS METHOD PROBABILITY AND STATISTICS COMBINATIONS PROBABILITY STIRLING’S APPROXIMATION STATISTICAL MECHANICS THE LAGRANGE MULTIPLIERS THE PARTITION FUNCTION MOLECULAR ENERGIES 1081 TRANSLATION 1082 ROTATION 1083 VIBRATION QUANTUM STATISTICS 1091 THE INDISTINGUISHABILITY OF IDENTICAL PARTICLES 1092 THE EXCLUSION PRINCIPLE 1093 FERMI–DIRAC STATISTICS 1094 BOSE–EINSTEIN STATISTICS 1010 ORTHO- AND PARA-HYDROGEN INTEGRAL TRANSFORMS THE FOURIER TRANSFORM 1111 CONVOLUTION 1112 FOURIER TRANSFORM PAIRS THE LAPLACE TRANSFORM 1121 EXAMPLES OF SIMPLE LAPLACE TRANSFORMS 1122 THE TRANSFORM OF DERIVATIVES 1123 SOLUTION OF DIFFERENTIAL EQUATIONS 1124 LAPLACE TRANSFORMS: CONVOLUTION AND INVERSION 1125 GREEN’S FUNCTIONS APPROXIMATION METHODS IN QUANTUM MECHANICS THE BORN–OPPENHEIMER APPROXIMATION PERTURBATION THEORY: STATIONARY STATES 1221 NONDEGENERATE SYSTEMS 1222 FIRST-ORDER APPROXIMATION 1223 SECOND-ORDER APPROXIMATION 1224 THE ANHARMONIC OSCILLATOR 1225 DEGENERATE SYSTEMS 1226 THE STARK EFFECT OF THE HYDROGEN ATOM TIME-DEPENDENT PERTURBATIONS 1231 THE SCHR¨ODINGER EQUATION 1232 INTERACTION OF LIGHT AND MATTER CONTENTS 1233 SPECTROSCOPIC SELECTION RULES THE VARIATION METHOD 1241 THE VARIATION THEOREM 1242 AN EXAMPLE: THE PARTICLE IN A BOX 1243 LINEAR VARIATION FUNCTIONS 1244 LINEAR COMBINATIONS OF ATOMIC ORBITALS (LCAO) 1245 THE H¨UCKEL APPROXIMATION NUMERICAL ANALYSIS ERRORS 1311 THE GAUSSIAN DISTRIBUTION 1312 THE POISSON DISTRIBUTION THE METHOD OF LEAST SQUARES POLYNOMIAL INTERPOLATION AND SMOOTHING 1341 THE DISCRETE FOURIER TRANSFORM (DFT) 1342 THE FAST FOURIER TRANSFORM (FFT) 1343 AN APPLICATION: INTERPOLATION AND SMOOTHING NUMERICAL INTEGRATION 1351 THE TRAPEZOID RULE 1352 SIMPSON’S RULE 1353 THE METHOD OF ROMBERG ZEROS OF FUNCTIONS 1361 NEWTON’S METHOD 1362 THE BISECTION METHOD 1363 THE ROOTS: AN EXAMPLE