Mathematical Methods for Physical and Analytical Chemistry -- Contents -- Preface -- List of Examples -- Greek Alphabet -- Part I. Calculus -- 1 Functions: General Properties -- 1.1 Mappings -- 1.2 Differentials and Derivatives -- 1.3 Partial Derivatives -- 1.4 Integrals -- 1.5 Critical Points -- 2 Functions: Examples -- 2.1 Algebraic Functions -- 2.2 Transcendental Functions -- 2.2.1 Logarithm and Exponential -- 2.2.2 Circular Functions -- 2.2.3 Gamma and Beta Functions -- 2.3 Functionals -- 3 Coordinate Systems -- 3.1 Points in Space -- 3.2 Coordinate Systems for Molecules -- 3.3 Abstract Coordinates -- 3.4 Constraints -- 3.4.1 Degrees of Freedom -- 3.4.2 Constrained Extrema -- 3.5 Differential Operators in Polar Coordinates -- 4 Integration -- 4.1 Change of Variables in Integrands -- 4.1.1 Change of Variable: Examples -- 4.1.2 Jacobian Determinant -- 4.2 Gaussian Integrals -- 4.3 Improper Integrals -- 4.4 Dirac Delta Function -- 4.5 Line Integrals -- 5 Numerical Methods -- 5.1 Interpolation -- 5.2 Numerical Differentiation -- 5.3 Numerical Integration -- 5.4 Random Numbers -- 5.5 Root Finding -- 5.6 Minimization -- 6 Complex Numbers -- 6.1 Complex Arithmetic -- 6.2 Fundamental Theorem of Algebra -- 6.3 The Argand Diagram -- 6.4 Functions of a Complex Variable -- 6.5 Branch Cuts -- 7 Extrapolation -- 7.1 Taylor Series -- 7.2 Partial Sums -- 7.3 Applications of Taylor Series -- 7.4 Convergence -- 7.5 Summation Approximants -- Part II. Statistics -- 8 Estimation -- 8.1 Error and Estimation -- 8.2 Probability Distributions -- 8.2.1 Probability Distribution Functions -- 8.2.2 The Normal Distribution -- 8.2.3 The Poisson Distribution -- 8.2.4 The Binomial Distribution -- 8.2.5 The Boltzmann Distribution -- 8.3 Outliers -- 8.4 Robust Estimation -- 9 Analysis of Significance -- 9.1 Confidence Intervals -- 9.2 Propagation of Error.

9.3 Monte Carlo Simulation of Error -- 9.4 Significance of Difference -- 9.5 Distribution Testing -- 10 Fitting -- 10.1 Method of Least Squares -- 10.1.1 Polynomial Fitting -- 10.1.2 Weighted Least Squares -- 10.1.3 Generalizations of the Least-Squares Method -- 10.2 Fitting with Error in Both Variables -- 10.2.1 Uncontrolled Error in x -- 10.2.2 Controlled Error in x -- 10.3 Nonlinear Fitting -- 11 Quality of Fit -- 11.1 Confidence Intervals for Parameters -- 11.2 Confidence Band for a Calibration Line -- 11.3 Outliers and Leverage Points -- 11.4 Robust Fitting -- 11.5 Model Testing -- 12 Experiment Design -- 12.1 Risk Assessment -- 12.2 Randomization -- 12.3 Multiple Comparisons -- 12.3.1 ANOVA -- 12.3.2 Post-Hoc Tests -- 12.4 Optimization -- Part III. Differential Equations -- 13 Examples of Differential Equations -- 13.1 Chemical Reaction Rates -- 13.2 Classical Mechanics -- 13.2.1 Newtonian Mechanics -- 13.2.2 Lagrangian and Hamiltonian Mechanics -- 13.2.3 Angular Momentum -- 13.3 Differentials in Thermodynamics -- 13.4 Transport Equations -- 14 Solving Differential Equations, I -- 14.1 Basic Concepts -- 14.2 The Superposition Principle -- 14.3 First-Order ODE's -- 14.4 Higher-Order ODE's -- 14.5 Partial Differential Equations -- 15 Solving Differential Equations, II -- 15.1 Numerical Solution -- 15.1.1 Basic Algorithms -- 15.1.2 The Leapfrog Method -- 15.1.3 Systems of Differential Equations -- 15.2 Chemical Reaction Mechanisms -- 15.3 Approximation Methods -- 15.3.1 Taylor Series -- 15.3.2 Perturbation Theory -- Part IV. Linear Algebra -- 16 Vector Spaces -- 16.1 Cartesian Coordinate Vectors -- 16.2 Sets -- 16.3 Groups -- 16.4 Vector Spaces -- 16.5 Functions as Vectors -- 16.6 Hilbert Spaces -- 16.7 Basis Sets -- 17 Spaces of Functions -- 17.1 Orthogonal Polynomials -- 17.2 Function Resolution -- 17.3 Fourier Series -- 17.4 Spherical Harmonics.

18 Matrices -- 18.1 Matrix Representation of Operators -- 18.2 Matrix Algebra -- 18.3 Matrix Operations -- 18.4 Pseudoinverse -- 18.5 Determinants -- 18.6 Orthogonal and Unitary Matrices -- 18.7 Simultaneous Linear Equations -- 19 Eigenvalue Equations -- 19.1 Matrix Eigenvalue Equations -- 19.2 Matrix Diagonalization -- 19.3 Differential Eigenvalue Equations -- 19.4 Hermitian Operators -- 19.5 The Variational Principle -- 20 Schrödinger's Equation -- 20.1 Quantum Mechanics -- 20.1.1 Quantum Mechanical Operators -- 20.1.2 The Wavefunction -- 20.1.3 The Basic Postulates -- 20.2 Atoms and Molecules -- 20.3 The One-Electron Atom -- 20.3.1 Orbitals -- 20.3.2 The Radial Equation -- 20.4 Hybrid Orbitals -- 20.5 Antisymmetry -- 20.6 Molecular Orbitals -- 21 Fourier Analysis -- 21.1 The Fourier Transform -- 21.2 Spectral Line Shapes -- 21.3 Discrete Fourier Transform -- 21.4 Signal Processing -- 21.4.1 Noise Filtering -- 21.4.2 Convolution -- A Computer Programs -- A.1 Robust Estimators -- A.2 FREML -- A.3 Nelder-Mead Simplex Optimization -- B Answers to Selected Exercises -- C Bibliography -- Index.