| |
| |
| Preface | |
| |
| |
| List of Examples | |
| |
| |
| Greek Alphabet | |
| |
| |
| |
| Calculus | |
| |
| |
| |
| Functions: General Properties | |
| |
| |
| |
| Mappings | |
| |
| |
| |
| Differentials and Derivatives | |
| |
| |
| |
| Partial Derivatives | |
| |
| |
| |
| Integrals | |
| |
| |
| |
| Critical Points | |
| |
| |
| |
| Functions: Examples | |
| |
| |
| |
| Algebraic Functions | |
| |
| |
| |
| Transcendental Functions | |
| |
| |
| |
| Logarithm and Exponential | |
| |
| |
| |
| Circular Functions | |
| |
| |
| |
| Gamma and Beta Functions | |
| |
| |
| |
| Functionals | |
| |
| |
| |
| Coordinate Systems | |
| |
| |
| |
| Points in Space | |
| |
| |
| |
| Coordinate Systems for Molecules | |
| |
| |
| |
| Abstract Coordinates | |
| |
| |
| |
| Constraints | |
| |
| |
| |
| Degrees of Freedom | |
| |
| |
| |
| Constrained Extrema* | |
| |
| |
| |
| Differential Operators in Polar Coordinates | |
| |
| |
| |
| Integration | |
| |
| |
| |
| Change of Variables in Integrands | |
| |
| |
| |
| Change of Variable: Examples | |
| |
| |
| |
| Jacobian Determinant | |
| |
| |
| |
| Gaussian Integrals | |
| |
| |
| |
| Improper Integrals | |
| |
| |
| |
| Dirac Delta Function | |
| |
| |
| |
| Line Integrals | |
| |
| |
| |
| Numerical Methods | |
| |
| |
| |
| Interpolation | |
| |
| |
| |
| Numerical Differentiation | |
| |
| |
| |
| Numerical Integration | |
| |
| |
| |
| Random Numbers | |
| |
| |
| |
| Root Finding | |
| |
| |
| |
| Minimization* | |
| |
| |
| |
| Complex Numbers | |
| |
| |
| |
| Complex Arithmetic | |
| |
| |
| |
| Fundamental Theorem of Algebra | |
| |
| |
| |
| The Argand Diagram | |
| |
| |
| |
| Functions of a Complex Variable* | |
| |
| |
| |
| Branch Cuts* | |
| |
| |
| |
| Extrapolation | |
| |
| |
| |
| Taylor Series | |
| |
| |
| |
| Partial Sums | |
| |
| |
| |
| Applications of Taylor Series | |
| |
| |
| |
| Convergence | |
| |
| |
| |
| Summation Approximants* | |
| |
| |
| |
| Statistics | |
| |
| |
| |
| Estimation | |
| |
| |
| |
| Error and Estimation | |
| |
| |
| |
| Probability Distributions | |
| |
| |
| |
| Probability Distribution Functions | |
| |
| |
| |
| The Normal Distribution | |
| |
| |
| |
| The Poisson Distribution | |
| |
| |
| |
| The Binomial Distribution* | |
| |
| |
| |
| The Boltzmann Distribution* | |
| |
| |
| |
| Outliers | |
| |
| |
| |
| Robust Estimation | |
| |
| |
| |
| Analysis of Significance | |
| |
| |
| |
| Confidence Intervals | |
| |
| |
| |
| Propagation of Error | |
| |
| |
| |
| Monte Carlo Simulation of Error | |
| |
| |
| |
| Significance of Difference | |
| |
| |
| |
| Distribution Testing* | |
| |
| |
| |
| Fitting | |
| |
| |
| |
| Method of Least Squares | |
| |
| |
| |
| Polynomial Fitting | |
| |
| |
| |
| Weighted Least Squares | |
| |
| |
| |
| Generalizations of the Least-Squares Method* | |
| |
| |
| |
| Fitting with Error in Both Variables | |
| |
| |
| |
| Uncontrolled Error in x | |
| |
| |
| |
| Controlled Error in x | |
| |
| |
| |
| Nonlinear Fitting | |
| |
| |
| |
| Quality of Fit | |
| |
| |
| |
| Confidence Intervals for Parameters | |
| |
| |
| |
| Confidence Band for a Calibration Line | |
| |
| |
| |
| Outliers and Leverage Points | |
| |
| |
| |
| Robust Fitting* | |
| |
| |
| |
| Model Testing | |
| |
| |
| |
| Experiment Design | |
| |
| |
| |
| Risk Assessment | |
| |
| |
| |
| Randomization | |
| |
| |
| |
| Multiple Comparisons | |
| |
| |
| |
| ANOVA* | |
| |
| |
| |
| Post-Hoc Tests* | |
| |
| |
| |
| Optimization* | |
| |
| |
| |
| Differential Equations | |
| |
| |
| |
| Examples of Differential Equations | |
| |
| |
| |
| Chemical Reaction Rates | |
| |
| |
| |
| Classical Mechanics | |
| |
| |
| |
| Newtonian Mechanics | |
| |
| |
| |
| Lagrangian and Hamiltonian Mechanics | |
| |
| |
| |
| Angular Momentum | |
| |
| |
| |
| Differentials in Thermodynamics | |
| |
| |
| |
| Transport Equations | |
| |
| |
| |
| Solving Differential Equations, I | |
| |
| |
| |
| Basic Concepts | |
| |
| |
| |
| The Superposition Principle | |
| |
| |
| |
| First-Order ODE's | |
| |
| |
| |
| Higher-Order ODE's | |
| |
| |
| |
| Partial Differential Equations | |
| |
| |
| |
| Solving Differential Equations, II | |
| |
| |
| |
| Numerical Solution | |
| |
| |
| |
| Basic Algorithms | |
| |
| |
| |
| The Leapfrog Method* | |
| |
| |
| |
| Systems of Differential Equations | |
| |
| |
| |
| Chemical Reaction Mechanisms | |
| |
| |
| |
| Approximation Methods | |
| |
| |
| |
| Taylor Series* | |
| |
| |
| |
| Perturbation Theory* | |
| |
| |
| |
| Linear Algebra | |
| |
| |
| |
| Vector Spaces | |
| |
| |
| |
| Cartesian Coordinate Vectors | |
| |
| |
| |
| Sets | |
| |
| |
| |
| Groups | |
| |
| |
| |
| Vector Spaces | |
| |
| |
| |
| Functions as Vectors | |
| |
| |
| |
| Hilbert Spaces | |
| |
| |
| |
| Basis Sets | |
| |
| |
| |
| Spaces of Functions | |
| |
| |
| |
| Orthogonal Polynomials | |
| |
| |
| |
| Function Resolution | |
| |
| |
| |
| Fourier Series | |
| |
| |
| |
| Spherical Harmonics | |
| |
| |
| |
| Matrices | |
| |
| |
| |
| Matrix Representation of Operators | |
| |
| |
| |
| Matrix Algebra | |
| |
| |
| |
| Matrix Operations | |
| |
| |
| |
| Pseudoinverse* | |
| |
| |
| |
| Determinants | |
| |
| |
| |
| Orthogonal and Unitary Matrices | |
| |
| |
| |
| Simultaneous Linear Equations | |
| |
| |
| |
| Eigenvalue Equations | |
| |
| |
| |
| Matrix Eigenvalue Equations | |
| |
| |
| |
| Matrix Diagonalization | |
| |
| |
| |
| Differential Eigenvalue Equations | |
| |
| |
| |
| Hermitian Operators | |
| |
| |
| |
| The Variational Principle* | |
| |
| |
| |
| Schr�dinger's Equation | |
| |
| |
| |
| Quantum Mechanics | |
| |
| |
| |
| Quantum Mechanical Operators | |
| |
| |
| |
| The Wavefunction | |
| |
| |
| |
| The Basic Postulates* | |
| |
| |
| |
| Atoms and Molecules | |
| |
| |
| |
| The One-Electron Atom | |
| |
| |
| |
| Orbitals | |
| |
| |
| |
| The Radial Equation* | |
| |
| |
| |
| Hybrid Orbitals | |
| |
| |
| |
| Antisymmetry* | |
| |
| |
| |
| Molecular Orbitals* | |
| |
| |
| |
| Fourier Analysis | |
| |
| |
| |
| The Fourier Transform | |
| |
| |
| |
| Spectral Line Shapes* | |
| |
| |
| |
| Discrete Fourier Transform* | |
| |
| |
| |
| Signal Processing | |
| |
| |
| |
| Noise Filtering* | |
| |
| |
| |
| Convolution* | |
| |
| |
| |
| Computer Programs | |
| |
| |
| |
| Robust Estimators | |
| |
| |
| |
| FREML | |
| |
| |
| |
| Nelder-Mead Simplex Optimization | |
| |
| |
| |
| Answers to Selected Exercises | |
| |
| |
| |
| Bibliography | |
| |
| |
| Index | |