Mathematical Methods for Physical and Analytical Chemistry – David Z. Goodson – 1st Edition

Description

Mathematical Methods for Physical and Analytical Chemistry presents mathematical and statistical methods to students of chemistry at the intermediate, post-calculus level.

The content includes a review of general calculus; a review of numerical techniques often omitted from calculus courses, such as cubic splines and Newton’s method; a detailed treatment of statistical methods for experimental data analysis; complex numbers; extrapolation; linear algebra; and differential equations.

With numerous example problems and helpful anecdotes, this text gives chemistry students the mathematical knowledge they need to understand the analytical and physical chemistry professional literature.

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  • Preface xiii
    List of Examples xv
    Greek Alphabet xix

    PART I. CALCULUS

    1 Functions: General Properties 3
    1.1 Mappings 3
    1.2 Differentials and Derivatives 4
    1.3 Partial Derivatives 7
    1.4 Integrals 9
    1.5 Critical Points 14

    2 Functions: Examples 19
    2.1 Algebraic Functions 19
    2.2 Transcendental Functions 21
    2.3 Functional 31

    3 Coordinate Systems 33
    3.1 Points in Space 33
    3.2 Coordinate Systems for Molecules 35
    3.3 Abstract Coordinates 37
    3.4 Constraints 39
    3.5 Differential Operators in Polar Coordinates 43

    4 Integration 47
    4.1 Change of Variables in Integrands 47
    4.2 Gaussian Integrals 51
    4.3 Improper Integrals 53
    4.4 Dirac Delta Function 56
    4.5 Line Integrals 57

    5 Numerical Methods 61
    5.1 Interpolation 61
    5.2 Numerical Differentiation 63
    5.3 Numerical Integration 65
    5.4 Random Numbers 70
    5.5 Root Finding 71
    5.6 Minimization* 74

    6 Complex Numbers 79
    6.1 Complex Arithmetic 79
    6.2 Fundamental Theorem of Algebra 81
    6.3 The Argand Diagram 83
    6.4 Functions of a Complex Variable* 87
    6.5 Branch Cuts* 89

    7 Extrapolation 93
    7.1 Taylor Series 93
    7.2 Partial Sums 97
    7.3 Applications of Taylor Series 99
    7.4 Convergence 102
    7.5 Summation Approximants* 104

    PART II. STATISTICS

    8 Estimation 111
    8.1 Error and Estimation Ill
    8.2 Probability Distributions 113
    8.3 Outliers 124
    8.4 Robust Estimation 126

    9 Analysis of Significance 131
    9.1 Confidence Intervals 131
    9.2 Propagation of Error 136
    9.3 Monte Carlo Simulation of Error 139
    9.4 Significance of Difference 140
    9.5 Distribution Testing* 144

    10 Fitting 151
    10.1 Method of Least Squares 151
    10.2 Fitting with Error in Both Variables 157
    10.3 Nonlinear Fitting 162

    11 Quality of Fit 165
    11.1 Confidence Intervals for Parameters 165
    11.2 Confidence Band for a Calibration Line 168
    11.3 Outliers and Leverage Points ' 171
    11.4 Robust Fitting* 173
    11.5 Model Testing 176

    12 Experiment Design 181
    12.1 Risk Assessment 181
    12.2 Randomization 185
    12.3 Multiple Comparisons 188
    12.4 Optimization* 195

    PART III. DIFFERENTIAL EQUATIONS

    13 Examples of Differential Equations 203
    13.1 Chemical Reaction Rates 203
    13.2 Classical Mechanics 205
    13.3 Differentials in Thermodynamics 212
    13.4 Transport Equations 213

    14 Solving Differential Equations, I 217
    14.1 Basic Concepts 217
    14.2 The Superposition Principle 220
    14.3 First-Order ODE's 222
    14.4 Higher-Order ODE's 225
    14.5 Partial Differential Equations 228

    15 Solving Differential Equations, II 231
    15.1 Numerical Solution 231
    15.2 Chemical Reaction Mechanisms 236
    15.3 Approximation Methods 239

    PART IV. LINEAR ALGEBRA

    16 Vector Spaces 247
    16.1 Cartesian Coordinate Vectors 247
    16.2 Sets 248
    16.3 Groups 249
    16.4 Vector Spaces 251
    16.5 Functions as Vectors 252
    16.6 Hilbert Spaces 253
    16.7 Basis Sets 256

    17 Spaces of Functions 261
    17.1 Orthogonal Polynomials 261
    17.2 Function Resolution 267
    17.3 Fourier Series 270
    17.4 Spherical Harmonics 275

    18 Matrices 279
    18.1 Matrix Representation of Operators 279
    18.2 Matrix Algebra 282
    18.3 Matrix Operations 284
    18.4 Pseudoinverse* 286
    18.5 Determinants 288
    18.6 Orthogonal and Unitary Matrices 290
    18.7 Simultaneous Linear Equations 292

    19 Eigenvalue Equations 297
    19.1 Matrix Eigenvalue Equations 297
    19.2 Matrix Diagonalization 301
    19.3 Differential Eigenvalue Equations 305
    19.4 Hermitian Operators 306
    19.5 The Variational Principle* 309

    20 Schrödinger's Equation 313
    20.1 Quantum Mechanics 313
    20.2 Atoms and Molecules 319
    20.3 The One-Electron Atom 321
    20.4 Hybrid Orbitals 325
    20.5 Antisymmetry* 327
    20.6 Molecular Orbitals* 329

    21 Fourier Analysis 333
    21.1 The Fourier Transform 333
    21.2 Spectral Line Shapes* 336
    21.3 Discrete Fourier Transform* 339
    21.4 Signal Processing 342

    A Computer Programs 351
    A.l Robust Estimators 351
    A.2 FREML 352
    A.3 Neider-Mead Simplex Optimization 352
    B Answers to Selected Exercises 355
    C Bibliography 367
    Index 373
  • Citation
    • Full Title: Mathematical Methods for Physical and Analytical Chemistry
    • Author/s:
    • ISBN-13: 9780470473542
    • Edition: 1st Edition
    • Topic: Math
    • Subtopic: Advanced Mathematics
    • File Type: eBook
    • Idioma: English

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