Symbolic Mathematics for Chemists

Professor Fred Senese is a computational chemist at Frostburg State University with a particular focus on chemical education. His research interests include applications of artificial intelligence in chemical education, development of web-based narratives and construction kits for chemical education, remote control and access of instrumentation, and environmental chemical analysis applied to problems in ethnobotany.

• Preface xiii1 Fundamentals 11.1 Getting Started With wxMaxima 11.1.1 Input Cells 21.1.2 The Toolbar 31.1.3 The Menus 31.1.4 Command History 41.1.5 Basic Arithmetic 51.1.6 Mathematical Functions 71.1.7 Assigning Variables 81.1.8 Defining Functions 101.1.9 Comments, Images, and Sectioning 121.2 A Tour of the General Math Pane 121.2.1 Basic Plotting 131.2.1.1 Plotting Multiple Curves 141.2.1.2 Parametric Plots 151.2.1.3 Discrete Plots 151.2.1.4 Three-Dimensional Plots 171.2.2 Basic Algebra 181.2.2.1 Equations 181.2.2.2 Substitutions 181.2.2.3 Simplification 201.2.2.4 Solving Equations 211.2.2.5 Simplifying Trigonometric and Exponential Functions 211.2.3 Basic Calculus 221.2.3.1 Limits 221.2.3.2 Differentiation 231.2.3.3 Series 241.2.3.4 Integration 251.2.4 Differential Equations 281.3 Controlling Execution 281.4 Using Packages 302 Storing and Transforming Data 332.1 Numbers 332.1.1 Floating Point Numbers 332.1.2 Integers and Rational Numbers 372.1.3 Complex Numbers 382.1.4 Constants 422.1.5 Units and Physical Constants 432.2 Boolean Expressions and Predicates 472.2.1 Relational Operators 472.2.2 Logical Operators 482.2.3 Predicates 492.3 Lists 512.3.1 List Assignments 512.3.2 Indexing List Items 522.3.3 Arithmetic with Lists 522.3.4 Building and Editing Lists 542.3.4.1 Adding Items 542.3.4.2 Deleting Items 552.3.5 Nested Lists 552.3.6 Sublists 562.4 Matrices 572.4.1 Row and Column Vectors 572.4.2 Indexing Matrices 582.4.3 Entering Matrices 592.4.4 Assigning Matrices 602.4.5 Editing Matrices 612.4.6 Reading and Writing Matrices From Files 632.4.7 Transforming Data in a Matrix 652.5 Strings 662.5.1 Using String Functions toWork with Files 673 Plotting Data and Functions 713.1 Plotting in Two Dimensions 713.1.1 Changing Plot Size and Resolution 713.1.2 Plotting Multiple Curves 733.1.3 Changing Axis Ranges 743.1.4 Plotting Complex Functions 743.1.5 Plotting Data 743.1.5.1 Plotting Data in Separate X, Y Lists 753.1.5.2 Plotting Data as Lists of X, Y Points 753.1.5.3 Plotting Data in Matrices 763.1.5.4 Plotting Data with Units 763.1.5.5 Plotting Functions and Data Together 773.1.6 Adding Text Labels to Graphs 773.1.7 Plotting Rapidly Rising Functions 783.1.7.1 Solving Axis Scaling Problems 813.1.7.2 Positioning the Legend 833.1.8 Parametric Plots 843.1.9 Implicit Plots 873.1.10 Histograms 893.2 Plotting inThree Dimensions 913.2.1 Plotting Functions of x, y, andz 913.2.2 Plotting Multiple Surfaces 933.2.3 Plotting in Spherical Coordinates 943.2.4 Plotting in Cylindrical Coordinates 953.2.5 Parametric Surface Plots 963.2.6 Plotting DiscreteThree-Dimensional Data 983.2.7 Contour Plotting 994 Programming Maxima 1034.1 Nouns and Verbs 1034.2 Writing Multiline Functions 1064.3 Decision Making 1084.4 Recursive Functions 1094.5 Contexts 1104.6 Iteration 1144.6.1 Indexed Loops 1144.6.2 Conditional Loops 1164.6.3 Looping Over Lists 1174.6.4 Nested Loops 1185 Algebra 1195.1 Series 1195.1.1 Simplifying Sums 1205.1.2 Reindexing and Combining Sums 1225.1.3 Applying Functions to Sums and Products 1235.2 Products 1245.3 Equations 1265.3.1 Simplifying Equations 1265.3.2 Simplifying Trigonometric and Exponential Functions 1275.3.3 Extracting Expressions From an Equation 1285.3.4 Expanding Expressions 1315.3.5 Factoring Expressions 1345.3.6 Substitution 1355.3.7 Solving an Equation Symbolically 1385.3.7.1 Handling Multiple Solutions 1395.3.8 Solving an Equation Numerically 1405.4 Systems of Equations 1415.4.1 Eliminating Variables 1415.4.2 Solving Systems of EquationsWithout Elimination 1435.5 Interpolation 1445.5.1 Piecewise Linear Interpolation 1465.5.2 Spline Interpolation 1476 Differentiation, Integration, and Minimization 1496.1.1 Limits for Discontinuous Functions 1516.1.2 Limits for Indefinite Functions 1526.2 Differentials 1536.3 Derivatives 1546.3.1 Explicit Partial and Total Derivatives 1566.3.2 Derivatives Evaluated at a Specific Point 1576.3.3 Higher-Order Derivatives 1586.3.4 Mixed Derivatives 1596.3.5 Assigning Partial Derivatives 1606.3.5.1 Partial Derivatives from Total Differential Expansions 1616.3.5.2 Writing Total Differential Expansions in Terms of New Variables 1616.3.6 Implicit Differentiation 1626.4 Maxima, Minima, and Inflection Points 1646.4.1 Critical Points of Surfaces 1676.4.2 Numerical Minimization 1696.5 Integration 1736.5.1 Integration Constants 1746.5.2 Definite Integration 1746.5.3 When Symbolic Integration Fails 1756.5.4 Numerical Integration 1786.5.4.1 Numerical Integration over Infinite Intervals 1796.5.4.2 Numerical Integration with Strongly Oscillating Integrands 1806.5.4.3 Numerical Integration with Discontinuous Integrands 1816.5.5 Multiple Integration 1826.5.6 Discrete Integration 1836.6 Power Series 1866.6.1 Testing Power Series for Convergence 1866.7 Taylor Series 1876.7.1 Exploring Function Properties with Taylor Series 1886.7.2 The Remainder Term 1906.7.3 Taylor Series for Multivariate Functions 1916.7.4 Approximating Taylor Series 1917 Matrices and Vectors 1937.1 Vectors 1937.1.1 Vector Arithmetic 1947.1.2 The Dot Product 1957.1.3 Vector Lengths and Angles 1967.1.4 The Cross Product 1977.1.5 Angular Momentum 1987.1.6 Vector Algebra 1997.2 Matrices 2007.2.1 Matrix Arithmetic 2017.2.2 The Transpose 2017.2.3 The Matrix Product 2027.2.4 Determinants 2037.2.5 The Inverse of a Matrix 2067.2.6 Matrix Algebra 2077.2.7 Eigenvalues and Eigenvectors 2117.2.7.1 Application: Energies and Molecular Orbitals of Ethylene 2127.2.7.2 Eigenvalues and Eigenvectors for Symmetric Matrices 2147.2.7.3 Matrix Diagonalization 2167.3 Vector Calculus 2177.3.1 Derivative of a Vector with Respect to a Scalar 2177.3.2 The Jacobian 2187.3.3 The Gradient 2207.3.4 The Laplacian 2227.3.5 The Divergence 2247.3.6 The Curl 2258 Error Analysis 2278.1 Classifying Experimental Errors 2278.1.1 Systematic Error 2298.1.2 Random Error 2308.2 Probability Density 2308.2.1 Discrete Probability Distributions 2308.2.2 The Poisson Distribution 2328.2.3 Continuous Probability Distributions 2358.2.4 The Normal Distribution 2368.3 Estimating Precision 2388.3.1 Standard Error of the Mean 2408.3.2 Confidence Interval of the Mean 2408.4 Hypothesis Testing 2418.4.1 Comparing a Mean with a True Value 2438.4.2 Comparing Variances 2448.4.3 Comparing Two Sample Means 2468.5 Propagation of Error 2498.5.1 Propagation of Independent Systematic Errors 2498.5.2 Propagation of Independent Random Errors 2518.5.3 Covariance and Correlation 2539 Fitting Data to a Straight Line 2579.1 The Ordinary Least-Squares Method 2599.1.1 Using Built-In Functions 2609.1.2 Error Estimates for the Slope and the Intercept 2639.1.3 The Determination Coefficient 2669.1.4 Residual Analysis 2689.1.5 Testing the Fit Parameters 2719.1.6 Testing for Lack-of-Fit 2729.2 Multiple Linear Regression 2749.2.1 Matrix Form of Multiple Linear Regression 2759.2.2 Estimating the Errors in the Fit Parameters in MLR 2779.2.3 Example: Microwave Rotational Spectrum of HCl 2789.2.4 Detecting and Dealing with Outliers 2819.3 WLS 2859.3.1 The Fit Parameters inWLS 2869.3.2 Error Estimates for theWLS Fit Parameters 2869.3.3 Finding theWeights 2879.3.4 Residual Analysis inWLS 2889.3.5 Evaluating Goodness-of-Fit 2889.4 Fitting Data to a Line with Errors in Both X and Y 2899.4.1 Finding Fit Parameters in TLS 2909.4.2 Error Estimates for the TLS Fit Parameters 2929.4.3 Assessing Goodness-of-Fit in TLS 2939.4.4 Multiple Linear Regression with TLS 2939.5 Calibration and Standard Additions 2949.5.1 Error Estimates for Calibrated Values 2949.5.2 Standard Additions 29510 Fitting Data to a Curve 29910.1 Transforming Data to a Linear Form 29910.2 Polynomial Least-Squares Fitting 30210.2.1 How Many Fit Parameters Are Needed? 30410.3 Nonlinear Least-Squares Models 30610.4 Estimating Error in Nonlinear Fit Parameters 31010.4.1 Estimating Parameter Errors with the Jackknife Method 31110.4.2 Estimating Parameter Errors with the Bootstrap Method 31311 Differential Equations 31711.1 Symbolic Solutions of ODEs 31811.1.1 Initial Value Problems 32011.1.2 Boundary Value Problems 32211.2 Power Series Solution of ODEs 32511.3 Direction Fields 32911.3.1 Direction Fields with Adjustable Parameters 33111.3.2 Direction Fields and Autonomous Equations 33211.4 Solving Systems of Linear Differential Equations 33511.5 Numerical Solution of ODEs 33811.6 Solving Partial Differential Equations 34012 Operators and Integral Transforms 34312.1 Defining Operators 34412.2 Fourier Series 34712.3 Fourier Transforms 35112.3.1 The Fast Fourier Transform 35512.4 The Laplace Transform 357Glossary 359References 367Index 371