An essential guide to using Maxima, a popular open source symbolic mathematics engine to solve problems, build models, analyze data and explore fundamental concepts
Symbolic Mathematics for Chemists offers students of chemistry a guide to Maxima, a popular open source symbolic mathematics engine that can be used to solve problems, build models, analyze data, and explore fundamental chemistry concepts. The author - a noted expert in the field - focuses on the analysis of experimental data obtained in a laboratory setting and the fitting of data and modeling experiments. The text contains a wide variety of illustrative examples and applications in physical chemistry, quantitative analysis and instrumental techniques.
Designed as a practical resource, the book is organized around a series of worksheets that are provided in a companion website. Each worksheet has clearly defined goals and learning objectives and a detailed abstract that provides motivation and context for the material. This important resource:
- Offers an text that shows how to use popular symbolic mathematics engines to solve problems
- Includes a series of worksheet that are prepared in Maxima
- Contains step-by-step instructions written in clear terms and includes illustrative examples to enhance critical thinking, creative problem solving and the ability to connect concepts in chemistry
- Offers hints and case studies that help to master the basics while proficient users are offered more advanced avenues for exploration
Written for advanced undergraduate and graduate students in chemistry and instructors looking to enhance their lecture or lab course with symbolic mathematics materials, Symbolic Mathematics for Chemists: A Guide for Maxima Users is an essential resource for solving and exploring quantitative problems in chemistry.
Table of Contents
Preface xiii
1 Fundamentals 1
1.1 Getting Started With wxMaxima 1
1.1.1 Input Cells 2
1.1.2 The Toolbar 3
1.1.3 The Menus 3
1.1.4 Command History 4
1.1.5 Basic Arithmetic 5
1.1.6 Mathematical Functions 7
1.1.7 Assigning Variables 8
1.1.8 Defining Functions 10
1.1.9 Comments, Images, and Sectioning 12
1.2 A Tour of the General Math Pane 12
1.2.1 Basic Plotting 13
1.2.1.1 Plotting Multiple Curves 14
1.2.1.2 Parametric Plots 15
1.2.1.3 Discrete Plots 15
1.2.1.4 Three-Dimensional Plots 17
1.2.2 Basic Algebra 18
1.2.2.1 Equations 18
1.2.2.2 Substitutions 18
1.2.2.3 Simplification 20
1.2.2.4 Solving Equations 21
1.2.2.5 Simplifying Trigonometric and Exponential Functions 21
1.2.3 Basic Calculus 22
1.2.3.1 Limits 22
1.2.3.2 Differentiation 23
1.2.3.3 Series 24
1.2.3.4 Integration 25
1.2.4 Differential Equations 28
1.3 Controlling Execution 28
1.4 Using Packages 30
2 Storing and Transforming Data 33
2.1 Numbers 33
2.1.1 Floating Point Numbers 33
2.1.2 Integers and Rational Numbers 37
2.1.3 Complex Numbers 38
2.1.4 Constants 42
2.1.5 Units and Physical Constants 43
2.2 Boolean Expressions and Predicates 47
2.2.1 Relational Operators 47
2.2.2 Logical Operators 48
2.2.3 Predicates 49
2.3 Lists 51
2.3.1 List Assignments 51
2.3.2 Indexing List Items 52
2.3.3 Arithmetic with Lists 52
2.3.4 Building and Editing Lists 54
2.3.4.1 Adding Items 54
2.3.4.2 Deleting Items 55
2.3.5 Nested Lists 55
2.3.6 Sublists 56
2.4 Matrices 57
2.4.1 Row and Column Vectors 57
2.4.2 Indexing Matrices 58
2.4.3 Entering Matrices 59
2.4.4 Assigning Matrices 60
2.4.5 Editing Matrices 61
2.4.6 Reading and Writing Matrices From Files 63
2.4.7 Transforming Data in a Matrix 65
2.5 Strings 66
2.5.1 Using String Functions toWork with Files 67
3 Plotting Data and Functions 71
3.1 Plotting in Two Dimensions 71
3.1.1 Changing Plot Size and Resolution 71
3.1.2 Plotting Multiple Curves 73
3.1.3 Changing Axis Ranges 74
3.1.4 Plotting Complex Functions 74
3.1.5 Plotting Data 74
3.1.5.1 Plotting Data in Separate X, Y Lists 75
3.1.5.2 Plotting Data as Lists of X, Y Points 75
3.1.5.3 Plotting Data in Matrices 76
3.1.5.4 Plotting Data with Units 76
3.1.5.5 Plotting Functions and Data Together 77
3.1.6 Adding Text Labels to Graphs 77
3.1.7 Plotting Rapidly Rising Functions 78
3.1.7.1 Solving Axis Scaling Problems 81
3.1.7.2 Positioning the Legend 83
3.1.8 Parametric Plots 84
3.1.9 Implicit Plots 87
3.1.10 Histograms 89
3.2 Plotting inThree Dimensions 91
3.2.1 Plotting Functions of x, y, andz 91
3.2.2 Plotting Multiple Surfaces 93
3.2.3 Plotting in Spherical Coordinates 94
3.2.4 Plotting in Cylindrical Coordinates 95
3.2.5 Parametric Surface Plots 96
3.2.6 Plotting DiscreteThree-Dimensional Data 98
3.2.7 Contour Plotting 99
4 Programming Maxima 103
4.1 Nouns and Verbs 103
4.2 Writing Multiline Functions 106
4.3 Decision Making 108
4.4 Recursive Functions 109
4.5 Contexts 110
4.6 Iteration 114
4.6.1 Indexed Loops 114
4.6.2 Conditional Loops 116
4.6.3 Looping Over Lists 117
4.6.4 Nested Loops 118
5 Algebra 119
5.1 Series 119
5.1.1 Simplifying Sums 120
5.1.2 Reindexing and Combining Sums 122
5.1.3 Applying Functions to Sums and Products 123
5.2 Products 124
5.3 Equations 126
5.3.1 Simplifying Equations 126
5.3.2 Simplifying Trigonometric and Exponential Functions 127
5.3.3 Extracting Expressions From an Equation 128
5.3.4 Expanding Expressions 131
5.3.5 Factoring Expressions 134
5.3.6 Substitution 135
5.3.7 Solving an Equation Symbolically 138
5.3.7.1 Handling Multiple Solutions 139
5.3.8 Solving an Equation Numerically 140
5.4 Systems of Equations 141
5.4.1 Eliminating Variables 141
5.4.2 Solving Systems of EquationsWithout Elimination 143
5.5 Interpolation 144
5.5.1 Piecewise Linear Interpolation 146
5.5.2 Spline Interpolation 147
6 Differentiation, Integration, and Minimization 149
6.1.1 Limits for Discontinuous Functions 151
6.1.2 Limits for Indefinite Functions 152
6.2 Differentials 153
6.3 Derivatives 154
6.3.1 Explicit Partial and Total Derivatives 156
6.3.2 Derivatives Evaluated at a Specific Point 157
6.3.3 Higher-Order Derivatives 158
6.3.4 Mixed Derivatives 159
6.3.5 Assigning Partial Derivatives 160
6.3.5.1 Partial Derivatives from Total Differential Expansions 161
6.3.5.2 Writing Total Differential Expansions in Terms of New Variables 161
6.3.6 Implicit Differentiation 162
6.4 Maxima, Minima, and Inflection Points 164
6.4.1 Critical Points of Surfaces 167
6.4.2 Numerical Minimization 169
6.5 Integration 173
6.5.1 Integration Constants 174
6.5.2 Definite Integration 174
6.5.3 When Symbolic Integration Fails 175
6.5.4 Numerical Integration 178
6.5.4.1 Numerical Integration over Infinite Intervals 179
6.5.4.2 Numerical Integration with Strongly Oscillating Integrands 180
6.5.4.3 Numerical Integration with Discontinuous Integrands 181
6.5.5 Multiple Integration 182
6.5.6 Discrete Integration 183
6.6 Power Series 186
6.6.1 Testing Power Series for Convergence 186
6.7 Taylor Series 187
6.7.1 Exploring Function Properties with Taylor Series 188
6.7.2 The Remainder Term 190
6.7.3 Taylor Series for Multivariate Functions 191
6.7.4 Approximating Taylor Series 191
7 Matrices and Vectors 193
7.1 Vectors 193
7.1.1 Vector Arithmetic 194
7.1.2 The Dot Product 195
7.1.3 Vector Lengths and Angles 196
7.1.4 The Cross Product 197
7.1.5 Angular Momentum 198
7.1.6 Vector Algebra 199
7.2 Matrices 200
7.2.1 Matrix Arithmetic 201
7.2.2 The Transpose 201
7.2.3 The Matrix Product 202
7.2.4 Determinants 203
7.2.5 The Inverse of a Matrix 206
7.2.6 Matrix Algebra 207
7.2.7 Eigenvalues and Eigenvectors 211
7.2.7.1 Application: Energies and Molecular Orbitals of Ethylene 212
7.2.7.2 Eigenvalues and Eigenvectors for Symmetric Matrices 214
7.2.7.3 Matrix Diagonalization 216
7.3 Vector Calculus 217
7.3.1 Derivative of a Vector with Respect to a Scalar 217
7.3.2 The Jacobian 218
7.3.3 The Gradient 220
7.3.4 The Laplacian 222
7.3.5 The Divergence 224
7.3.6 The Curl 225
8 Error Analysis 227
8.1 Classifying Experimental Errors 227
8.1.1 Systematic Error 229
8.1.2 Random Error 230
8.2 Probability Density 230
8.2.1 Discrete Probability Distributions 230
8.2.2 The Poisson Distribution 232
8.2.3 Continuous Probability Distributions 235
8.2.4 The Normal Distribution 236
8.3 Estimating Precision 238
8.3.1 Standard Error of the Mean 240
8.3.2 Confidence Interval of the Mean 240
8.4 Hypothesis Testing 241
8.4.1 Comparing a Mean with a True Value 243
8.4.2 Comparing Variances 244
8.4.3 Comparing Two Sample Means 246
8.5 Propagation of Error 249
8.5.1 Propagation of Independent Systematic Errors 249
8.5.2 Propagation of Independent Random Errors 251
8.5.3 Covariance and Correlation 253
9 Fitting Data to a Straight Line 257
9.1 The Ordinary Least-Squares Method 259
9.1.1 Using Built-In Functions 260
9.1.2 Error Estimates for the Slope and the Intercept 263
9.1.3 The Determination Coefficient 266
9.1.4 Residual Analysis 268
9.1.5 Testing the Fit Parameters 271
9.1.6 Testing for Lack-of-Fit 272
9.2 Multiple Linear Regression 274
9.2.1 Matrix Form of Multiple Linear Regression 275
9.2.2 Estimating the Errors in the Fit Parameters in MLR 277
9.2.3 Example: Microwave Rotational Spectrum of HCl 278
9.2.4 Detecting and Dealing with Outliers 281
9.3 WLS 285
9.3.1 The Fit Parameters inWLS 286
9.3.2 Error Estimates for theWLS Fit Parameters 286
9.3.3 Finding theWeights 287
9.3.4 Residual Analysis inWLS 288
9.3.5 Evaluating Goodness-of-Fit 288
9.4 Fitting Data to a Line with Errors in Both X and Y 289
9.4.1 Finding Fit Parameters in TLS 290
9.4.2 Error Estimates for the TLS Fit Parameters 292
9.4.3 Assessing Goodness-of-Fit in TLS 293
9.4.4 Multiple Linear Regression with TLS 293
9.5 Calibration and Standard Additions 294
9.5.1 Error Estimates for Calibrated Values 294
9.5.2 Standard Additions 295
10 Fitting Data to a Curve 299
10.1 Transforming Data to a Linear Form 299
10.2 Polynomial Least-Squares Fitting 302
10.2.1 How Many Fit Parameters Are Needed? 304
10.3 Nonlinear Least-Squares Models 306
10.4 Estimating Error in Nonlinear Fit Parameters 310
10.4.1 Estimating Parameter Errors with the Jackknife Method 311
10.4.2 Estimating Parameter Errors with the Bootstrap Method 313
11 Differential Equations 317
11.1 Symbolic Solutions of ODEs 318
11.1.1 Initial Value Problems 320
11.1.2 Boundary Value Problems 322
11.2 Power Series Solution of ODEs 325
11.3 Direction Fields 329
11.3.1 Direction Fields with Adjustable Parameters 331
11.3.2 Direction Fields and Autonomous Equations 332
11.4 Solving Systems of Linear Differential Equations 335
11.5 Numerical Solution of ODEs 338
11.6 Solving Partial Differential Equations 340
12 Operators and Integral Transforms 343
12.1 Defining Operators 344
12.2 Fourier Series 347
12.3 Fourier Transforms 351
12.3.1 The Fast Fourier Transform 355
12.4 The Laplace Transform 357
Glossary 359
References 367
Index 371
