Symbolic Mathematics for Chemists. A Guide for Maxima Users

  • ID: 4518532
  • Book
  • 400 Pages
  • John Wiley and Sons Ltd
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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.

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Preface

1 Fundamentals

1.1 Getting Started With wxMaxima

1.1.1 Input Cells

1.1.2 The Toolbar

1.1.3 The Menus

1.1.4 Command History

1.1.5 Basic Arithmetic

1.1.6 Mathematical Functions

1.1.7 Assigning Variables

1.1.8 Defining Functions

1.1.9 Comments, Images, and Sectioning

1.2 A Tour of the General Math Pane

1.2.1 Basic Plotting

1.2.1.1 Plotting Multiple Curves

1.2.1.2 Parametric Plots

1.2.1.3 Discrete Plots

1.2.1.4 Three Dimensional Plots

1.2.2 Basic Algebra

1.2.2.1 Equations

1.2.2.2 Substitutions

1.2.2.3 Simplification

1.2.2.4 Solving equations

1.2.2.5 Simplifying trigonometric and exponential functions

1.2.3 Basic Calculus

1.2.3.1 Limits

1.2.3.2 Differentiation

1.2.3.3 Series

1.2.3.4 Integration

1.2.4 Differential Equations

1.3 Controlling Execution

1.4 Using Packages

2 Storing and Transforming Data

2.1 Numbers

2.1.1 Floating Point Numbers

2.1.2 Integers and Rational Numbers

2.1.3 Complex Numbers

2.1.4 Constants

2.1.5 Units and Physical Constants

2.2 Boolean Expressions and Predicates

2.2.1 Relational Operators

2.2.2 Logical Operators

2.2.3 Predicates

2.3 Lists

2.3.1 List Assignments

2.3.2 Indexing List Items

2.3.3 Arithmetic With Lists

2.3.4 Building and Editing Lists

2.3.4.1 Adding Items

2.3.4.2 Deleting Items

2.3.5 Nested Lists

2.3.6 Sublists

2.4 Matrices

2.4.1 Row and Column Vectors

2.4.2 Indexing Matrices

2.4.3 Entering Matrices

2.4.4 Assigning Matrices

2.4.5 Editing Matrices

2.4.6 Reading and Writing Matrices From Files

2.4.7 Transforming Data in a Matrix

2.5 Strings

2.5.1 Using String Functions to Work with Files

3 Plotting Data and Functions

3.1 Plotting in Two Dimensions

3.1.1 Changing Plot Size and Resolution

3.1.2 Plotting Multiple Curves

3.1.3 Changing Axis Ranges

3.1.4 Plotting Complex Functions

3.1.5 Plotting Data

3.1.5.1 Plotting Data in Separate X, Y Lists

3.1.5.2 Plotting Data as Lists of X, Y Points

3.1.5.3 Plotting Data in Matrices

3.1.5.4 Plotting Data with Units

3.1.5.5 Plotting Functions and Data Together

3.1.6 Adding Text Labels to Graphs

3.1.7 Plotting Rapidly Rising Functions

3.1.7.1 Solving Axis Scaling Problems

3.1.7.2 Positioning the Legend

3.1.8 Parametric Plots

3.1.9 Implicit Plots

3.1.10 Histograms

3.2 Plotting in Three Dimensions

3.2.1 Plotting Functions of x, y, and z

3.2.2 Plotting Multiple Surfaces

3.2.3 Plotting in Spherical Coordinates

3.2.4 Plotting in Cylindrical Coordinates

3.2.5 Parametric Surface Plots

3.2.6 Plotting Discrete Three–Dimensional Data

3.2.7 Contour Plotting

4 Programming Maxima

4.1 Nouns and Verbs

4.2 Writing Multi–Line Functions

4.3 Decision Making

4.4 Recursive Functions

4.5 Contexts

4.6 Iteration

4.6.1 Indexed Loops

4.6.2 Conditional Loops

4.6.3 Looping Over Lists

4.6.4 Nested Loops

5 Algebra

5.1 Series

5.1.1 Simplifying Sums

5.1.2 Reindexing and Combining Sums

5.1.3 Applying Functions to Sums and Products

5.2 Products

5.3 Equations

5.3.1 Simplifying Equations

5.3.2 Simplifying Trigonometric and Exponential Functions

5.3.3 Extracting Expressions From an Equation

5.3.4 Expanding Expressions

5.3.5 Factoring Expressions

5.3.6 Substitution

5.3.7 Solving an Equation Symbolically

5.3.7.1 Handling Multiple Solutions

5.3.8 Solving an Equation Numerically

5.4 Systems of Equations 1

5.4.1 Eliminating Variables

5.4.2 Solving Systems of Equations Without Elimination

5.5 Interpolation

5.5.1 Piecewise Linear Interpolation

5.5.2 Spline Interpolation

6 Differentiation, Integration, and Minimization

6.1 Limits

6.1.1 Limits for Discontinuous Functions

6.1.2 Limits for Indefinite Functions

6.2 Differentials

6.3 Derivatives

6.3.1 Explicit Partial and Total Derivatives

6.3.2 Derivatives Evaluated at a Specific Point

6.3.3 Higher Order Derivatives

6.3.4 Mixed Derivatives

6.3.5 Assigning Partial Derivatives

6.3.5.1 Partial Derivatives from Total Differential Expansions

6.3.5.2 Writing Total Differential Expansions in Terms of New Variables

6.3.6 Implicit Differentiation

6.4 Maxima, Minima, and Inflection Points

6.4.1 Critical Points of Surfaces

6.4.2 Numerical Minimization

6.5 Integration

6.5.1 Integration Constants

6.5.2 Definite Integration

6.5.3 When Symbolic Integration Fails

6.5.4 Numerical Integration

6.5.4.1 Numerical Integration over Infinite Interval

6.5.4.2 Numerical Integration with Strongly Oscillating Integrands

6.5.4.3 Numerical Integration with Discontinuous Integrands

6.5.5 Multiple Integration

6.5.6 Discrete Integration

6.6 Power Series

6.6.1 Testing Power Series for Convergence

6.7 Taylor Series

6.7.1 Exploring Function Properties with Taylor Series

6.7.2 The Remainder Term

6.7.3 Taylor Series for Multivariate Functions

6.7.4 Approximating Taylor Series

7 Matrices and Vectors

7.1 Vectors

7.1.1 Vector Arithmetic

7.1.2 The Dot Product

7.1.3 Vector Lengths and Angles

7.1.4 The Cross Product

7.1.5 Angular Momentum

7.1.6 Vector Algebra

7.2 Matrices

7.2.1 Matrix Arithmetic

7.2.2 The Transpose

7.2.3 The Matrix Product

7.2.4 Determinants

7.2.5 The Inverse of a Matrix

7.2.6 Matrix Algebra

7.2.7 Eigenvalues and Eigenvectors

7.2.7.1 Application: Energies and Molecular Orbitals of Ethylene

7.2.7.2 Eigenvalues and Eigenvectors for Symmetric Matrices

7.2.7.3 Matrix Diagonalization

7.3 Vector Calculus

7.3.1 Derivative of a Vector With Respect to a Scalar

7.3.2 The Jacobian

7.3.3 The Gradient

7.3.4 The Laplacian

7.3.5 The Divergence

7.3.6 The Curl

8 Error Analysis

8.1 Classifying Experimental Errors

8.1.1 Systematic Error

8.1.2 Random Error

8.2 Probability Density

8.2.1 Discrete Probability Distributions

8.2.2 The Poisson Distribution

8.2.3 Continuous Probability Distributions

8.2.4 The Normal Distribution

8.3 Estimating Precision

8.3.1 Standard Error of the Mean

8.3.2 Confidence Interval of the Mean

8.4 Hypothesis Testing

8.4.1 Comparing a Mean with a True Value

8.4.2 Comparing Variances

8.4.3 Comparing Two Sample Means

8.5 Propagation of Error

8.5.1 Propagation of Independent Systematic Errors

8.5.2 Propagation of Independent Random Errors

8.5.3 Covariance and Correlation

9 Fitting Data to a Straight Line

9.1 The Ordinary Least–Squares Method

9.1.1 Using Built–In Functions

9.1.2 Error Estimates for the Slope and the Intercept

9.1.3 The Determination Coefficient

9.1.4 Residual Analysis

9.1.5 Testing the Fit Parameters

9.1.6 Testing for Lack–Of–Fit

9.2 Multiple Linear Regression

9.2.1 Matrix Form of Multiple Linear Regression

9.2.2 Estimating the Errors in the Fit Parameters in MLR

9.2.3 Example: Microwave Rotational Spectrum of HCl

9.2.4 Detecting and Dealing with Outliers

9.3 Weighted Least Squares

9.3.1 The Fit Parameters In WLS

9.3.2 Error Estimates for the WLS Fit Parameters

9.3.3 Finding the Weights

9.3.4 Residual Analysis in WLS

9.3.5 Evaluating Goodness–Of–Fit

9.4 Fitting Data to a Line With Errors in Both X and Y

9.4.1 Finding Fit Parameters in TLS

9.4.2 Error Estimates for the TLS Fit Parameters

9.4.3 Assessing Goodness–Of–Fit in TLS

9.4.4 Multiple Linear Regression With TLS

9.5 Calibration and Standard Additions

9.5.1 Error Estimates for Calibrated Values

9.5.2 Standard Additions

10 Fitting Data to a Curve

10.1 Transforming Data to a Linear Form

10.2 Polynomial Least–Squares Fitting

10.2.1 How Many Fit Parameters Are Needed?

10.3 Nonlinear Least–Squares Models

10.4 Estimating Error in Nonlinear Fit Parameters

10.4.1 Estimating Parameter Errors With the Jackknife Method

10.4.2 Estimating Parameter Errors with the Bootstrap Method

11 Differential Equations

11.1 Symbolic Solutions of ODEs

11.1.1 Initial Value Problems

11.1.2 Boundary Value Problems

11.2 Power Series Solution of ODEs

11.3 Direction Fields

11.3.1 Direction Fields with Adjustable Parameters

11.3.2 Direction Fields and Autonomous Equations

11.4 Solving Systems of Linear Differential Equations

11.5 Numerical Solution of ODEs

11.6 Solving Partial Differential Equations

12 Operators and Integral Transforms

12.1 Defining Operators

12.2 Fourier Series

12.3 Fourier Transforms

12.3.1 The Fast Fourier Transform

12.4 The Laplace Transform

Glossary

Index

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Fred Senese
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