Contents
Preface
1 Problem Solving and Numerical Mathematics
1.1 Problem Solving
1.2 Measurements
1.3 Numerical Mathematical Operations
1.3.1 Binary Arithmetic Operations
1.3.2 Additional Numerical Operations
1.4 Units of Measurement
1.5 The Factor-Label Method
1.6 Measurements, Accuracy, and Significant Digits
1.6.1 Scientific Notation
1.6.2 Rounding
1.6.3 Significant Digits in a Calculated Quantity
1.6.3.1 Multiplication and Division
1.6.3.2 Addition and Subtraction
Significant Digits With Other Operations
1.7 Problems
2 Mathematical Functions
2.1 Mathematical Functions in Physical Chemistry
2.1.1 Functions in Thermodynamics
2.1.2 Functions in Quantum Mechanics
2.1.3 Function Notation
2.1.4 Continuity
2.1.5 Graphs of Functions
2.1.6 Graphing With Excel
2.2 Important Families of Functions
2.2.1 Linear Functions
2.2.2 Quadratic Functions
2.2.3 Cubic Functions
2.2.4 Logarithms
2.2.4.1 Common Logarithms
2.2.4.2 Natural Logarithms
2.2.4.3 Logarithm Identities
2.2.5 Exponentials
2.2.6 Trigonometric Functions
2.2.6.1 Trigonometric Identities
2.2.6.2 A Useful Approximation
2.2.6.3 General Properties of Trigonometric Functions
2.2.7 Inverse Trigonometric Functions
2.2.8 Hyperbolic Functions
2.2.9 Significant Digits in Logarithms, Exponentials, and Trigonometric Functions
2.3 Generating Approximate Graphs
2.4 Problems
3 Symbolic Mathematics: Algebra
3.1 The Algebra of Real Scalar Variables
3.2 Coordinate Systems in Two Dimensions
3.3 Coordinate Systems in Three Dimensions
3.3.1 Cartesian Coordinates
3.3.2 Spherical Polar Coordinates
3.3.3 Cylindrical Polar Coordinates
3.4 Imaginary and Complex Numbers
3.4.1 Mathematical Operations With Complex Numbers
3.4.2 The Argand Diagram
3.4.3 The Complex Conjugate
3.4.4 The Magnitude of a Complex Quantity
3.4.5 Roots of a Complex Number
3.5 Problem Solving and Symbolic Mathematics
3.6 Problems
4 Vectors and Vector Algebra
4.1 Vectors in Two Dimensions
4.1.1 The Sum and Difference of Two Vectors
4.1.2 The Product of a Vector and a Scalar
4.1.3 Unit Vectors
4.1.3.1 The Product of a Scalar and a Vector in Terms of Components
4.1.3.2 The Sum of Two Vectors in Terms of Components
4.1.4 The Scalar Product of Two Vectors
4.1.5 The Magnitude of a Vector
4.2 Vectors in Three Dimensions
4.2.1 Unit Vectors in Three Dimensions
4.2.2 The Magnitude of a Vector
4.2.3 The Sum and Difference of Two Vectors
4.2.4 The Product of a Scalar and a Vector
4.2.5 The Scalar Product of Two Vectors
4.2.6 The Vector Product of Two Vectors
4.3 Physical Examples of Vector Products
4.3.1 Magnetic Force
4.3.2 Electrostatic Force
4.3.3 Angular Momentum
4.4 Problems
5 The Solution of Algebraic Equations
5.1 Algebraic Methods for Solving One Equation With One Unknown
5.1.1 Polynomial Equations
5.1.1.1 Linear Equations
5.1.1.2 Quadratic Equations
5.1.2 Approximate Solutions to Equations
5.1.2.1 Approximation by Use of Simplifying Assumptions
5.1.2.2 Solution by Successive Approximations
5.1.2.3 Approximation by Linearization
5.2 Numerical Solution of Algebraic Equations
5.2.1 Graphical Solution of Algebraic Equations
5.2.2 Trial and Error
5.2.3 The Method of Bisection
5.2.4 Solving Equations Numerically With Excel
5.3 A Brief Introduction to Mathematica
5.3.1 Numerical Calculations With Mathematica
5.3.2 Symbolic Algebra With Mathematica
5.3.3 Solving Equations With Mathematica
5.3.4 Graphing With Mathematica
5.4 Simultaneous Equations: Two Equations With Two Unknowns
5.4.1 The Method of Substitution
5.4.2 The Method of Elimination
5.4.3 Consistency and Independence in Simultaneous Equations
5.4.4 Homogeneous Linear Equations
5.4.5 Using Mathematica to Solve Simultaneous Equations
5.4.6 Problems
6 Differential Calculus
6.1 The Tangent Line and the Derivative of a Function
6.1.1 The Derivative
6.1.2 Derivatives of Specific Functions
6.2 Differentials
6.3 Some Derivative Identities
6.3.1 The Derivative of a Constant
6.3.2 The Derivative of a Function Times a Constant
6.3.3 The Derivative of a Product of Two Functions
6.3.4 The Derivative of the Sum of Two Functions
6.3.5 The Derivative of the Difference of Two Functions
6.3.6 The Derivative of the Quotient of Two Functions
6.3.7 The Derivative of a Function of a Function (the Chain Rule)
6.4 Higher-Order Derivatives
6.4.1 The Curvature of a Function
6.5 Newton's Method
6.6 Maximum-Minimum Problems
6.7 Locating Inflection Points
6.8 Limiting Values of Functions
6.9 L'Hôpital's Rule
6.9.1 Problems
7 Integral Calculus
7.1 The Antiderivative of a Function
7.2 The Process of Integration
7.2.1 The Definite Integral as an Area
7.2.2 Facts About Integrals
7.2.3 Derivatives of Definite Integrals
7.3 Tables of Indefinite Integrals
7.4 Improper Integrals
7.5 Techniques of Integration
7.5.1 The Method of Substitution
7.5.2 Integration by Parts
7.5.3 The Method of Partial Fractions
7.5.4 Integration With Mathematica
7.6 Numerical Integration
7.6.1 The Bar-Graph Approximation
7.6.2 The Trapezoidal Approximation
7.6.3 Simpson's Rule
7.6.4 Numerical Integration With Mathematica
Problems
8 Differential Calculus With Several Independent Variables
8.1 Functions of Several Independent Variables
8.2 Changes in a Function of Several Variables. Partial Derivatives
8.3 Change of Variables
8.4 Useful Partial Derivative Identities
8.4.1 The Reciprocal Identity
8.4.2 The Euler Reciprocity Relation
8.4.3 The Maxwell Relations
8.4.4 The Cycle Rule
8.4.5 The Chain Rule
8.5 Thermodynamic Variables Related to Partial Derivatives
8.6 Exact and Inexact Differentials
8.6.1 Integrating Factors
8.7 Maximum and Minimum Values of Functions of Several Variables
8.7.1 Constrained Maximum/Minimum Problems
8.7.2 Lagrange's Method of Undetermined Multipliers
8.8 Vector Derivative Operators
8.8.1 Vector Derivatives in Cartesian Coordinates
8.8.1.1 The Gradient
8.8.1.2 The Divergence
8.8.1.3 The Curl
8.8.1.4 The Laplacian
8.8.2 Vector Derivatives in Other Coordinate Systems
8.8.2.1 The Gradient
8.8.2.2 The Divergence
8.8.2.3 The Curl
8.8.2.4 The Laplacian
8.8.3 Problems
9 Integral Calculus With Several Independent Variables
9.1 Line Integrals
9.1.1 Line Integrals of Exact Differentials
9.1.2 Line Integrals of Inexact Differentials
9.1.3 Line Integrals With Three Integration Variables
9.1.4 Line Integrals in Thermodynamics
9.2 Multiple Integrals
9.2.1 Double Integrals
9.2.2 The Double Integral Representing a Volume
9.2.3 Triple Integrals
9.2.4 Changing Variables in Multiple Integrals
9.2.4.1 Double Integrals
9.2.4.2 Triple Integrals
9.2.4.3 The Jacobian
9.2.5 Problems
10 Mathematical Series
10.1 Constant Series
10.1.1 Some Convergent Constant Series
10.1.2 The Harmonic Series
10.1.3 Tests for Convergence
10.2 Power Series
10.2.1 Maclaurin Series
10.2.2 Taylor Series
10.2.3 The Convergence of Power Series
10.2.4 Power Series in Physical Chemistry
10.3 Mathematical Operations on Series
10.4 Power Series With More Than One Independent Variable
10.4.1 Problems
11 Functional Series and Integral Transforms
11.1 Fourier Series
11.1.1 Finding the Coefficients of a Fourier Series—Orthogonality
11.1.2 One-Sided Fourier Series
11.1.3 Fourier Series With Complex Exponential Basis Functions
11.2 Other Functional Series With Orthogonal Basis Sets
11.2.1 Hilbert Space
11.2.2 Determining the Expansion Coefficients
11.3 Integral Transforms
11.3.1 Fourier Transforms (Fourier Integrals)
11.3.2 Laplace Transforms
11.3.3 Problems
12 Differential Equations
12.1 Differential Equations and Newton's Laws of Motion
12.2 Ordinary Linear Differential Equations With Constant Coefficients
12.2.1 The Harmonic Oscillator
12.2.1.1 The Equation of Motion of the Harmonic Oscillator
12.2.1.2 Solution of the Equation of Motion
12.2.1.3 The Vibration of a Diatomic Molecule
12.2.1.4 The Energy of a Harmonic Oscillator
12.2.2 The Damped Harmonic Oscillator—A Nonconservative System
12.2.2.1 Greater Than Critical Damping
12.2.2.2 Less Than Critical Damping
12.2.2.3 Critical Damping
12.2.3 The Forced Harmonic Oscillator—Inhomogeneous Linear Differential Equations
12.2.3.1 Variation of Parameters Method
12.3 Differential Equations With Separable Variables
12.4 Exact Differential Equations
12.5 Inexact Differential Equations
12.6 Partial Differential Equations
12.6.1 Waves in a String
12.6.2 The Schrödinger Equation
12.7 Solution of Differential Equations Using Laplace Transforms
12.8 Numerical Solution of Differential Equations
12.8.1 Euler's Method
12.8.2 The Runge–Kutta Method
12.8.3 Solution of Differential Equations With Mathematica
12.8.3.1 Symbolic Solution
12.8.3.2 Numerical Solution
12.8.4 Problems
13 Operators, Matrices, and Group Theory
13.1 Mathematical Operators
13.1.1 Eigenfunctions and Eigenvalues
13.1.2 Operator Algebra
13.1.3 Operators in Quantum Mechanics
13.2 Symmetry Operators
13.3 The Operation of Symmetry Operators on Functions
13.4 Matrices and Matrix Algebra
13.4.1 Equality of Two Matrices
13.4.2 Sum of Two Matrices
13.4.3 Product of a Scalar and a Matrix
13.4.4 Product of Two Matrices
13.4.5 The Properties of Matrix Multiplication
13.4.6 The Identity Matrix
13.4.7 The Inverse of a Matrix
13.4.8 Matrix Terminology
13.5 Matrices in Quantum Mechanics
13.6 Determinants
13.7 Matrix Algebra With Mathematica
13.8 An Elementary Introduction to Group Theory
13.9 Matrix Representations of Symmetry Operators
13.9.1 Problems
14 The Solution of Simultaneous Algebraic Equations With More Than Two Unknowns
14.1 Cramer's Rule
14.2 Linear Dependence and Inconsistency
14.3 Solution by Matrix Inversion
14.4 Gauss–Jordan Elimination
14.5 Linear Homogeneous Equations
14.6 Matrix Eigenvalues and Eigenvectors
14.7 The Use of Mathematica to Solve Simultaneous Equations
14.8 The Use of Mathematica to Find Matrix Eigenvalues and Eigenvectors
14.8.1 Problems
15 Complex Variables
15.1 Mapping in the Complex Plane
15.1.1 Polar Form
15.2 Cauchy–Riemann Equations
15.3 Contour Integration
15.4 Cauchy's Theorem
15.5 Cauchy's Integral Formula
15.6 Taylor Series
15.7 Laurent Expansions
15.8 Calculus of Residues
15.9 Trigonometric Integrals
15.10 Improper Integrals
15.11 Multivalued Functions
15.12 Integrals Using Branch Cuts
15.13 Problems
16 Probability, Statistics, and Experimental Errors
16.1 Experimental Errors in Measured Quantities
16.1.1 Systematic and Random Errors
16.2 Probability Theory
16.2.1 Properties of a Population
16.2.1.1 Discrete Probability Distributions
16.2.2 Continuous Probability Distributions
16.2.3 The Uniform Probability Distribution
16.2.4 The Gaussian Distribution
16.2.5 Probability Distributions in Quantum Mechanics
16.2.6 Probability Distributions in Gas Kinetic Theory
16.2.7 Time Averages
16.3 Statistics and the Properties of a Sample
16.4 Numerical Estimation of Random Errors
16.4.1 Rejection of Discordant Data
16.4.2 Problems
17 Data Reduction and the Propagation of Errors
17.1 The Combination of Errors
17.1.1 The Combination of Random and Systematic Errors
17.1.2 Error Propagation in Data Reduction With a Formula
17.2 Curve Fitting
17.2.1 The Method of Least Squares (Regression)
17.2.2 Linear Least Squares (Linear Regression)
17.2.3 The Correlation Coefficient and the Covariance
17.2.4 Error Propagation in Linear Least Squares
17.2.5 Carrying Out Least Squares Fits With Excel
17.2.5.1 Linear Least Squares Fits in a Worksheet
17.2.5.2 Least Squares Fits on a Graph
17.2.6 Some Warnings About Least-Squares Procedures
17.2.7 Weighting Factors in Linear Least Squares
17.2.8 Linear Least Squares With Fixed Slope or Intercept
17.3 Data Reduction With a Derivative
17.3.1 Problems
18 Mathematica: Advanced Applications
The Basic Math Assistant
Symbolic Mathematics
Functions and Plots
Other Types of Plots
2D Graphics
Plain English Instructions
External Data
Appendices
Appendix A. Values of Physical Constants
Some Conversion Factors
Appendix B. Some Mathematical Formulas and Identities
Appendix C. Infinite Series
Series With Constant Terms
Power Series
Appendix D. A Short Table of Derivatives
Appendix E. A Short Table of Indefinite Integrals
Appendix F. A Short Table of Definite Integrals
Appendix G. Some Integrals With Exponentials in the Integrands: The Error Function
The Error Function
Appendix H. Answers to Selected Numerical Exercises and Odd-Numbered Problems
Chapter 1
Exercises
Problems
Chapter 2
Exercises
Problems
Chapter 3
Exercises
Problems
Chapter 4
Exercises
Problems
Chapter 5
Exercises
Problems
Chapter 6
Exercises
Problems
Chapter 7
Exercises
Problems
Chapter 8
Exercises
Problems
Chapter 9
Exercises
Problems
Chapter 10
Exercises
Problems
Chapter 11
Chapter 12
Exercises
Problems
Chapter 13
Exercises
Problems
Chapter 14
Exercises
Problems
Chapter 15
Exercises
Problems
Chapter 16
Exercises
Problems
Chapter 17
Exercises
Problems
Additional Reading
Books on Mathematics for Science
Calculus Textbooks
Books on Numerical Analysis
Advanced Mathematics Books
Books on Group Theory
Books on Experimental Data Analysis
Computer Books
Problem-Solving and Problem Books
Mathematical Tables
Index