Applied Mathematics for Physical Chemistry

A "how to do it" review and learn book on advanced mathematics necessary to physical chemistry. Coordinate systems, functions and graphs, logarithms, differential calculus, integral calculus, infinite series, differential equations, scalars and vectors, matrices and determinants, operators, numerical methods and the use of the computer, and mathematical methods in the laboratory. Educators, Technicians, and other professionals using mathematics in physical chemistry.

Preface

vii

1 Coordinate Systems

1

(10)

1.1 Introduction

1

(1)

1.2 Cartesian Coordinates

2

(2)

1.3 Plane Polar Coordinates

4

(1)

1.4 Spherical Polar Coordinates

5

(2)

1.5 The Complex Plane

7

(4)

2 Functions and Graphs

11

(13)

2.1 Functions

11

(1)

2.2 Graphical Representation of Functions

12

(8)

2.3 Roots to Polynomial Equations

20

(4)

3 Logarithms

24

(6)

3.1 Introduction

24

(1)

3.2 General Properties of Logarithms

24

(1)

3.3 Common Logarithms

25

(2)

3.4 Natural Logarithms

27

(3)

4 Differential Calculus

30

(24)

4-1 Introduction

30

(1)

4-2 Functions of Single Variables

31

(4)

4-3 Functions of Several Variables-Partial Derivatives

35

(2)

4-4 The Total Differential

37

(3)

4-5 Derivative as a Ratio of Infinitesimally Small Changes

40

(3)

4-6 Geometric Properties of Derivatives

43

(4)

4-7 Constrained Maxima and Minima

47

(7)

5 Integral Calculus

54

(15)

5-1 Introduction

54

(1)

5-2 Integral as an Antiderivative

55

(1)

5-3 General Methods of Integration

55

(2)

5-4 Special Methods of Integration

57

(2)

5-5 The Integral as a Summation of Infinitesimally Small Elements

59

(2)

5-6 Line Integrals

61

(2)

5-7 Double and Triple Integrals

63

(6)

6 Differential Equations

69

(25)

6-1 Introduction

69

(1)

6-2 Linear Combinations

70

(1)

6-3 First-Order Differential Equations

71

(3)

6-4 Second-Order Differential Equations with Constant Coefficients

74

(3)

6-5 General Series Method of Solution

77

(2)

6-6 Special Polynomial Solutions to Differential Equations

79

(5)

6-7 Exact and Inexact Differentials

84

(3)

6-8 Integrating Factors

87

(1)

6-9 Partial Differential Equations

88

(6)

7 Infinite Series

94

(16)

7-1 Introduction

94

(1)

7-2 Tests for Convergence and Divergence

95

(3)

7-3 Power Series Revisited

98

(1)

7-4 Maclaurin and Taylor Series

99

(2)

7-5 Fourier Series and Fourier Transforms

101

(9)

8 Scalars and Vectors

110

(10)

8.1 Scalars

110

(1)

8.2 Vectors and their Addition

111

(3)

8.3 Multiplication of Vectors

114

(3)

8.4 Applications

117

(3)

9 Matrices and Determinants

120

(15)

9.1 Introduction

120

(1)

9.2 Square Matrices and Determinants

121

(2)

9.3 Matrix Algebra

123

(3)

9.4 Solution of Systems of Linear Equations

126

(2)

9.5 Characteristic Equation of a Matrix

128

(7)

10 Operators

135

(15)

10.1 Introduction

135

(2)

10.2 Vector Operators

137

(1)

10.3 Eigenvalue Equations Revisited

138

(2)

10.4 Hermitian Operators

140

(2)

10.5 Rotational Operators

142

(3)

10.6 Transformation of XXX(2) to Plane Polar Coordinates

145

(5)

11 Numerical Methods and the Use of the Computer

150

(29)

11.1 Introduction

150

(2)

11.2 Programming a Computer

152

(11)

11.3 Programming Examples

163

(6)

11.4 Numerical Integration

169

(5)

11.5 Roots to Equations

174

(5)

12 Mathematical Methods in the Laboratory

179

(13)

12.1 Introduction

179

(1)

12.2 Probability

180

(2)

12.3 Experimental Errors

182

(3)

12.4 Propagation of Errors

185

(2)

12.5 Preparation of Graphs

187

(1)

12.6 Tongents and Areas

188

(4)

Appendices

192

(18)

I Table of Physical Constants

192

(1)

II Table of Integrals

192

(14)

III Transformation of XXX(2) to Spherical Polar Coordinates

206

(2)

IV Stirling's Approximation

208

(2)

Answers

210

(11)

Index

221

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Applied Mathematics for Physical Chemistry

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Table of Contents

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