Mathematics for Physical Chemistry

Front Cover......Page 0
Half Title......Page 2
Title Page......Page 4
Copyright......Page 5
Dedication......Page 6
Contents......Page 8
Preface......Page 12
1.2 Numbers and Measurements......Page 14
1.3.2 Additional Numerical Operations......Page 15
1.4 Units of Measurement......Page 16
1.6 Measurements, Accuracy, and Significant Digits......Page 18
1.6.2 Rounding......Page 19
1.6.3 Significant Digits in a Calculated Quantity......Page 20
2.1.1 Functions in Thermodynamics......Page 24
2.1.5 Graphs of Functions......Page 25
2.2.1 Linear Functions......Page 28
2.2.4 Logarithms......Page 29
2.2.5 Exponentials......Page 30
2.2.6 Trigonometric Functions......Page 31
2.2.7 Inverse Trigonometric Functions......Page 34
2.3 Generating Approximate Graphs......Page 35
3.1 The Algebra of Real Scalar Variables......Page 38
3.2 Coordinate Systems in Two Dimensions......Page 39
3.3.2 Spherical Polar Coordinates......Page 40
3.3.3 Cylindrical Polar Coordinates......Page 41
3.4.2 The Argand Diagram......Page 42
3.4.4 The Magnitude of a Complex Quantity......Page 44
3.5 Problem Solving and Symbolic Mathematics......Page 45
4.1.1 The Sum and Difference of Two Vectors......Page 48
4.1.3 Unit Vectors......Page 49
4.1.4 The Scalar Product of Two Vectors......Page 50
4.2.2 The Magnitude of a Vector......Page 51
4.2.6 The Vector Product of Two Vectors......Page 52
4.3.1 Magnetic Force......Page 53
4.3.3 Angular Momentum......Page 54
5.1.1 Polynomial Equations......Page 56
5.1.2 Approximate Solutions to Equations......Page 57
5.2.1 Graphical Solution of Algebraic Equations......Page 60
5.2.4 Solving Equations Numerically with Excel......Page 61
5.3.1 Numerical Calculations with Mathematica......Page 62
5.3.2 Symbolic Algebra with Mathematica......Page 64
5.3.3 Solving Equations with Mathematica......Page 65
5.4.1 The Method of Substitution......Page 66
5.4.4 Homogeneous Linear Equations......Page 67
5.4.5 Using Mathematica to Solve Simultaneous Equations......Page 68
6.1 The Tangent Line and the Derivative of a Function......Page 72
6.1.1 The Derivative......Page 73
6.2 Differentials......Page 74
6.3.7 The Derivative of a Function of a Function (the Chain Rule)......Page 76
6.4 Newton's Method......Page 77
6.5 Higher-Order Derivatives......Page 78
6.6 Maximum–Minimum Problems......Page 79
6.7 Limiting Values of Functions......Page 80
6.8 l'Hôpital's Rule......Page 81
7.1.1 Position, Velocity, and Acceleration......Page 86
7.2 The Process of Integration......Page 87
7.2.2 Facts about Integrals......Page 89
7.3 Tables of Indefinite Integrals......Page 91
7.4 Improper Integrals......Page 92
7.5.2 Integration by Parts......Page 93
7.5.3 The Method of Partial Fractions......Page 94
7.6.2 The Trapezoidal Approximation......Page 96
7.6.3 Simpson's Rule......Page 97
7.6.4 Numerical Integration with Mathematica......Page 98
8.1 Functions of Several Independent Variables......Page 102
8.2.1 Differentials......Page 104
8.3 Change of Variables......Page 105
8.4.1 The Variable-Change Identity......Page 106
8.4.4 The Maxwell Relations......Page 107
8.5 Thermodynamic Variables Related to Partial Derivatives......Page 108
8.6 Exact and Inexact Differentials......Page 109
8.6.1 Integrating Factors......Page 110
8.7 Maximum and Minimum Values of Functions of Several Variables......Page 111
8.7.2 Lagrange's Method of Undetermined MultipliersNamed for Joseph Louis Lagrange (born Guisepps Lodovico Lagrangia), 1736–1813, French-Italian physicist and mathematician.......Page 112
8.8.1 Vector Derivatives in Cartesian Coordinates......Page 114
8.8.2 Vector Derivatives in Other Coordinate Systems......Page 116
9.1 Line Integrals......Page 120
9.1.1 Line Integrals of Exact Differentials......Page 121
9.1.3 Line Integrals with Three Integration Variables......Page 122
9.1.4 Line Integrals in Thermodynamics......Page 123
9.2.1  Double Integrals......Page 124
9.2.2 The Double Integral Representing a Volume......Page 125
9.2.4 Changing Variables in Multiple Integrals......Page 126
10.1 Constant Series......Page 132
10.1.2 The Geometric Series......Page 133
10.1.4 Tests for Convergence......Page 134
10.2.1 Maclaurin Series......Page 135
10.2.2 Taylor Series......Page 136
10.2.3 The Convergence of Power Series......Page 137
10.2.4 Power Series in Physical Chemistry......Page 138
10.4 Power Series with More than One Independent Variable......Page 139
11.1.1 Finding the Coefficients of a Fourier Series—Orthogonality......Page 142
11.2.1 Hilbert Space......Page 145
11.2.2 Determining the Expansion Coefficients......Page 146
11.3.1 Fourier Transforms (Fourier Integrals)......Page 147
11.3.2 Laplace Transforms......Page 149
12.1 Differential Equations and Newton's Laws of Motion......Page 152
12.2.1 The Harmonic Oscillator......Page 154
12.2.2 The Damped Harmonic Oscillator—A Nonconservative System......Page 157
12.3.1 Variation of Parameters Method......Page 160
12.5 Exact Differential Equations......Page 162
12.6 Solution of Inexact Differential Equations Using Integrating Factors......Page 163
12.7.2 Solution by Separation of Variables......Page 164
12.8 Solution of Differential Equations using Laplace Transforms......Page 167
12.9.1 Euler's Method......Page 168
12.9.3 Solution of Differential Equations with Mathematica......Page 169
13.1 Mathematical Operators......Page 174
13.1.2 Operator Algebra......Page 175
13.1.3 Operators in Quantum Mechanics......Page 177
13.2 Symmetry Operators......Page 178
13.3 The Operation of Symmetry Operators on Functions......Page 180
13.4.4 The Product of Two Matrices......Page 182
13.4.5 The Identity Matrix......Page 183
13.4.6 The Inverse of a Matrix......Page 184
13.5 Determinants......Page 185
13.6 Matrix Algebra with Mathematica......Page 187
13.7 An Elementary Introduction to Group Theory......Page 188
13.8 Symmetry Operators and Matrix Representations......Page 190
14.1 Cramer's Rule......Page 196
14.3 Solution by Matrix Inversion......Page 198
14.5 Linear Homogeneous Equations......Page 199
14.6 Matrix Eigenvalues and Eigenvectors......Page 200
14.8 The Use of Mathematica to Find Matrix Eigenvalues and Eigenvectors......Page 202
15.1.1 Systematic and Random Errors......Page 204
15.2.1 Properties of a Population......Page 205
15.2.3 The Gaussian Distribution......Page 209
15.2.4 Probability Distributions in Quantum Mechanics......Page 211
15.2.5 Probability Distributions in Gas Kinetic Theory......Page 212
15.3 Statistics and the Properties of a Sample......Page 214
15.4 Numerical Estimation of Random Errors......Page 215
16.1.1 The Combination of Random and Systematic Errors......Page 220
16.1.2 Error Propagation in Data Reduction with a Formula......Page 221
16.2 Curve Fitting......Page 222
16.2.1 The Method of Least Squares (Regression)......Page 223
16.2.2 Linear Least Squares (Linear Regression)......Page 224
16.2.3 The Correlation Coefficient and the Covariance......Page 226
16.2.4 Error Propagation in Linear Least Squares......Page 227
16.2.5 Carrying Out Least-Squares Fits with Excel......Page 229
16.2.7 Weighting Factors in Linear Least Squares......Page 230
16.3 Data Reduction with  a Derivative......Page 232
Appendices......Page 236
Additional Reading......Page 250
C......Page 254
F......Page 255
L......Page 256
O......Page 257
R......Page 258
V......Page 259
W......Page 260