Mathematics for Physical Chemistry (eBook)

Cover 1
Contents 8
Preface 12
1 Numbers, Measurements, and Numerical Mathematics 14
Numbers and Measurements 15
Numerical Mathematical Operations 18
Units of Measurement 24
Numerical Calculations 27
2 Symbolic Mathematics and Mathematical Functions 34
Algebraic Operations on Real Scalar Variables 35
Trigonometric Functions 37
Inverse Trigonometric Functions 42
Vectors and Coordinate Systems 44
Imaginary and Complex Numbers 57
Problem Solving and Symbolic Mathematics 65
3 The Solution of Algebraic Equations 70
Algebraic Methods for Solving One Equation with One Unknown 71
Graphical Solution of Equations 77
Numerical Solution of Algebraic Equations 83
Simultaneous Equations: Two Equations with Two Unknowns 92
4 Mathematical Functions and Differential Calculus 102
Mathematical Functions 103
The Tangent Line and the Derivative of a Function 111
Differentials 115
Some Useful Facts About Derivatives 117
Higher-Order Derivatives 121
Maximum-Minimum Problems 123
Limiting Values of Functions: L’Hôpital’s Rule 126
5 Integral Calculus 134
The Antiderivative of a Function 135
The Process of Integration 137
Indefinite Integrals: Tables of Integrals 145
Improper Integrals 147
Methods of Integration 149
Numerical Integration 154
Probability Distributions and Mean Values 158
6 Mathematical Series and Transforms 171
Constant Series 172
Functional Series 178
Fourier Series 185
Mathematical Operations on Series 191
Integral Transforms 193
7 Calculus With Several Independent Variables 202
Functions of Several Independent Variables 203
Change of Variables 209
Additional Useful Relations Between Partial Derivatives 211
Exact and Inexact Differentials 215
Line Integrals 218
Multiple Integrals 223
Vector Derivative Operators 230
Maximum and Minimum Values of Functions of Several Variables 237
8 Differential Equations 247
Differential Equations and Newton’s Laws of Motion 248
The Harmonic Oscillator: Linear Differential Equations with Constant Coefficients 251
Differential Equations with Separable Variables 262
Exact Differential Equations 264
Solution of Inexact Differential Equations by the Use of Integrating Factors 265
Partial Differential Equations: Waves in a String 266
Solution of Differential Equations with Laplace Transforms 271
Numerical Solutions of Differential Equations 273
9 Operators, Matrices, and Group Theory 281
Operators and Operator Algebra 282
Symmetry Operators 288
Matrix Algebra 295
Matrix Algebra with Mathematica 305
An Elementary Introduction to Group Theory 307
10 The Solution of Simultaneous Algebraic Equations 318
Simultaneous Equations with More than Two Unknowns 319
Cramer’s Rule 319
Solution by Matrix Inversion 322
The Use of Mathematica to Solve Simultaneous Equations 326
11 The Treatment of Experimental Data 331
Experimental Errors in Measured Quantities 332
Statistical Treatment of Random Errors 335
Data Reduction and the Propagation of Errors 342
Graphical and Numerical Data Reduction 346
Numerical Curve Fitting: The Method of Least Squares (Regression) 352
Appendixes 377
A Values of Physical Constants1 378
B Some Mathematical Formulas and Identities 380
C Infinite Series 383
Series with Constant Terms 383
Power Series 383
D A Short Table of Derivatives 386
E A Short Table of Indefinite Integrals 388
F A Short Table of Definite Integrals 392
G Some Integrals with Exponentials in the Integrands: The Error Function 396
Index 400