Derivations of Applied Mathematics Thaddeus H. Black Revised 14 December 2006 ii Derivations of Applied Mathematics. 14 December 2006. c 1983–2006 by Thaddeus H. Black
[email protected] Copyright Published by the Debian Project [7]. This book is free software. You can redistribute and/or modify it under the terms of the GNU General Public License [11], version 2. Contents Preface xiii 1 Introduction 1.1 Applied mathematics . . . . . . . . . . . 1.2 Rigor . . . . . . . . . . . . . . . . . . . . 1.2.1 Axiom and definition . . . . . . . 1.2.2 Mathematical extension . . . . . 1.3 Complex numbers and complex variables 1.4 On the text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 2 2 4 5 5 2 Classical algebra and geometry 2.1 Basic arithmetic relationships . . . . . . . . . . . . 2.1.1 Commutivity, associativity, distributivity . 2.1.2 Negative numbers . . . . . . . . . . . . . . 2.1.3 Inequality . . . . . . . . . . . . . . . . . . . 2.1.4 The change of variable . . . . . . . . . . . . 2.2 Quadratics . . . . . . . . . . . . . . . . . . . . . . 2.3 Notation for series sums and products . . . . . . . 2.4 The arithmetic series . . . . . . . . . . . . . . . . . 2.5 Powers and roots . . . . . . . . . . . . . . . . . . . 2.5.1 Notation and integral powers . . . . . . . . 2.5.2 Roots . . . . . . . . . . . . . . . . . . . . . 2.5.3 Powers of products and powers of powers . 2.5.4 Sums of powers . . . . . . . . . . . . . . . . 2.5.5 Summary and remarks . . . . . . . . . . . . 2.6 Multiplying and dividing power series . . . . . . . 2.6.1 Multiplying power series . . . . . . . . . . . 2.6.