Preface

Translator's note

Introduction

1  Real and complex numbers

  1.1   Rational numbers

  1.2   The existence of irrational numbers

  1.3   A description of real numbers

  1.4   Limit

  1.5   The Bolzano-Weierstrass theorem

  1.6   Definitions for complex numbers and vectors

  1.7   Polar coordinates and multiplication

  1.8   De Moivre's theorem

  1.9   Completeness of the complex numbers

  1.10  Introduction to quaternions

  Supplement

  1.11  Binary arithmetics

  1.12  Periodic decimals

  1.13  Rational approximations to real numbers

  1.14  Error terms

  1.15  Solutions to cubic and quartic equations

2  Vector algebra

  2.1   Space coordinates and vectors

  2.2   Addition of vectors

  2.3   The decomposition of a vector

  2.4   Inner product (scalar product)

  2.5   Vector product (outer product)

  2.6   Multiple products

  2.7   Change of coordinates

  2.8   Planes

  2.9   Equation for a line in space

2  Supplement

  2.10  Main formulae in spherical trigonometry

  2.11  Duality principle

  2.12  Right-angled and right-sided triangles

  2.13  Forces, systems and equivalent systems

  2.14  Combination of parallel forces

  2.15  Moments

  2.16  Couples

  2.17  Standard form for a system

  2.18  Equilibrium and its applications

3  Functions and graphs

  3.1   Variables

  3.2   Functions

  3.3   Implicit functions

  3.4   Functions represented by graphs and tables

  3.5   Several elementary functions

  3.6   Functions with simple special properties

  3.7   Periodic functions

  3.8   Representations for a complex function

  3.9   Line of regression

  3.10  Lagrange's interpolation formula

  3.11  Other interpolation formulae

  3.12  Experimental formulae

  3.13  Family of curves

4  Limits

  4.1   Limits of sequences

  4.2   Sequences without limits

  4.3   Series

  4.4   Conditionally convergent series

  4.5   The method of Zu Chongzhi in calculating π

  4.6   Archimedes' method for the area of a parabolic region

  4.7   Calculating pressure on a boundary

  4.8   The number e

  4.9   Taking limit in the continuum

  4.10  On several important limits

  4.11  Some examples

  4.12  Orders of infinity

  4.13  The symbols ~, O and o

  4.14  Continuous functions

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