1.1 Introduction
1.2 Cavalieri's Method of Indivisbles
1.3 Wallis' Law for Integration
of Polynomials
1.4 Fermat's Approach to Integration
2. History of the
Differential from the 17th Century
2.1 Introduction
2.2 Roberval's Method of Tangent
Lines using Instantaneous Motion
2.3 Fermat's Maxima and Tangent
2.4 Newton and Leibniz
2.5 The Ellusive Inverses – the
Integral and Differential
3. Selected Problems
from the History of the Infinite Series
3.1 Introduction
3.2 James Gregory's Infinite Series
for arctan
3.3 Leibniz's Early Infinite Series
3.4 Leibniz and the Infinite Series
for Trigonometric Functions
3.5 Euler's Sum of the Reciprocals
of the Squares of the Natural Numbers
4.1 Introduction
4.2 Gregory of St. Vincent (1584-1667)
4.3 Rene Descartes (1596-1650)
4.4 Bonaventura Cavalieri (1598-1647)
4.5 Pierre de Fermat (1601-1665)
4.6 Gilles Persone de Roberval (1602-1675)
4.7 John Wallis (1616-1703)
4.8 Blaise Pascal (1623-1662)
4.9 Christiaan Huygens (1629-1695)
4.10 Isaac Barrow (1630-1677)
4.11 James Gregory (1638-1675)
4.12 Sir Isaac Newton (1642 - 1727)
4.13 Gottfried Wilhelm Leibniz (1646
- 1716)
4.14 Leonhard Euler (1707 - 1783)
5.1 Introduction
- What is a Computer Algebra System?
5.2 Data Structures
5.2.1 Introduction
5.2.2 Polynomials in one variable
- Coefficients
5.2.3 Polynomials in one variable
- Terms
5.2.4 Polynomials in one variable
- Recursive definition
5.2.5 Multivariate Polynomials
5.2.6 The Syntax Tree - Our Data
Structure Implementation
5.2.7 The Syntax Tree - Advantages
5.2.8 The Syntax Tree - Disadvantages
5.3 Simplification
5.3.1 General
Issues in Simplification
5.3.2 The Steps of Simplification
- Our Approach
5.3.3 Transforming Negatives
5.3.4 Leveling Operators
5.3.5 Simplifying Rational Expressions
5.3.6 Collecting Like Terms
5.3.7 Folding Constants
5.3.8 Canonical Order
5.3.9 Full Simplification
5.4.1 Introduction
5.4.2 Differentiation
5.4.3 Integration
5.4.3 Differentiation - Our Implementation