The History of the Calculus and the Development of Computer Algebra Systems


0. Introduction
1. History of the Integral from the 17th Century

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. Short Biographies

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. Computer Algebra Systems

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 Advanced Operations

5.4.1 Introduction
5.4.2 Differentiation
5.4.3 Integration
5.4.3 Differentiation - Our Implementation

6. Conclusions
A. Bibliography
 


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