Computable Calculus

by Aberth

• ISBN13: 9780120417520
• ISBN10: 0120417529
• Edition: CD
• Format: Paperback
• Copyright: 2001-05-21
• Publisher: Elsevier Science

Summary

Computable Calculus treats the fundamental topic of calculus in a novel way that is more in tune with today's computer age. Comprising 11 chapters, the book presents mathematical analysis that has been created to deal with constructively defined concepts. The book's "show your work" approach makes it easier to understand the pitfalls of various computations and, more importantly, how to avoid these pitfalls.

Table of Contents

Preface	xi
Acknowledgments	xiii
Introduction	1	(6)
What Is Computable Calculus?	1	(2)
What Are the Advantages of Computable Calculus, If Any?	3	(1)
A Brief Description of Computable Calculus	4	(3)
The Real Numbers	7	(14)
Definition of a Real Number	7	(3)
An Ordering of Intervals	10	(4)
Interval Arithmetic	14	(1)
The Real Number Operations +, -, x, ÷	15	(3)
The Absolute Value of a Real Number	18	(3)
Solvable Problems and Nonsolvable Problems	21	(14)
Introduction	21	(1)
Turning's Resolution of the Halting Problem	22	(2)
A Certain Computation Problem	24	(2)
Deciding Whether a Number is Zero	26	(1)
A List of Nonsolvable Problems	27	(1)
Solvable Problems	28	(2)
Key Nonsolvable Problems	30	(1)
Deciding Whether a Number is Rational or Irrational	31	(1)
Deciding Which of Two Real Numbers is Larger	32	(3)
Sequences and Functions	35	(16)
Sequences of Real Numbers	35	(1)
The Cantor Counting Theory	36	(1)
Functions	37	(3)
The nth Root of a Real Number	40	(1)
An Algebra of Functions	41	(1)
The Function sgn(x)	42	(2)
Functions Defined on Intervals	44	(2)
Semifunctions	46	(1)
Other Calculus Concepts	47	(4)
The Ideal Computer	51	(42)
The Goal of This Chapter	51	(1)
The Various Methods of Proof	52	(2)
Definition of the Ideal Computer	54	(1)
The Ideal Computer Steps	55	(4)
Viewing, Compiling, and Executing Ideal Computer Programs	59	(3)
More Ideal Computer Details	62	(2)
Programming the Ideal Computer to Add Natural Integers	64	(4)
The Addition of Two General Integers	68	(2)
The Subtraction, Multiplication, and Division of Integers	70	(1)
Rational Number Arithmetic	71	(2)
Interval Arithmetic	73	(1)
The Retrieval of a Program from Disk Memory	74	(2)
The Real Number π	76	(1)
Changing and Approximation Algorithm	77	(2)
Rational Numbers Converted to Real Numbers	79	(1)
nth Roots of Real Numbers	79	(1)
Arithmetic Operations on Real Numbers	80	(1)
Rational Sequence Arithmetic	81	(1)
Function Programs	82	(1)
The Exponential Function ex	83	(1)
Two Semifunction Examples	84	(2)
The Beginning of the Nonsolvability Proof for Problem 3.1	86	(2)
Programs That Decide Whether a Real Number is Zero	88	(5)
Limits	93	(26)
Limit of a Sequence	93	(2)
Monotone Sequences	95	(2)
The Specker Theorem	97	(2)
Consequences of Specker's Theorem	99	(1)
Limit of a Function	99	(6)
Using Limits to Extend a Function's Domain	105	(2)
A Standard Result for Functions	107	(3)
Unbounded Continuous Functions on [a, b]	110	(2)
Limits of Sequences of Functions	112	(2)
Limits of Functions with More Than One Variable	114	(5)
Uniformaly Continuous Functions	119	(12)
Introduction	119	(5)
Bounds of Uniformly Continuous Functions	124	(7)
The Derivative	131	(12)
A Difficulty with the Derivative Definition	131	(4)
Rules of Differentiation	135	(2)
A Computation Problem	137	(1)
The Mean Value Theorem	138	(5)
The Riemann Integral	143	(8)
Riemann Sums	143	(1)
The Integration of Uniformly Continuous Functions	144	(2)
Properties of Integrals	146	(2)
Defining a Function by Means of an Integral	148	(2)
A Mean Value Theorem for Integrals	150	(1)
Functions of Two Variables	151	(12)
Partial Derivatives	151	(3)
The Chain Rule	154	(3)
Equality of Cross Derivatives	157	(6)
The Differential Equation y'=f(x,y)	163	(12)
Introduction	163	(2)
The Lipschitz Condition	165	(5)
The Possibility of No Solution	170	(5)
Ideal Computer Simulation	175	(10)
The Extended Ideal Computer	175	(1)
Call Loops	176	(2)
Composing and Deleting a Program	178	(2)
Managing the Ideal Computer's Input and Output	180	(2)
Possible Projects	182	(3)
References	185	(4)
About the CD	189	(2)
Index	191