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| ISBN-13: | 9780486151014 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 12/17/2013 |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 144 |
| File size: | 10 MB |
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Infinitesimal Calculus
By James M. Henle, Eugene M. Kleinberg
Dover Publications, Inc.
Copyright © 1979 The Massachusetts Institute of TechnologyAll rights reserved.
ISBN: 978-0-486-15101-4
CHAPTER 1
Introduction
The history of modern mathematics is to an astonishing degree the history of the calculus. The calculus was the first great achievement of mathematics since the Greeks and it dominated mathematical exploration for centuries. The questions it answered and the questions it raised lay at the heart of man's understanding of not only geometry and number, but also space and time and mathematical truth. It began with the surprising unification of two rather different geometrical problems, and almost immediately its ideas bore fruit in dozens of seemingly unrelated areas. The methods it developed gave the physical sciences an impetus without parallel in history, for through them natural science was born, and without them physics could not have progressed much further than the mystical vortices of Descartes.
In the beginning there were two calculi, the differential and the integral. The first had been developed to determine the slopes of tangents to certain curves, the second to determine the areas of certain regions bounded by curves. Algebra, geometry, and trigonometry were simply insufficient to solve general problems of this sort, and prior to the late seventeenth century mathematicians could at best handle only special cases.
The general idea of the calculus, its fundamental theorem, and its first applications to the outstanding problems of mathematics and the natural sciences are due independently to Isaac Newton (1642–1727) and Gottfried Leibniz (1646–1716).
Their work was certainly built on foundations laid by others, but their penetrating insights represented what is easily the most significant mathematical breakthrough since the Greeks. Remarkably, the powerful methods developed by these two men solved the same class of problems and proved many of the same theorems yet were based on different theories. Newton thought in terms of limits whereas Leibniz thought in terms of infinitesimals, and although Newton's theory was formalized long before Leibniz's, it is far easier to work with Leibniz's techniques.
The approach to the calculus we shall employ is based on Leibniz's ideas as formalized by Abraham Robinson in 1961 under the name of "nonstandard analysis." Simply stated, our approach will involve expanding the real number system by introducing new numbers called "infinitesimals."
These new numbers will have the property that although different from 0, each is smaller than every positive real number and larger than every negative real number. Of course our infinitesimals cannot themselves be real numbers, but so what? This sort of expansion of a number system through the introduction of new numbers which themselves correspond to nothing in the real world is common in mathematics. Negative numbers and imaginary numbers have no direct physical presence in the real world, yet both serve an essential role in solving problems about the real world. These new infinitesimals, once suitably defined, will enable us to solve general problems of slopes of tangents and areas of regions with extraordinary ease. Here's an example:
We shall find the slope of the tangent to y = x2 at the point (1, 1).
How can we approach this problem? We know how to find the slope of a line given two points on the line, but here we are given only one point plus the information that the line is tangent to y = x2 at that point. Our solution is simple: We let [??] be an infinitesimal (positive, say) and consider two points on the curve of y = x2 which are infinitely close to one another, (1, 1) and (1 + [??], (1 + [??])2):
[ILLUSTRATION OMITTED]
We can certainly find the slope of the line going through these two points:
[ILLUSTRATION OMITTED]
It is
[ILLUSTRATION OMITTED]
Similarly, the slope of the chord going through (1, 1) and (1 – [??], (1 – [??])) is 2 – [??].
[ILLUSTRATION OMITTED]
So, what is the slope of our desired tangent? Well, the slope of the tangent must be a real number (2 + [??] and 2 – [??] are not real numbers), and it must fall between the slope of the chords BA and AC.
[ILLUSTRATION OMITTED]
Two is such a real number, and it is easy to see that 2 is the only such number: Since 0 < [??] r for every real r > 0, there can be no reals between 2 and 2 + [??], and similarly, since – r< – [??] < 0 for every real r > 0, there can be no reals between 2 – [??] and 2. Thus 2 is the only real number between 2 – and 2 + [??], and so 2 must be our desired slope.
The expansion of number systems has occurred often in the history of mathematics and has usually marked a major turning point. Our problem of desiring a new number, an infinitesimal, is entirely similar to that, say, of the early algebraists struggling to solve equations. To these men the known number system consisted of 0, the positive rational numbers, and possibly a few irrationals like [square root of 2], [square root of 3], and π. When faced with an equation such as 2x + 10 = 6, they recognized no possible solution. Later generations of mathematicians invented symbols to represent the solutions to such equations, solutions we would call negative numbers, but the inventors still denied their existence. They recognized them only as symbols which could be manipulated in equations, but which were not actually numbers.
The adoption of negative numbers was a slow process which was not completed for centuries. During this period they were tested thoroughly for consistency to see if they could be used without harm. Only when a number system was solidly constructed which contained both positive and negative numbers were they finally accepted. The most interesting point is that these numbers had a formal existence long before they became accepted. At first they were mere symbols, and then only after centuries of use did they become numbers.
A similar story can be told of the birth of complex numbers. As early as 1545, Cardan had formal symbols for them which he enjoyed manipulating, but which he regarded as fictitious. Soon everyone was using them, but again only in a formal way. The great mathematician of the eighteenth century, Leonhard Euler, remarked of them: "... and of such numbers we may truly assert that they are neither nothing, nor greater than nothing, which necessarily constitutes them imaginary or impossible...." Recognition and acceptance of the complex numbers as numbers wasn't actually final until the nineteenth century.
The history of algebra is, in large part, the history of number. From the earliest conception of 1, 2, 3, ... our idea of number has grown slowly and painfully. At each step the growth was the result of a need. The need for numbers to represent the solution to equations such as 2x + 10 = 6 led to the negative numbers. The need for numbers to represent 10 ÷ 4 and 1 ÷ 3 led to the rational numbers. The need for numbers to represent the diagonal of a square of side 1
[ILLUSTRATION OMITTED]
and the circumference of a circle of radius 1
[ILLUSTRATION OMITTED]
led to the irrational numbers. The need for a root to the equation x2 + 1 = 0 led to the complex numbers. Finally, the need for the calculus led to the infinitesimals. At each step expansion of the number system met with opposition, and at each step the new numbers were formally accepted long before they were given the status of numbers. As late as the 1880s, there was a distinguished mathematician, Leopold Kronecker, who philosophically disputed the existence of irrational numbers. At each step the numbers went through a period of experimentation and trial and were finally accepted only when some mathematician was able to develop a consistent system which contained both the old and the new numbers.
The same pattern holds for the infinitesimals. Even after their existence had been denied, they were in constant use as formal symbols. After 300 years of such usage, their existence finally became established when Robinson developed a system containing both infinitesimals and the real numbers. Robinson's system is now called the hyperreal number system, and using it he was able to completely justify the Leibniz approach to the calculus.
Often the new numbers which mathematicians invent shed light on the old numbers. For example, complex numbers were very useful in understanding real numbers. We will find that this is precisely the case with the hyperreals. They will be used exclusively for proving the theorems of calculus, theorems about real numbers. The power and beauty of this method, compared to the theory of limits, is sometimes astonishing. As Leibniz knew, the method of infinitesimals is the easy, natural way to attack these problems, while the theory of limits represents the lengths to which mathematicians were willing to go to avoid them.
Before actually launching ourselves into the details of the calculus, let us conclude this introductory section by considering an example of the second sort of physical problem solvable by calculus, namely the problem of finding the area of a plane figure. We might as well use our function from before, so let A denote the region bounded by the curve y = x2 and the lines x = 1, x = 2, and y = 0. We will find the area of A:
[ILLUSTRATION OMITTED]
Our approach to the problem is very simple: Since we know how to find areas of rectangles, we will simply approximate the area of A by placing a great many thin rectangles over the region and adding up these areas. How thin should these rectangles be? Infinitesimally thin! We proceed as follows:
[ILLUSTRATION OMITTED]
If we divide the area up into rectangles of thickness h, where h is a real number, not an infinitesimal, we can use formulas from high school mathematics to show that the sum of the areas of the rectangles is
7/3 + 3h/2 + h2/6.
Does this formula still hold if h is [??], an infinitesimal? As it happens, it does, and so to find out the desired area, we note that if [??] is an infinitesimal, so are 3[??]/2 and [??]2/6. Thus
7/3 + 3[??]/2 +[??]2/6
is infinitely close to 7/3. But since the area of A must be a real number, we argue, as we did in the tangent example, that the area must be 7/3.
There is one crucial point in our area calculation that bears repeating. We derived the formula for the approximation
7/3 + 3h/2 + h2/6
for h an actual real number, but we then assumed the formula was true even if h were [??], an infinitesimal. With this the proof was simple and easy. In the next chapter we will construct what we call the hyperreal number system by adding new numbers, infinitesimals, to the reals, and the most important part of our work will be to guarantee that formulas that work for reals also work for hyperreals, including infinitesimals.
To accomplish this, we will have to have a very good idea of what we mean by "formula," and this is where the techniques of mathematical logic will come in. Earlier mathematicians who attempted to build the hyperreals were defeated by this idea. There are many, many formulas, in fact, infinitely many, and they seem to be in complete disorder. It was only with the concept of a mathematical language that Robinson was able to bring order out of chaos. With this one key idea and its twin concept, mathematical structure, it will be relatively easy for us to construct the hyperreals. We will do it slowly and carefully, leaving no loose ends, and when we are done we can attack the problems of the calculus with directness and ease.
Ironically, Leibniz, in addition to his development of the calculus and his contributions to law, religion, philosophy, and diplomacy, also proposed another calculus, a calculus ratiocinator. It would consist, he imagined, of "a general method in which all truths of the reason would be reduced to a kind of calculation." In this dream Leibniz anticipated by almost 200 years the birth of mathematical logic.
CHAPTER 2Language and Structure
The job before us is to build the hyperreal numbers. At first glance this does not seem so difficult; all we have to do is take the real numbers and add some infinitesimals. There is, however, a hitch. As mentioned in Chapter 1, we want something very special out of the hyperreals. We want every formula that is true in the reals to be true in the hyperreals. This will not be easy. There are infinitely many formulas, one more complicated than the next, and yet we must construct our system in only a few pages.
This is where mathematical logic comes in. Without it, mathematicians tried in vain for 300 years to construct the hyperreals. With it, Robinson was able to surmount the difficulties with astonishing ease.
Language
... the Symboles serve only to make men go faster about, as greater Winde to a Winde-mill.
Thomas Hobbes (1588–1679)
The key to our construction is to study first the language of the formulas. In this way we can organize the formulas and see how they arise. It sounds a little messy, but it isn't. When we actually build the hyperreals, it will be quite smooth and natural.
There are many different so-called mathematical systems, and each system has an appropriate language. The best way to understand them is to look at a number of examples.
Example 1. Suppose we wish to study the diagram above and, in particular, the relationships between the different regions. The first requirement for our language is that it have symbols for the oddly shaped areas. Let's use the following symbols:
a b c d e f g
(Such symbols are commonly known as constant symbols.) Let's assume that we are primarily interested in the question of which areas are next to which. For this we use a relation symbol N(... , ---) to mean "... is next to ---." For example, N (c, b) says that c is next to b. Using these symbols we can make all manner of statements — true: N(e,d), N (g, b); and false:N (c, g), N (f,e). We will agree that no area is next to itself, so N (c, c) is considered false.
To make longer and more intricate statements, we use the following connectives:
^ [??] ->
The symbol ^ means "and." For example, N (c,d) ^ N (f, a) is read "c is next to d and f is next to a." We can take any two statements P and Q and put them together to get P ^ Q. This new statement is true if and only if both P and Q are true. For example,
N(c, d) [??] N(f, a) is true. N(b, f) [??] N(d, e) is false.
[??] means "or." For example, N (c, f) [??] N (a, b) is read "either c is next to f or a is next to b." Again, we can take any two statements P and Q and put them together to get P [??] Q, and this new statement is true if and only if at least one of the two statements P or Q is true. For example,
N(b, f) [??] N(d, e) is true. N(c, d) [??] N(b, a) is true. N(b, f) [??] N(c, f) is false.
The symbol -> means "implies." For example, N (c, d) ->N (g,f) is read "c is next to d implies g is next to f," or equivalently "if c is next to d, then g must be next to f." Once again, we can take any two statements P and Q and put them together to get P ->Q. This statement is true if and only if whenever P is true, so is Q. Strictly speaking, the statement is only false when P is true and Q is false. For example,
N(a, f) ->N(e, f) is false. N(a, f) ->N(d, f) is true. N(e, f) ->N(d, f) is true. N(e, f) ->N(b, e) is true.
Using parentheses ( ) the connectives can be used to make longer statements. For example,
N(c, d) ->(N (c, g) [??] N(f, a)) (true). N(c, f) [??] (N(b, g) -> N(b, f)) (false). (N(b, a) [??] N(e, f)) -> (N(c, g) ^ N(e, a)) (false).
Finally, another symbol is useful. The symbol ~ means "not" or "it is not true that." The symbol ~ changes a statement from false to true or from true to false. For example,
~ N(d,f) is false. ~ N(g, c) is true.
EXERCISES
Are the following statements true or false?
1. N (d, e) ^ ~ N(b, f).
2. N(c, e).
3. ~ N(c, e).
4. ~ ~ N(c, e).
5. ~ ~ ~N(c, e).
6. ~ (N(g, c) -> N(g, f)).
7. ~ (~ N(f, c) ^ ~ N(g, b])).
8. (N(b, c) [??] ~ N(b, c) -> (Nb, c) ^ ~ N(b, c)).
9. N(a, b) -> N(b, c) -> N(c, d) -> N(d, e).
10. ~((~N(g, a) -> N(e, f)) [??] ~ (~ N(c, d) ^ ~ N(d, a) ^ ~ N(f, b)).
(Continues...)
Excerpted from Infinitesimal Calculus by James M. Henle, Eugene M. Kleinberg. Copyright © 1979 The Massachusetts Institute of Technology. Excerpted by permission of Dover Publications, Inc..
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Table of Contents
Contents
Preface, vi,1 Introduction, 3,
2 Language and Structure, 13,
3 The Hyperreal Numbers, 25,
4 The Hyperreal Line, 32,
5 Continuous Functions, 42,
6 Integral Calculus, 52,
7 Differential Calculus, 65,
8 The Fundamental Theorem, 77,
9 Infinite Sequences and Series, 84,
10 Infinite Polynomials, 95,
11 The Topology of the Real Line, 106,
12 Standard Calculus and Sequences of Functions, 116,
Appendix A Defining Quasi-big Sets, 126,
Appendix B The Proof of Theorem 3.1, 128,
Subject Index, 130,
Name Index, 134,