无穷小微积分，入门三道坎儿

上世纪30年代，哥德尔证明了紧致性定理，到了60年代，塔尔斯基创立模型论，借此鲁宾逊创立非标准分析，随后，1976年，J.Keisler发表无穷小微积分教科书。

该教材复活了300年前莱布尼兹当初创立微积分的思想，给微积分注入了创新活力。

无穷小微积分，入门三道坎儿，意思是说：延伸原则、转移原则与取超实数标准部分原则（所谓“三道坎儿”）。过了这三道坎儿，微积分教学被大大简化，50学时即可严格导出微积分学基本定理。厉害！

有兴趣的读者可参阅该教材第一章5节。

附：该教材第1.5节原文。

Let us sum

marize our intuitive description of the hyperreal numbers from Section 1.4.

The real line is a subset of the hyperreal line; that is, each real number belongs to the set of hyperreal numbers. Surrounding each real number r, we introduce a collection of hyperreal numbers infinitely close to r. The hyperreal numbers infinitely

close to zero are called infinitesimals. The reciprocals of nonzero infinitesimals are infinite hyperreal numbers. The collection of all hyperreal numbers satisfies the

same algebraic laws as the real numbers. In this section we describe the hyperreal

numbers more precisely and develop a facility for computation with them.

This entire calculus course is developed from three basic principles relating the real and hyperreal numbers: the Extension Principle, the Transfer Principle, and the Standard Part Principle. The first two principles are presented in this section,

and the third principle is in the next section.

We begin with the Extension Principle, which gives us new numbers called

hyperreal numbers and extends all real functions to these numbers. The Extension Principle will deal with hyperreal functions as well as real functions. Our discussion

of real functions in Section 1.2 can readily be carried over to hyperreal functions.

Recall that for each real number a, a real function! of one variable either associates

another real number b = f(a) or is undefined. Now, for each hyperreal number H, a hyperreal function F of one variable either associates another hyperreal number

K = F(H) or is undefined. For each pair of hyperreal numbers Hand J, a hyperreal

function G of two variables either associates another hyperreal number K = G(H, J)

or is undefined. Hyperreal functions of three or more variables are defined in a milar way.

?I The ?Extension ?PRINCIPLE

(a) The real numbers form a subset of the hyperreal numbers, and the order relation x < y for the real numbers is a subset of the order relation for

h IT l n hers.

?(b) There is a hyperreal number that is greater than zero but less than every

positive real number.

hyperreal function f* of the same number of variables. f* is

called the natural extension off

Part (a) of the Extension Principle says that the real line is a part of the

hyperreal line. To explain part (b) of the Extension Principle, we give a careful

definition of an infinitesimal.

DEFINITION

A hyperreal number b is said to be:

positive infinitesimal if b is positive but less than every positive real number.

negative infinitesimal if b is negative but greater than every negative real

number.

infinitesimal if b is either positive infinitesimal, negative infinitesimal, or zero.

With this definition, part (b) of the Extension Principle says that there is at

least one positive infinitesimal. We shall see later that there are infinitely many

positive infinitesimals. A positive infinitesimal is a hyperreal number but cannot be

a real number, so part (b) ensures that there are hyperreal numbers that are not real numbers.

Part (c) of the Extension Principle allows us to apply real functions to

hyperreal numbers. Since the addition function + is a real function of two variables,

its natural extension + * is a hyperreal function of two variables. If x and y are

hyperreal numbers, the sum of x and y is the number x + * y formed by using the

natural extension of +. Similarly, the product of x andy is the number x ·* y formed

by using the natural extension of the product function ?. To make things easier

to read, we shall drop the asterisks and write simply x + y and x ? y for the sum

and product of two hyperreal numbers x and y. Using the natural extensions of

the sum and product functions, we will be able to develop algebra for hyperreal

numbers. Part (c) of the Extension Principle also allows us to work with expressions

such as cos (x) or sin (x + cos (y)), which involve one or more real functions. We

call such expressions real expressions. These expressions can be used even when

x and yare hyperreal numbers instead of real numbers. For example, when x and y

are hyperreal, sin (x +cos (y)) will mean sin* (x +cos* (y)), where sin* and cos*

are the natural extensions of sin and cos. The asterisks are dropped as before.

We now state the Transfer Principle, which allows us to carry out computations

with the hyperreal numbers in the same way as we do for real numbers.

Intuitively, the Transfer Principle says that the natural extension of each real function has the same properties as the original function.

II. TRANSFER PRINCIPLE

Every real statement that holds for one or more particular real functions holds

for the hyperrealnatural extensions of these functions.

Here are seven examples that illustrate what we mean by a real statement.

In general, by a real statement we mean a combination of equations or inequalities

about real expressions, and statements specifying whether a real expression is defined

or undefined. A real statement will involve real variables and particular real functions.

(1) Closure law for addition: for any x andy, the sum x + y is defined.

(2) Commutative law for addition: x + y = y + x.

(3) A rule for order: If 0 < x < y, then 0 < 1/y < 1/x.

(4) Division by zero is never allowed: x/0 is undefined.

(5) An algebraic identity: (x - y)2 = x 2

- 2xy + y2

(6) A trigonometric identity: sin2 x + cos2 x = 1.

(7) A rule for logarithms: If x > 0 and y > 0, then log10 (xy) = log10 x

+ loglo y.

Each example has two variables, x and y, and holds true whenever x and y are real

numbers. The Transfer Principle tells us that each example also holds whenever x

and y are hyperreal numbers. For instance, by Example (3), x/0 is undefined, even

for hyperreal x. By Example (6), sin2 x + cos2 x = 1, even for hyperreal x.

Notice that the first five examples involve only the sum, difference, product,

and quotient functions. However, the last two examples are real statements involving

the transcendental functions sin, cos, and log 10. The Transfer Principle extends all

the familiar rules of trigonometry, exponents, and logarithms to the hyperreal

numbers.

In calculus we frequently make a computation involving one or more

unknown real numbers. The Transfer Principle allows us to compute in exactly

the same way with hyperreal numbers. It "transfers" facts about the real numbers

to facts about the hyperreal numbers. In particular, the Transfer Principle implies

that a real function and its natural extension always give the same value when applied

to a real number. This is why we are usually able to drop the asterisks when computing

with hyperreal numbers.

A real statement is often used to define a new real function from old real

functions. By the Transfer Principle, whenever a real statement defines a real function,

the same real statement also defines the hyperreal natural extension function. Here

are three more examples.

(8) The square root function is defined by the real statement y = Jx if,

and only if, l = x and y 2: 0.

(9) The absolute value function is defined by the real statement y = lxl

if, and only if, y = p.

(10) The common logarithm function is defined by the real statement

y = log10 x if, and only if, lOY = x.

In each case, the hyperreal natural extension is the function defined by the given

real statement when x and y vary over the hyperreal numbers. For example, the

hyperreal natural extension ofthe square root function, J *,is defined by Example (8)

when x andy are hyperreal.

An important use of the Transfer Principle is to carry out computations

with infinitesimals. For example, a computation with infinitesimals was used in the

slope calculation in Section 1.4. The Extension Principle tells us that there is at

least one positive infinitesimal hyperreal number, say e. Starting from e, we can use

the Transfer Principle to construct infinitely many other positive infinitesimals. For

example, e2 is a positive infinitesimal that is smaller than e, 0 < e2 < e. (This

follows from the Transfer Principle because 0 < x 2 < x for all real x between 0

and 1.) Here are several positive infinitesimals, listed in increasing order:

, e2

, e/100, e, 75e, Je, e + Je.

30 1 REAL AND HYPER REAL NUMBERS

We can also construct negative infinitesimals, such as - 8 and -8

, and other hyperreal

numbers such as 1 + Je, (10 - 8f, and 1/8.

We shall now give a list of rules for deciding whether a given hyperreal

number is infinitesimal, finite, or infinite. All these rules follow from the Transfer

Principle alone. First, look at Figure 1.5.1, illustrating the hyperrealline.

Infinitesimal

----- ------- -----41--<>--+---_____________________ _

Negative

infinite

Figure 1.5.1

DEFINITION

-3 -2 -1 0 I 2 3

Finite

A hype!Teal number b is said to be:

finite if b is between two real numbers.

positive infinite if b is greater than every real number.

negative infinite if b is less than every real number.

Positive

infinite

Notice that each infinitesimal number is finite. Before going through the

whole list of rules, let us take a close look at two of them.

If 8 is infinitesimal and a is finite, then the product a ? 8 is infinitesimal. For

example, k -68, 10008, (5 - 8)8 are infinitesimal. This can be seen intuitively from

Figure 1.5.2; an infinitely thin rectangle of length a has infinitesimal area.

If 8 is positive infinitesimal, then 1/8 is positive infinite. From experience we

know that reciprocals of small numbers are large, so we intuitively expect 1/8 to

be positive infinite. We can use the Transfer Principle to prove 1/8 is positive infinite.

Let r be any positive real number. Since 8 is positive infinitesimal, 0 < 8 < 1/r.

Applying the Transfer Principle, 1/8 > r > 0. Therefore, l/8 is positive infinite.

cc===========================~ Area=a·E

Figure 1.5.2

RULES FOR INFINITESIMAL, FINITE, AND INFINITE NUMBERS Assume that 8, 6

are infinitesima/s; b, care hype!Tea/ numbers that are finite but not infinitesimal;

and H, K are infinite hype1Teal numbers.

(i) Real numbe1·s:

The only infinitesimal real number is 0.

Every real number is finite.

(ii) Negatives:

1.5 INFINITESIMAL, FINITE, AND INFINITE NUMBERS 31

-b is finite but not infinitesimal.

-His infinite.

(iii) Reciprocals:

If e =I= 0, lfe is infinite.

1/ b is finite but not infinitesimal.

1/H is infinitesimal.

(iv) Sums:

e + l5 is infinitesimal.

b + e is finite but not infinitesimal.

b + c is finite (possibly infinitesimal).

H + e and H + bare infinite.

(v) Products:

(j ? e and b ? e are infinitesimal.

b ? c is finite but not infinitesimal.

H ? band H ? K are infinite.

(vi) Quotients:

efb, ef H, and bf Hare infinitesimal.

bf c is finite but not infinitesimal.

bfe, Hfe, and Hfb are infinite, provided that e -=f. 0.

(vii) Roots:

If e > 0, ~ is infinitesimal.

If b > 0, jZi is finite but not infinitesimal.

If H > 0, '..:jii is infinite.

Notice that we have given no rule for the following combinations:

efl5, the quotient of two infinitesimals.

HfK, the quotient of two infinite numbers.

He, the product of an infinite number and an infinitesimal.

H + K, the sum of two infinite numbers.

Each of these can be either infinitesimal, finite but not infinitesimal, or infinite,

depending on what e, l5, H, and K are. For this reason, they are called indeterminate

forms.

Here are three very different quotients of infinitesimals.

~is infinitesimal (equal to e).

!. is finite but not infinitesimal (equal to 1).

e . . fi . ( 1 1) e2 ts m mte equa to "i .

Table 1.5.1 on the following page shows the three possibilities for each indeterminate

form. Here are some examples which show how to use our rules.

EXAMPLE 1 Consider (b - 3e)f(c + 26). e is infinitesimal, so - 3e is infinitesimal,

and b - 3e is finite but not infinitesimal. Similarly, c + 2(5 is finite but not

infinitesimal. Therefore the quotient

is finite but not infinitesimal.

b - 3e

c + 2(5……（全文完）

。。。

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