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.（以下省略）