国内数学守旧派为何惧怕超实微积分?
(2019-08-24 14:52:58)国内数学守旧派为何惧怕超实微积分?
国内数学守旧派,故步自封,沉迷于十九世纪数学,“亲”不够,惧怕
“对接”现时代超实微积分。
袁萌
附件:
Hyperreal calculus
Abstract This
project deals with doing calculus not by using epsilons and deltas,
but by using a number system called the hyperreal numbers. The
hyperreal numbers is an extension of the normal real numbers with
both innitely small and innitely large numbers added. We will
rst show how this
systemcanbecreated,andthenshowsomebasicprop
Contents 1 1 Construction of the hyperreal numbers 3 1.1 Intuitive construction .. . . . .. . 3
1.2 Ultralters . 3
1.3 Formal construction . . . . . . .. . . 4
1.4 Innitely small and large numbers . . . . . . . . . . . . . . . . . 5
1.5 Enlarging sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.6 Extending functions . . . . . . . . . . . . . . . . . . 6
2 The transfer principle 6
2.1 Stating the transfer principle . . . . . . . . 6
2.2 Using the transfer principle . . . . . . . . . . . . . . . 7
3 Properties of the hyperreals 8
3.1 Terminology and notation . . . . . . . . . . . . . . 8
3.2 Arithmetic of
hyperreals .
3.3 Halos . . . . . . . . . . . . . . . . . . . . . . . 9
3.4 Shadows . . . . . . . . . . . . . . . . . . . 10
4 Convergence 11
4.1 Convergence in hyperreal calculus. . . . . . .. 11 4.2 Monotone
convergence . . . .
5 Continuity 13