17小时前 北大袁萌最新实数探索的深处

实数探索水很深，不是糊涂一下就行了。
比如，可构造性实数、代数数与可计算性实数，等等。

请见本文附件。

Definable real number

The square root of 2 is equal to the length of the hypotenuse of a right triangle with legs of length 1 and is therefore a constructible number
Informally, a definable real number is a real number that can be uniquely specified by its description. The description may be expressed as a construction or as a formula of a formal language. For example, the positive square root of 2,
2 {\displaystyle {\sqrt {2}}}
, can be defined as the unique positive solution to the equation
x 2 = 2 {\displaystyle x^{2}=2}
, and it can be constructed with a compass and straightedge.
Different choices of a formal language or its interpretation can give rise to different notions of definability. Specific varieties of definable numbers include the constructible numbers of geometry, the algebraic numbers, and the computable numbers.

Contents
1
Constructible numbers
2
Algebraic numbers
3
Computable real numbers
4
Definability in arithmetic
5
Definability in models of ZFC
6
7
References
8
Constructible numbers
One way of specifying a real number uses geometric techniques. A real number r is a constructible number if there is a method to construct a line segment of length r using a compass and straightedge, beginning with a fixed line segment of length 1.
Each positive integer, and each positive rational number, is constructible. The positive square root of 2 is constructible. However, the cube root of 2 is not constructible; this is related to the impossibility of doubling the cube.
Algebraic numbers

Algebraic numbers on the complex plane colored by degree (red=1, green=2, blue=3, yellow=4)
A real number r is called an algebraic number if there is a polynomial p(x), with only integer coefficients, so that r is a root of p, that is, p(r)=0. Each algebraic number can be defined individually using the order relation on the reals. For example, if a polynomial q(x) has 5 roots, the third one can be defined as the unique r such that q(r) = 0 and such that there are two distinct numbers less than r for which q is zero.
All rational numbers are algebraic, and all constructible numbers are algebraic. There are numbers such as the cube root of 2 which are algebraic but not constructible.
The algebraic numbers form a subfield of the real numbers. This means that 0 and 1 are algebraic numbers and, moreover, if a and b are algebraic numbers, then so are a+b, a−b, ab and, if b is nonzero, a/b.
The algebraic numbers also have the property, which goes beyond being a subfield of the reals, that for each positive integer n and each algebraic number a, all of the nth roots of a that are real numbers are also algebraic.
There are only countably many algebraic numbers, but there are uncountably many real numbers, so in the sense of cardinality most real numbers are not algebraic. This nonconstructive proof that not all real numbers are algebraic was first published by Georg Cantor in his 1874 paper "On a Property of the Collection of All Real Algebraic Numbers".
Non-algebraic numbers are called transcendental numbers. Specific examples of transcendental numbers include π and Euler's number e.
Computable real numbers
A real number is a computable number if there is an algorithm that, given a natural number n, produces a decimal expansion for the number accurate to n decimal places. This notion was introduced by Alan Turing in 1936.
The computable numbers include the algebraic numbers along with many transcendental numbers including π and e. Like the algebraic numbers, the computable numbers also form a subfield of the real numbers, and the positive computable numbers are closed under taking nth roots for each positive n.
Not all real numbers are computable. The entire set of computable numbers is countable, so most reals are not computable. Specific examples of noncomputable real numbers include the limits of Specker sequences, and algorithmically random real numbers such as Chaitin's Ω numbers.
Definability in arithmetic
Another notion of definability comes from the formal theories of arithmetic, such as Peano arithmetic. The language of arithmetic has symbols for 0, 1, the successor operation, addition, and multiplication, intended to be interpreted in the usual way over the natural numbers. Because no variables of this language range over the real numbers, a different sort of definability is needed to refer to real numbers. A real number a is definable in the language of arithmetic (or arithmetical) if its Dedekind cut can be defined as a predicate in that language; that is, if there is a first-order formula φ in the language of arithmetic, with three free variables, such that
∀m∀n∀p ( φ ( n , m , p )( − 1 ) p⋅n m + 1 < a ) . {\displaystyle \forall m\,\forall n\,\forall p\left(\varphi (n,m,p)\iff {\frac {(-1)^{p}\cdot n}{m+1}}

Here m, n, and p range over nonnegative integers.
The second-order language of arithmetic is the same as the first-order language, except that variables and quantifiers are allowed to range over sets of naturals. A real that is second-order definable in the language of arithmetic is called analytical.
Every computable real number is arithmetical, and the arithmetical numbers form a subfield of the reals, as do the analytical numbers. Every arithmetical number is analytical, but not every analytical number is arithmetical. Because there are only countably many analytical numbers, most real numbers are not analytical, and thus also not arithmetical.
Every computable number is arithmetical, but not every arithmetical number is computable. For example, the limit of a Specker sequence is an arithmetical number that is not computable.
The definitions of arithmetical and analytical reals can be stratified into the arithmetical hierarchy and analytical hierarchy. In general, a real is computable if and only if its Dedekind cut is at level
Δ1 0 {\displaystyle \Delta _{1}^{0}}
of the arithmetical hierarchy, one of the lowest levels. Similarly, the reals with arithmetical Dedekind cuts form the lowest level of the analytical hierarchy.
Definability in models of ZFC
A real number a is first-order definable in the language of set theory, without parameters, if there is a formula φ in the language of set theory, with one free variable, such that a is the unique real number such that φ(a) holds (see Kunen 1980, p. 153). This notion cannot be expressed as a formula in the language of set theory.
All analytical numbers, and in particular all computable numbers, are definable in the language of set theory. Thus the real numbers definable in the language of set theory include all familiar real numbers such as 0, 1, π, e, et cetera, along with all algebraic numbers. Assuming that they form a set in the model, the real numbers definable in the language of set theory over a particular model of ZFC form a field.
Each set model M of ZFC set theory that contains uncountably many real numbers must contain real numbers that are not definable within M (without parameters). This follows from the fact that there are only countably many formulas, and so only countably many elements of M can be definable over M. Thus, if M has uncountably many real numbers, we can prove from "outside" M that not every real number of M is definable over M.
This argument becomes more problematic if it is applied to class models of ZFC, such as the von Neumann universe (Hamkins 2010). The argument that applies to set models cannot be directly generalized to class models in ZFC because the property "the real number x is definable over the class model N" cannot be expressed as a formula of ZFC. Similarly, the question whether the von Neumann universe contains real numbers that it cannot define cannot be expressed as a sentence in the language of ZFC. Moreover, there are countable models of ZFC in which all real numbers, all sets of real numbers, functions on the reals, etc. are definable (Hamkins, Linetsky & Reitz 2013).
Constructible universe
Entscheidungsproblem
Ordinal definable set
Tarski's undefinability theorem
References
Hamkins, Joel David (October 2010), "Is the analysis as taught in universities in fact the analysis of definable numbers?", MathOverflow, retrieved 2016-03-05.
Hamkins, Joel David; Linetsky, David; Reitz, Jonas (2013), "Pointwise Definable Models of Set Theory", Journal of Symbolic Logic, 78 (1): 139–156, arXiv:1105.4597, doi:10.2178/jsl.7801090.
Kunen, Kenneth (1980), Set Theory: An Introduction to Independence Proofs, Amsterdam: North-Holland, ISBN 978-0-444-85401-8.
Turing, A.M. (1936), "On Computable Numbers, with an Application to the Entscheidungsproblem", Proceedings of the London Mathematical Society, 2 (published 1937), 42 (1), pp. 230–65, doi:10.1112/plms/s2-42.1.230 (and Turing, A.M. (1938), "On Computable Numbers, with an Application to the Entscheidungsproblem: A correction", Proceedings of the London Mathematical Society, 2 (published 1937), 43 (6), pp. 544–6, doi:10.1112/plms/s2-43.6.544). Computable numbers (and Turing's a-machines) were introduced in this paper; the definition of computable numbers uses infinite decimal sequences.
Can each number be specified by a finite text?
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Natural numbers (
N {\displaystyle \mathbb {N} }
)Integers (
Z {\displaystyle \mathbb {Z} }
)Rational numbers (
Q {\displaystyle \mathbb {Q} }
)Constructible numbersAlgebraic numbers (
A {\displaystyle \mathbb {A} }
)PeriodsComputable numbersDefinable real numbersArithmetical numbersGaussian integers
Division algebras
Real numbers (
R {\displaystyle \mathbb {R} }
)Complex numbers (
C {\displaystyle \mathbb {C} }
)Quaternions (
H {\displaystyle \mathbb {H} }
)Octonions (
O {\displaystyle \mathbb {O} }
)
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composition algebras
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R {\displaystyle \mathbb {R} }
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C {\displaystyle \mathbb {C} }
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Printable version 1天前 北大袁萌实数系统的公理化

2019-12-10

# 实数系统的公理化

随起来很是好笑，无穷小微积分教材给全国高校传统微积分教材提供了实数系统的公理化。

传统微积分

教材需要不需要公理化？糊涂下去是死路一条。

请见本文附件。

袁萌   陈启清   12月10日

附件：

无穷小微积分教材的结束语摘要

Chapter 1. The axioms for the real numbers come in three sets: the Algebraic Axioms, the Order Axioms, and the Completeness Axiom. All the familiar facts about the real numbers can be proved using only these axioms.

EPILOGUE

905

I. ALGEBRAIC AXIOMS FOR THE REAL NUMBERS

A Closure laws 0 and 1 are real numbers. If a and b are real numbers, then so are a + b, ab, and -a. If a is a real number and a # 0, then 1/a is a real number. B Commutative laws a + b = b + a ab = ba. C Associative laws a + (b + c) = (a + b) + c a(bc) = (ab)c.

0 Identity Jaws

E Inverse laws

F Distributive law

DEFINITION

O+a=a

a+(-a)=O

1·a =a.

If a # 0, a ·- = 1. a a • (b + c) = ab + ac.

The positive integers are the real numbers 1, 2 = 1 + 1, 3 = 1 + 1 + 1, 4 = 1 + 1 + 1 + 1, and so on.

II. ORDER AXIOMS FOR THE REAL NUMBERS

A 0 < 1. B Transitive law If a < b and b < c then a < c. C Trichotomy law Exactly one of the relations a < b, a = b, b < a, holds. 0 Sum law If a < b, then a + c < b + c.

E Product law If a < b and 0 < c, then ac < be.

F Root axiom For every real number a > 0 and every positive integer n, there is a real number b > 0 such that b" = a.

Ill. COMPLETENESS AXIOM 2天前 北大袁萌学微积，讲历史，不做迷路人

目前，国内微积分教材，讲微积分发展史，一般讲到牛顿、莱布尼兹为止，其实相差十万八千里，甚胡

请见本文附件。

Epllogue（无穷小微积分课程的结束语）

How does the infinitesimal calculus as developed in this book relate to the traditional (or e, 3) calculus? To get the proper perspective we shall sketch the history of the calculus. Many problems involving slopes, areas, and volumes, which we would today call calculus problems, were solved by the ancient Greek mathematicians. The greatest of them was Archimedes (287-212 B.C.). Archimedes anticipated both the infinitesimal and the   ε,δ）approach to calculus. He sometimes discovered his results by reasoning with infinitesimals, but always published his proofs using the "method of exhaustion," which is similar to the ε,δ）  approach. Calculus problems became important in the early 1600's with the development of physics and astronomy. The basic rules for differentiation and integration were discovered in that period by informal reasoning with infinitesimals. Kepler, Galileo, Fermat, and Barrow were among the contributors. In the 1660's and 1670's Sir Isaac Newton and Gottfried Wilhelm Leibniz independently "invented" the calculus. They took the major step of recognizing the importance of a collection of isolated results and organizing them into a whole. Newton, at different times, described the derivative of y (which he called the "fluxion" of y) in three different ways, roughly

(1) The ratio of an infinitesimal change in y to an infinitesimal change in x. (The infinitesimal method.) (2) The limit of the ratio of the change in y to the change in x, l'ly/ l'lx, as l'lx approaches zero. (The limit method.) (3) The velocity of y where x denotes time. (The velocity method.)

In his later writings Newton sought to avoid infinitesimals and emphasized the methods (2) and (3). Leibniz rather consistently favored the infinitesimal method but believed (correctly) that the same results could be obtained using only real numbers. He regarded the infinitesimals as "ideal" numbers like the imaginary numbers. To justify them he proposed his law of continuity: "In any supposed transition, ending in any terminus, it is permissible to institute a general reasoning, in which the terminus may

EPILOGUE 903

also be included."1 This "law" is far too imprecise by present standards. But it was a remarkable forerunner of the Transfer Principle on which modern infinitesimal calculus is based. Leibniz was on the right track, but 300 years too soon! The notation developed by Leibniz is still in general use today, even though it was meant to suggest the infinitesimal method: dyjdx for the derivative (to suggest an infinitesimal change in y divided by an infinitesimal change in x), and s~ f(x) dx for the integral (to suggest the sum of infinitely many infinitesimal quantities f(x) dx). All three approaches had serious inconsistencies which were criticized most effectively by Bishop Berkeley in 1734. However, a precise treatment of the calculus was beyond the state of the art at the time, and the three intuitive descriptions (1H3) of the derivative competed with each other for the next two hundred years. Until sometime after 1820, the infinitesimal method (1) of Leibniz was dominant on the European continent, because of its intuitive appeal and the convenience of the Leibniz notation. In England the velocity method (3) predominated; it also has intuitive appeal but cannot be made rigorous. In 1821 A. L. Cauchy published a forerunner of the modern treatment of the calculus based on the limit method (2). He defined the integral as well as the derivative in terms of limits, namely

f

b b f(x) dx = lim If(x) Llx. a Ax-o+ a

He still used infinitesimals, regarding 4天前 北大袁萌微积分发展历史，为何断裂？

微积分发展历史，在时间轴上，表现为一个区间，其中不该发生断裂。
微积分思想老祖宗阿基米德预见到解决微积分问题的两种方法：无穷小与 （ε,δ）极限方法。这是历史的事实。
到了十九世纪，（ε,δ）极限理论护犊子彻底驱除无穷小方法，致使微积分发展历史出现断裂。
到了二十世纪六十年代，鲁宾逊恢复了无穷小的名誉，弥补了微积分发展历史上的断裂。
请见本文附件。

Epllogue（无穷小微积分课程的结束语）
How does the infinitesimal calculus as developed in this book relate to the traditional (or e, 3) calculus? To get the proper perspective we shall sketch the history of the calculus. Many problems involving slopes, areas, and volumes, which we would today call calculus problems, were solved by the ancient Greek mathematicians. The greatest of them was Archimedes (287-212 B.C.). Archimedes anticipated both the infinitesimal and the   （ε,δ）approach to calculus. He sometimes discovered his results by reasoning with infinitesimals, but always published his proofs using the "method of exhaustion," which is similar to the （ε,δ）  approach. Calculus problems became important in the early 1600's with the development of physics and astronomy. The basic rules for differentiation and integration were discovered in that period by informal reasoning with infinitesimals. Kepler, Galileo, Fermat, and Barrow were among the contributors. In the 1660's and 1670's Sir Isaac Newton and Gottfried Wilhelm Leibniz independently "invented" the calculus. They took the major step of recognizing the importance of a collection of isolated results and organizing them into a whole. Newton, at different times, described the derivative of y (which he called the "fluxion" of y) in three different ways, roughly
(1) The ratio of an infinitesimal change in y to an infinitesimal change in x. (The infinitesimal method.) (2) The limit of the ratio of the change in y to the change in x, l'ly/ l'lx, as l'lx approaches zero. (The limit method.) (3) The velocity of y where x denotes time. (The velocity method.)
In his later writings Newton sought to avoid infinitesimals 6天前 北大袁萌学微积，辨真假，不做糊涂人

近日，短文“塔尔斯基的天才思想”发表之后，读者知道数学命题的真与假的正确概念。

进入二十世纪，卡尔纳普“数学语言分析”兴起。在此发展潮流之下，塔尔斯基给出了数学命题的真理理论。

实际上，J.Keisler精心撰写的无穷小微积分课程的“结束语”中，对此说得明明白白。

真假不辨就是糊涂。学微积，辨真假，不做糊涂人。 9天前 北大袁萌塔尔斯基的天才思

Tarski’s Truth Definitions

First published Sat Nov 10, 2001; substantive revision Mon Aug 20, 2018

In 1933 the Polish logician Alfred Tarski published a paper in which he discussed the criteria that a definition of ‘true sentence’ should meet, and gave examples of several such definitions for particular formal languages. In 1956 he and his colleague Robert Vaught published a revision of one of the 1933 truth definitions, to serve as a truth definition for model-theoretic languages. This entry will simply review the definitions and make no attempt to explore the implications of Tarski’s work for semantics (natural language or programming languages) or for the philosophical study of truth. (For those implications, see the entries on truth and Alfred Tarski.)

1. The 1933 programme and the semantic conception

1.1 Object language and metalanguage

1.2 Formal correctness

2. Some kinds of truth definition on the 1933 pattern

2.1 The standard truth definitions

2.2 The truth definition by quantifier elimination

3. The 1956 definition and its offspring

Bibliography

Other Internet Resources

Related Entries

1. The 1933 programme and the semantic conception

In the late 1920s Alfred Tarski embarked on a project to give rigorous definitions for notions useful in scientific methodology. In 1933 he published (in Polish) his analysis of the notion of a true sentence. This long paper undertook two tasks: first to say what should count as a satisfactory definition of ‘true sentence’ for a given formal language, and second to show that there do exist satisfactory definitions of ‘true sentence’ for a range of formal languages. We begin with the first task; Section 2 will consider the second.

We say that a language is fully interpreted if all its sentences have meanings that make them either true or false. All the languages that Tarski considered in the 1933 paper were fully interpreted, with one exception described in Section 2.2 below. This was the main difference between the 1933 definition and the later model-theoretic definition of 1956, which we shall examine in Section 3.

Tarski described several conditions that a satisfactory definition of truth should meet.

1.1 Object language and metalanguage

If the language under discussion (the object language) is

L

L

, then the definition should be given in another language known as the metalanguage, call it

M

M

. The metalanguage should contain a copy of the object language (so that anything one can say in

L

L

can be said in

M

M

too), and

M

M

should also be able to talk about the sentences of

L

L

and their syntax. Finally Tarski allowed

M

M

to contain notions from set theory, and a 1-ary predicate symbol True with the intended reading ‘is a true sentence of 12天前 北大袁萌微积分“小糊涂”，难担大任

众所周知，

如此老化陈旧的微积分教学大纲培养出的微积分“小糊涂”，难担三十年之后的社会大任。

—— 课程团队

课程概述 13天前 北大袁萌数学发展史上的几个里程碑

进入二十世纪，世界数学发展史上出现几个里程碑事件。

1.2001年，希尔伯特“几何基础”

2.2010年，罗素“数学原理”（“      Principia Mathematica

”）

3.2033年，哥德尔“不完全性定理”

4.2060年，鲁宾逊“非标准分析”，

52076年，Keisler“初等微积分”课程。

袁萌  陈启清   1128 16天前 北大袁萌微积分公理化的最好范例

众所周知，进入二十世纪，希尔伯特倡导的数学公理化（公理系统）大行其道，无人可挡，我国除外。

从公理系统的视角来看，微积分公理化的的结果是“包容”初等数学，而不是“高居于”初等数学。换言之，公理化微积分应当定义出全部初等函数。

请见第七章目录：

7 TRIGONOMETRIC FUNCTIONS  365

7.1 Trigonometry 365

7.2 Derivatives of Trigonometric Functions 373

7.3 Inverse Trigonometric Functions 381

7.4 Integration by Parts 391

7.5 Integrals of Powers of Trigonometric Functions 397

7.6 Trigonometric Substitutions 402

7.7 Polar Coordinates 406

7.8 Slopes and Curve Sketching in Polar Coordinates 412

7.9 Area in Polar Coordinates 420

CONTENTS ix

7.10 Length of a Curve in Polar Coordinates 425 Extra Problems for Chapter 7  428

8 EXPONENTIAL AND LOGARITHMIC FUNCTIONS 431

8.1 Exponential Functions 431

8.2 Logarithmic Functions 436

8.3 Derivatives of Exponential Functions and the Number e 441

8.4 Some Uses of Exponential Functions 449 8.5 Natural Logarithms 454

8.6 Some Differential Equations 461

8.7 Derivatives and Integrals Involving In x 469

8.8 Integration of Rational Functions 474 8.9 Methods of Integration 481 Extra Problems for Chapter 8  489

CONTENTS

INTRODUCTION  xiii

1 REAL AND HVPERREAL NUMBERS 1

1.1 The Real Line 1

1.2 Functions of Real Numbers 6

1.3 Straight Lines 16

1.4 Slope and Velocity; The Hyperreal Line 21

1.5 Infinitesimal, Finite, and Infinite Numbers 27

1.6 Standard Parts 35 Extra Problems for Chapter I 41

2 DIFFERENTIATION 43

2.1 Derivatives 43

2.2 Differentials and Tangent Lines 53

2.3 Derivatives of Rational Functions 60

2.4 Inverse Functions 70

2.5 Transcendental Functions 78

2.6 Chain Rule 85

2.7 Higher Derivatives 94

2.8 Implicit Functions 97 Extra Problems for Chapter 2  103

3 CONTINUOUS FUNCTIONS 105

3.1 How to Set Up a Problem 105

3.2 Related Rates 110

3.3 Limits 117

3.4 Continuity 124

3.5 Maxima and Minima 134

3.6 Maxima and Minima - Applications 144

3.7 Derivatives and Curve Sketching 151

vii

viii CONTENTS

3.8 Properties of Continuous Functions 159 Extra Problems for Chapter 3  171

4 INTEGRATION  175

4.1 The Definite Integral 175

4.2 Fundamental Theorem of Calculus 186

4.3 Indefinite Integrals 198

4.4 Integration by Change of Variables 209

4.5 Area between Two Curves 218

4.6 Numerical Integration 224 Extra Problems for Chapter 4  234

5 LIMITS, ANALYTIC GEOMETRY, AND APPROXIMATIONS 237

5.1 Infinite Limits 237

5.2 L'Hospital's Rule 242

5.3 Limits and Curve Sketching 248 5.4 Parabolas 256

5.5 Ellipses and Hyperbolas 264

5.6 Second Degree Curves 272

5.7 Rotation of Axes 276

5.8 The e, 8 Condition for Limits 282

5.9 Newton's Method 289

5.10 Derivatives and Increments 294 Extra Problems for Chapter 5  300

6 APPLICATIONS OF THE INTEGRAL 302

6.1 Infinite Sum Theorem 302

6.2 Volumes of Solids of Revolution 308

6.3 Length of a Curve 319

6.4 Area of a Surface of Revolution 327

6.5 Averages 336

6.6 Some Applications to Physics 341 6.7 Improper Integrals 351 Extra Problems for Chapter 6  362

7 TRIGONOMETRIC FUNCTIONS  365

7.1 Trigonometry 365

7.2 Derivatives of Trigonometric Functions 373

7.3 Inverse Trigonometric Functions 381

7.4 Integration by Parts 391

7.5 Integrals of Powers of Trigonometric Functions 397

7.6 Trigonometric Substitutions 402

7.7 Polar Coordinates 406

7.8 Slopes and Curve Sketching in Polar Coordinates 412

7.9 Area in Polar Coordinates 420

CONTENTS ix

7.10 Length of a Curve in Polar Coordinates 425 Extra Problems for Chapter 7  428

8 EXPONENTIAL AND LOGARITHMIC FUNCTIONS 431

8.1 Exponential Functions 431

8.2 Logarithmic Functions 436

8.3 Derivatives of Exponential Functions and the Number e 441

8.4 Some Uses of Exponential Functions 449 8.5 Natural Logarithms 454

8.6 Some Differential Equations 461

8.7 Derivatives and Integrals Involving In x 469

8.8 Integration of Rational Functions 474 8.9 Methods of Integration 481 Extra Problems for Chapter 8  489

9 INFINITE SERIES 492 9.1 Sequences 492

9.2 Series 501

9.3 Properties of Infinite Series 507

9.4 Series with Positive Terms 511

9.5 Alternating Series 517

9.6 Absolute and Conditional Convergence  521

9.7 Power Series 528

9.8 Derivatives and Integrals of Power Series 533

9.9 Approximations by Power Series 540 9.10 Taylor's Formula 547 9.11 Taylor Series 554 Extra Problems for Chapter 9  561

10 VECTORS 564

10.1 Vector Algebra 564 10.2 Vectors and Plane Geometry 576

10.3 Vectors and Lines in Space 585 10.4 Products of Vectors 593

10.5 Planes in Space 604 10.6 Vector Valued Functions 615

10.7 Vector Derivatives 620 10.8 Hyperreal Vectors 627 Extra Problems for Chapter I 0  635

11 PARTIAL DIFFERENTIATION 639

II. I Surfaces 639

11.2 Continuous Functions of Two or More Variables 651

11.3 Partial Derivatives 656

11.4 Total Differentials and Tangent Planes 662

X CONTENTS

11.5

11.6 11.7 11.8

Chain Rule Implicit Functions Maxima and Minima Higher Partial Derivatives Extra Problems for Chapter II

12 MULTIPLE INTEGRALS 12.1

12.2 12.3

12.4 12.5

12.6

12.7 Double Integrals Iterated Integrals Infinite Sum Theorem and Volume Applications to Physics Double Integrals in Polar Coordinates Triple Integrals Cylindrical and Spherical Coordinates Extra Problems for Chapter 12

13 VECTOR CALCULUS 13.1

13.2

13.3

13.4

13.5

13.6 Directional Derivatives and Gradients Line Integrals Independence of Path Green's Theorem Surface Area and Surface Integrals Theorems of Stokes and Gauss Extra Problems for Chapter 13

14 DIFFERENTIAL EQUATIONS 14.1

14.2

14.3

14.4

14.5

14.6

14.7 Equations with Separable Variables First Order Homogeneous Linear Equations First Order Linear Equations Existence and Approximation of Solutions Complex Numbers Second Order Homogeneous Linear Equations Second Order Linear Equations Extra Problems for Chapter 14

EPILOGUE 18天前 北大袁萌公理化微积分课程堪称一绝

今年1122日，按照教育部高教司的安排，微积分课程进入数学评论的聚光灯下，任由人们评说。

我们推荐的微积分课程是公理化微积分课程，独一无二，堪称一绝也。

公理化不是人为标签，而是实质性的定语。

CONTENTS

INTRODUCTION  xiii

1

REAL AND HVPERREAL NUMBERS 1 1.1 The Real Line 1

1.2 Functions of Real Numbers 6

1.3 Straight Lines 16

1.4 Slope and Velocity; The Hyperreal Line 21

1.5 Infinitesimal, Finite, and Infinite Numbers 27

1.6 Standard Parts 35 Extra Problems for Chapter I 41

2 DIFFERENTIATION 43

2.1 Derivatives 43

2.2 Differentials and Tangent Lines 53

2.3 Derivatives of Rational Functions 60

2.4 Inverse Functions 70

2.5 Transcendental Functions 78

2.6 Chain Rule 85

2.7 Higher Derivatives 94

2.8 Implicit Functions 97 Extra Problems for Chapter 2  103

3 CONTINUOUS FUNCTIONS 105

3.1 How to Set Up a Problem 105 3.2 Related Rates 110

3.3 Limits 117

3.4 Continuity 124

3.5 Maxima and Minima 134

3.6 Maxima and Minima - Applications 144

3.7 Derivatives and Curve Sketching 151

vii

viii CONTENTS

3.8 Properties of Continuous Functions 159 Extra Problems for Chapter 3  171

4 INTEGRATION  175

4.1 The Definite Integral 175

4.2 Fundamental Theorem of Calculus 186

4.3 Indefinite Integrals 198

4.4 Integration by Change of Variables 209

4.5 Area between Two Curves 218

4.6 Numerical Integration 224 Extra Problems for Chapter 4  234

5 LIMITS, ANALYTIC GEOMETRY, AND APPROXIMATIONS 237

5.1 Infinite Limits 237

5.2 L'Hospital's Rule 242

5.3 Limits and Curve Sketching 248 5.4 Parabolas 256

5.5 Ellipses and Hyperbolas 264

5.6 Second Degree Curves 272

5.7 Rotation of Axes 276

5.8 The e, 8 Condition for Limits 282

5.9 Newton's Method 289

5.10 Derivatives and Increments 294 Extra Problems for Chapter 5  300

6 APPLICATIONS OF THE INTEGRAL 302

6.1 Infinite Sum Theorem 302

6.2 Volumes of Solids of Revolution 308

6.3 Length of a Curve 319

6.4 Area of a Surface of Revolution 327

6.5 Averages 336

6.6 Some Applications to Physics 341 6.7 Improper Integrals 351 Extra Problems for Chapter 6  362

7 TRIGONOMETRIC FUNCTIONS 365

7.1 Trigonometry 365

7.2 Derivatives of Trigonometric Functions 373 7.3 Inverse Trigonometric Functions 381

7.4 Integration by Parts 391

7.5 Integrals of Powers of Trigonometric Functions 397

7.6 Trigonometric Substitutions 402

7.7 Polar Coordinates 406

7.8 Slopes and Curve Sketching in Polar Coordinates 412

7.9 Area in Polar Coordinates 420

CONTENTS ix

7.10 Length of a Curve in Polar Coordinates 425 Extra Problems for Chapter 7  428

8 EXPONENTIAL AND LOGARITHMIC FUNCTIONS 431

8.1 Exponential Functions 431

8.2 Logarithmic Functions 436

8.3 Derivatives of Exponential Functions and the Number e 441

8.4 Some Uses of Exponential Functions 449 8.5 Natural Logarithms 454

8.6 Some Differential Equations 461

8.7 Derivatives and Integrals Involving In x 469

8.8 Integration of Rational Functions 474 8.9 Methods of Integration 481 Extra Problems for Chapter 8  489

9 INFINITE SERIES 492 9.1 Sequences 492

9.2 Series 501

9.3 Properties of Infinite Series 507

9.4 Series with Positive Terms 511

9.5 Alternating Series 517

9.6 Absolute and Conditional Convergence  521

9.7 Power Series 528

9.8 Derivatives and Integrals of Power Series 533

9.9 Approximations by Power Series 540 9.10 Taylor's Formula 547 9.11 Taylor Series 554 Extra Problems for Chapter 9  561

10 VECTORS 564

10.1 Vector Algebra 564 10.2 Vectors and Plane Geometry 576

10.3 Vectors and Lines in Space 585 10.4 Products of Vectors 593

10.5 Planes in Space 604 10.6 Vector Valued Functions 615

10.7 Vector Derivatives 620 10.8 Hyperreal Vectors 627 Extra Problems for Chapter I 0  635

11 PARTIAL DIFFERENTIATION 639

II. I Surfaces 639

11.2 Continuous Functions of Two or More Variables 651

11.3 Partial Derivatives 656

11.4 Total Differentials and Tangent Planes 662

X CONTENTS

11.5

11.6 11.7 11.8

Chain Rule Implicit Functions Maxima and Minima Higher Partial Derivatives Extra Problems for Chapter II

12 MULTIPLE INTEGRALS 12.1

12.2 12.3

12.4 12.5

12.6

12.7 Double Integrals Iterated Integrals Infinite Sum Theorem and Volume Applications to Physics Double Integrals in Polar Coordinates Triple Integrals Cylindrical and Spherical Coordinates Extra Problems for Chapter 12

13 VECTOR CALCULUS 13.1

13.2

13.3

13.4

13.5

13.6 Directional Derivatives and Gradients Line Integrals Independence of Path Green's Theorem Surface Area and Surface Integrals Theorems of Stokes and Gauss Extra Problems for Chapter 13

14 DIFFERENTIAL EQUATIONS 14.1

14.2

14.3

14.4

14.5

14.6

14.7 Equations with Separable Variables First Order Homogeneous Linear Equations First Order Linear Equations Existence and Approximation of Solutions Complex Numbers Second Order Homogeneous Linear Equations Second Order Linear Equations Extra Problems for Chapter 14

EPILOGUE 19天前 北大袁萌微积分教学对比实践说明了什么？

50年前，美国芝加哥地区高校，进行了一项“教学对比”实验。

简单地说，指定14个班级。分成两个组：A组与B组。

假定A组按照本文附件的大纲内容进行课堂教学；B组按照传统微积分教学大纲进行。

实验的统计结果如何呢？答案是明显的（有论文资料可查）。

如今，50年过去了，在我国能否重复这一“教学对比”实验？这就要看有关领导的决心了。。

CONTENTS

INTRODUCTION xiii

REAL AND HVPERREAL NUMBERS 1

1.1 The Real Line 1

1.2 Functions of Real Numbers 6

1.3 Straight Lines 16

1.4 Slope and Velocity;

The Hyperreal Line 21

1.5 Infinitesimal, Finite, and Infinite Numbers 27

1.6 Standard Parts 35

Extra Problems 20天前 北大袁萌无穷小微积分摆在数学评论的聚光灯下

近日，按照国家教育部的安排，无穷小微积分将摆在数学评论的聚光灯下，以便找出一个可接受普的说法。

我们相信，这是无穷小微积分的大好机会。

为达此目的，首先请见本文附件。

CONTENTS

INTRODUCTION xiii

REAL AND HVPERREAL NUMBERS 1 1.1 The Real Line 1

1.2 Functions of Real Numbers 6

1.3 Straight Lines 16

1.4 Slope and Velocity; The Hyperreal Line 21

1.5 Infinitesimal, Finite, and Infinite Numbers 27

1.6 Standard Parts 35 Extra Problems for Chapter I 41

DIFFERENTIATION 43

2.1 Derivatives 43

2.2 Differentials and Tangent Lines 53

2.3 Derivatives of Rational Functions 60

2.4 Inverse Functions 70

2.5 Transcendental Functions 78

2.6 Chain Rule 85

2.7 Higher Derivatives 94

2.8 Implicit Functions 97 Extra Problems for Chapter 2 103

CONTINUOUS FUNCTIONS 105

3.1 How to Set Up a Problem 105 3.2 Related Rates 110

3.3 Limits 117

3.4 Continuity 124

3.5 Maxima and Minima 134

3.6 Maxima and Minima - Applications 144

3.7 Derivatives and Curve Sketching 151

vii

viii CONTENTS

3.8 Properties of Continuous Functions 159 Extra Problems for Chapter 3 171

4 INTEGRATION 175

4.1 The Definite Integral 175

4.2 Fundamental Theorem of Calculus 186

4.3 Indefinite Integrals 198

4.4 Integration by Change of Variables 209

4.5 Area between Two Curves 218

4.6 Numerical Integration 224 Extra Problems for Chapter 4 234

5 LIMITS, ANALYTIC GEOMETRY, AND APPROXIMATIONS 237

5.1 Infinite Limits 237

5.2 L'Hospital's Rule 242

5.3 Limits and Curve Sketching 248 5.4 Parabolas 256

5.5 Ellipses and Hyperbolas 264

5.6 Second Degree Curves 272

5.7 Rotation of Axes 276

5.8 The e, 8 Condition for Limits 282

5.9 Newton's Method 289

5.10 Derivatives and Increments 294 Extra Problems for Chapter 5 300

6 APPLICATIONS OF THE INTEGRAL 302

6.1 Infinite Sum Theorem 302

6.2 Volumes of Solids of Revolution 308

6.3 Length of a Curve 319

6.4 Area of a Surface of Revolution 327

6.5 Averages 336

6.6 Some Applications to Physics 341 6.7 Improper Integrals 351 Extra Problems for Chapter 6 362

7 TRIGONOMETRIC FUNCTIONS 365

7.1 Trigonometry 365

7.2 Derivatives of Trigonometric Functions 373 7.3 Inverse Trigonometric Functions 381

7.4 Integration by Parts 391

7.5 Integrals of Powers of Trigonometric Functions 397

7.6 Trigonometric Substitutions 402

7.7 Polar Coordinates 406

7.8 Slopes and Curve Sketching in Polar Coordinates 412

7.9 Area in Polar Coordinates 420

CONTENTS ix

7.10 Length of a Curve in Polar Coordinates 425 Extra Problems for Chapter 7 428

8 EXPONENTIAL AND LOGARITHMIC FUNCTIONS 431

8.1 Exponential Functions 431 8.2 Logarithmic Functions 436

8.3 Derivatives of Exponential Functions and the Number e 441

8.4 Some Uses of Exponential Functions 449 8.5 Natural Logarithms 454

8.6 Some Differential Equations 461 8.7 Derivatives and Integrals Involving In x 469

8.8 Integration of Rational Functions 474 8.9 Methods of Integration 481 Extra Problems for Chapter 8 489

9 INFINITE SERIES 492 9.1 Sequences 492

9.2 Series 501 9.3 Properties of Infinite Series 507

9.4 Series with Positive Terms 511

9.5 Alternating Series 517

9.6 Absolute and Conditional Convergence 521

9.7 Power Series 528

9.8 Derivatives and Integrals of Power Series 533

9.9 Approximations by Power Series 540 9.10 Taylor's Formula 547 9.11 Taylor Series 554 Extra Problems for Chapter 9 561

10 VECTORS 564

10.1 Vector Algebra 564 10.2 Vectors and Plane Geometry 576

10.3 Vectors and Lines in Space 585 10.4 Products of Vectors 593

10.5 Planes in Space 604 10.6 Vector Valued Functions 615 10.7 Vector Derivatives 620 10.8 Hyperreal Vectors 627 Extra Problems for Chapter I 0 635

11 PARTIAL DIFFERENTIATION 639 II. I Surfaces 639 11.2 Continuous Functions of Two or More Variables 651 11.3 Partial Derivatives 656 11.4 Total Differentials and Tangent Planes 662

X CONTENTS

11.5 11.6 11.7 11.8

Chain Rule Implicit Functions Maxima and Minima Higher Partial Derivatives Extra Problems for Chapter II

12 MULTIPLE INTEGRALS 12.1 12.2 12.3 12.4 12.5 12.6 12.7 Double Integrals Iterated Integrals Infinite Sum Theorem and Volume Applications to Physics Double Integrals in Polar Coordinates Triple Integrals Cylindrical and Spherical Coordinates Extra Problems for Chapter 12

13 VECTOR CALCULUS 13.1 13.2 13.3 13.4 13.5 13.6 Directional Derivatives and Gradients Line Integrals Independence of Path Green's Theorem Surface Area and Surface Integrals Theorems of Stokes and Gauss Extra Problems for Chapter 13

14 DIFFERENTIAL EQUATIONS 14.1 14.2 14.3 14.4 14.5 14.6 14.7 Equations with Separable Variables First Order Homogeneous Linear Equations First Order Linear Equations Existence and Approximation of Solutions Complex Numbers Second Order Homogeneous Linear Equations Second Order Linear Equations Extra Problems for Chapter 14

EPILOGUE

APPENDIX: TABLES I Trigonometric Functions II Greek Alphabet III Exponential Functions IV Natural Logarithms V Powers and Roots 21天前 北大袁萌函数糊涂概念扩散何时了？

上世纪30年代，法国布尔巴基数学学派给出了函数的序偶标准定义。

反观国内本科微积分教材，关于函数“y=fx）”的传统定义，千篇一律，糊里糊涂（… …），谈何世界一流水平？

我们认为，建设世界水平一流高校本科数学课程，函数的序偶定义是必不可少的，也是回避不了的，“躲”不过去。

请见本文附件：函数的序偶定义。

A real fimction of one variable is a set f of ordered pairs of real numbers such that for every real number a one of the following two things happens:

(i) There is exactly one real number b for which the ordered pair (a, b) is a member of f. In this case we say that f(a) is defined and we write f(a) = b. The number b is called the value of f at a.

(ii) There is no real number b for which the ordered pair (a, b) is a member of f. In this case we say that f(a) is undefined.

Thus f(a) = b means that the ordered pair (a, b) is an element of f. Here is one way to visualize a function. Imagine a black box labeled f as in Figure 1.2.1.

ich

函数的序偶定义

袁萌 陈启清  1120

附件：

DEFINITION

A real fimction of one variable is a set f of ordered pairs of real numbers such that for every real number a one of the following two things happens:

(i) There is exactly one real number b for which the ordered pair (a, b) is a member of f. In this case we say that f(a) is defined and we write f(a) = b. The number b is called the value of f at a.

(ii) There is no real number b for which the ordered pair (a, b) is a member of f. In this case we say that f(a) is undefined.

Thus f(a) = b means that the ordered pair (a, b) is an element of f. Here is one way to visualize a function. Imagine a black box labeled f as in Figure 1.2.1.

ich

近日，国家教育部发文，要求全国高校组织力量建设一批世界水平本科微积分课程。

上世纪30年代，法国布尔巴基数学学派给出了函数的序偶标准定义。

A real fimction of one variable is a set f of ordered pairs of real numbers such that for every real number a one of the following two things happens:

(i) There is exactly one real number b for which the ordered pair (a, b) is a member of f. In this case we say that f(a) is defined and we write f(a) = b. The number b is called the value of f at a.

(ii) There is no real number b for which the ordered pair (a, b) is a member of f. In this case we say that f(a) is undefined.

Thus f(a) = b means that the ordered pair (a, b) is an element of f. Here is one way to visualize a function. Imagine a black box labeled f as in Figure 1.2.1.

ich

函数的序偶定义

袁萌 陈启清  1120

附件：

DEFINITION

A real fimction of one variable is a set f of ordered pairs of real numbers such that for every real number a one of the following two things happens:

(i) There is exactly one real number b for which the ordered pair (a, b) is a member of f. In this case we say that f(a) is defined and we write f(a) = b. The number b is called the value of f at a.

(ii) There is no real number b for which the ordered pair (a, b) is a member of f. In this case we say that f(a) is undefined.

Thus f(a) = b means that the ordered pair (a, b) is an element of f. Here is one way to visualize a function. Imagine a black box labeled f as in Figure 1.2.1.

ich • 匿名人士 江苏省南通市 (153.37.114.*): 重复三遍四遍？   (2019-11-24 17:55:09)

23天前 北大袁萌现代微积分已经扎根中国

当今，现代微积分的“分界桩”已经牢牢地竖立在中国的大地上。围绕着它，大批守护者跳舞又唱歌。

面对此种情景，数学守旧者低头叹息，沉默不语。

现代微积分“分界桩”位于何处?答案是：无穷小专业网站也。下载“ElemenTARY CALCULUS”（“分界桩”） 24天前 北大袁萌现代微积分的宣言书

四十多年前，中科院计算所张锦文研究员托人从国外带回J.Keisler精心撰写的“基础微积分”教科书。

该书第一版“序言”看后令人感到十分震惊。这是现代微积分的宣言书！

请见本文附件。

The calculus was originally developed using the intuitive concept of an infinitesimal, or an infinitely small number. But for the past one hundred years infinitesimals have been banished from the calculus course for reasons of mathematical rigor. Students have had to learn the subject without the original intuition. This calculus book is based on the work of Abraham Robinson, who in 1960 found a way to make infinitesimals rigorous. While the traditional course begins with the difficult limit concept, this course begins with the more easily understood infinitesimals. It is aimed at the average beginning calculus student and covers the usual three or four semester sequence. The infinitesimal approach has three important advantages for the student. First, it is closer to the intuition which originally led. to the calculus. Second, the central concepts of derivative and integral become easier for the student to understand and use. Third, it teaches both the infinitesimal and traditional approaches, giving the student an extra tool which may become increasingly important in the future. Before describing this book, I would like to put Robinson's work in historical perspective. In the 1670's, Leibniz and Newton developed the calculus based on the intuitive notion of infinitesimals. Infinitesimals were used for another two hundred years, until the first rigorous treatment of the calculus was perfected by Weierstrass in the 1870's. The standard calculus course of today is still based on the "a, 6 definition" of limit given by Weierstrass. In 1960 Robinson solved a three hundred year old problem by giving a precise treatment of the calculus using infinitesimals. Robinson's achievement will probably rank as one of the major mathematical advances of the twentieth century. Recently, infinitesimals have had exciting applications outside mathematics, notably in the fields of economics and physics. Since it is quite natural to use infinitesimals in modelling physical and social processes, such applications seem certain to grow in variety and importance. This is a unique opportunity to find new uses for mathematics, but at present few people are prepared by training to take advantage of this opportunity. Because the approach to calculus is new, some instructors may need additional background material. An instructor's volume, "Foundations of Infinitesimal

PREFACE TO THE FIRST EDITION v

Calculus," gives the necessary background and develops the theory in detail. The instructor's volume is keyed to this book but is self-contained and is intended for the general mathematical public. This book contains all the ordinary calculus topics, including the traditional hmit definition, plus one exua tool-the infinitesimals. Thus the student will be prepared for more advanced courses as they are now taught. In Chapters 1 through 4 the basic concepts of derivative, continuity, and integral are developed quickly using infinitesimals. The traditional limit concept is put off until Chapter 5, where it is motivated by approximation problems. The later chapters develop transcendental functions, series, vectors, partial derivatives, and multiple .integrals. The theory differs from the traditional course, but the notation and methods for solving practical problems are the same. There is a variety of applications to both natural and social sciences. I have included the following innovation for instructors who wish to introduce the transcendental functions early. At the end of Chapter 2 on derivatives, there is a section beginning an alternate track on transcendental functions, and each of Chapters 3 through 6 have alternate track problem sets on transcendental functions. This alternate track can be used to provide greater variety in the early problems, or can be skipped in order to reach the integral as soon as possible. In Chapters 7 and 8 the transcendental functions are developed anew at a more leisurely pace. The book is written for average students. The problems preceded by a square box go somewhat beyond the examples worked out in the text and are intended for the more adventuresome. I was originally led to write this book when it became clear that Robinson's infinitesimal calculus col}ld be made available to college freshmen. The theory is simply presented; for example, Robinson's work used mathematical logic, but this book does not. I first used an early draft of this book in a one-semester course at the University of Wisconsin in 1969. In 1971 a two-semester experimental version was published. It has been used at several colleges and at Nicolet High School near Milwaukee, and was tested at five schools in a controlled experiment by Sister Kathleen Sullivan in 1972-1974. The results (in her 1974 Ph.D. thesis at the University of Wisconsin) show the viability of the infinitesimal approach and will be summarized in an article in the American Mathematical Monthly. I am indebted to many colleagues and students who have given me encouragement and advice, and have carefully read and used various stages of the manuscript. Special thanks are due to Jon Barwise, University of Wisconsin; G. R. Blakley, Texas A & M University; Kenneth A. Bowen, Syracuse University; William P. Francis, Michigan Technological University; A. W. M. Glass, Bowling Green University; Peter Loeb, University of Illinois at Urbana; Eugene Madison and Keith Stroyan, University of Iowa; Mark Nadel, Notre Dame University; Sister Kathleen Sullivan, Barat College; and Frank Wattenberg, University of Massachusetts.

H. Jerome Keisler

CONTENTS PREFACE TO THE FIRST EDITION

The calculus was originally developed using the intuitive concept of an infinitesimal, or an infinitely small number. But for the past one hundred years infinitesimals have been banished from the calculus course for reasons of mathematical rigor. Students have had to learn the subject without the original intuition. This calculus book is based on the work of Abraham Robinson, who in 1960 found a way to make infinitesimals rigorous. While the traditional course begins with the difficult limit concept, this course begins with the more easily understood infinitesimals. It is aimed at the average beginning calculus student and covers the usual three or four semester sequence. The infinitesimal approach has three important advantages for the student. First, it is closer to the intuition which originally led. to the calculus. Second, the central concepts of derivative and integral become easier for the student to understand and use. Third, it teaches both the infinitesimal and traditional approaches, giving the student an extra tool which may become increasingly important in the future. Before describing this book, I would like to put Robinson's work in historical perspective. In the 1670's, Leibniz and Newton developed the calculus based on the intuitive notion of infinitesimals. Infinitesimals were used for another two hundred years, until the first rigorous treatment of the calculus was perfected by Weierstrass in the 1870's. The standard calculus course of today is still based on the "a, 6 definition" of limit given by Weierstrass. In 1960 Robinson solved a three hundred year old problem by giving a precise treatment of the calculus using infinitesimals. Robinson's achievement will probably rank as one of the major mathematical advances of the twentieth century. Recently, infinitesimals have had exciting applications outside mathematics, notably in the fields of economics and physics. Since it is quite natural to use infinitesimals in modelling physical and social processes, such applications seem certain to grow in variety and importance. This is a unique opportunity to find new uses for mathematics, but at present few people are prepared by training to take advantage of this opportunity. Because the approach to calculus is new, some instructors may need additional background material. An instructor's volume, "Foundations of Infinitesimal

PREFACE TO THE FIRST EDITION v

Calculus," gives the necessary background and develops the theory in detail. The instructor's volume is keyed to this book but is self-contained and is intended for the general mathematical public. This book contains all the ordinary calculus topics, including the traditional hmit definition, plus one exua tool-the infinitesimals. Thus the student will be prepared for more advanced courses as they are now taught. In Chapters 1 through 4 the basic concepts of derivative, continuity, and integral are developed quickly using infinitesimals. The traditional limit concept is put off until Chapter 5, where it is motivated by approximation problems. The later chapters develop transcendental functions, series, vectors, partial derivatives, and multiple .integrals. The theory differs from the traditional course, but the notation and methods for solving practical problems are the same. There is a variety of applications to both natural and social sciences. I have included the following innovation for instructors who wish to introduce the transcendental functions early. At the end of Chapter 2 on derivatives, there is a section beginning an alternate track on transcendental functions, and each of Chapters 3 through 6 have alternate track problem sets on transcendental functions. This alternate track can be used to provide greater variety in the early problems, or can be skipped in order to reach the integral as soon as possible. In Chapters 7 and 8 the transcendental functions are developed anew at a more leisurely pace. The book is written for average students. The problems preceded by a square box go somewhat beyond the examples worked out in the text and are intended for the more adventuresome. I was originally led to write this book when it became clear that Robinson's infinitesimal calculus col}ld be made available to college freshmen. The theory is simply presented; for example, Robinson's work used mathematical logic, but this book does not. I first used an early draft of this book in a one-semester course at the University of Wisconsin in 1969. In 1971 a two-semester experimental version was published. It has been used at several colleges and at Nicolet High School near Milwaukee, and was tested at five schools in a controlled experiment by Sister Kathleen Sullivan in 1972-1974. The results (in her 1974 Ph.D. thesis at the University of Wisconsin) show the viability of the infinitesimal approach and will be summarized in an article in the American Mathematical Monthly. I am indebted to many colleagues and students who have given me encouragement and advice, and have carefully read and used various stages of the manuscript. Special thanks are due to Jon Barwise, University of Wisconsin; G. R. Blakley, Texas A & M University; Kenneth A. Bowen, Syracuse University; William P. Francis, Michigan Technological University; A. W. M. Glass, Bowling Green University; Peter Loeb, University of Illinois at Urbana; Eugene Madison and Keith Stroyan, University of Iowa; Mark Nadel, Notre Dame University; Sister Kathleen Sullivan, Barat College; and Frank Wattenberg, University of Massachusetts.

H. Jerome Keisler 25天前 北大袁萌现代微积分包容极限理论，而不是相反

近日，教育部发文要求建设一批世界水平一流本科（基础数学）课程，引发思考。

首先，我们需要搞清楚的是：现代微积分包容极限理论，而不是相反。

这一断语的证据就是Keisler的微积分教科书。请见本文附件。

袁萌  陈启清  1117

附件：

Keisler's textbook]

Keisler's Elementary Calculus: An Infinitesimal Approach defines continuity on page 125 in terms of infinitesimals, to the exclusion of epsilon, delta methods. The derivative is defined on page 45 using infinitesimals rather than an epsilon-delta approach. The integral is defined on page 183 in terms of infinitesimals. Epsilon, delta definitions are introduced on page 282. 27天前 北大袁萌无穷小放飞互联网七年有余

关于现代无穷小理论的博文，七年来我们发表了大约两千余篇，这是值得纪念的。

对于一个国家而言，出现几个现代无穷小理论的守护者是一件非常可喜的事情。

时至今日，国内数学守旧者，为何沉默不语？国内数学守旧派，前景堪忧。 29天前 北大袁萌微分是无穷小吗？

近日，教育部发文要求高校建设一批世界水平一流本科数学课程，取消微积‘分“水课’。·

S is said tobe the slope of f at a if

*   S = st（（f(a+x) – f(a)/x

for every nonzero infinitesmal x

b]，我们将其无限等分，得到无限多个“分点”，做出黎曼和，再取其标准部分，即导函数f'[ab]上的定积分等于

st( f'(x)dx) 1月前 北大袁萌高校名师讲“水课”微积，让笑掉大牙

近日，教育部发文（请见附件），从此以后，高校名师讲“水课”微积，就让笑掉大牙了。

袁萌  陈启清  119

附件：

（二）目标导向，课程优起来。以目标为导向加强课程建设。立足经济社会发展需求和人才培养目标，优化重构教学内容与课程体系，破除课程千校一面，杜绝必修课因人设课，淘汰“水课”，立起课程建设新标杆。“双一流”建设高校、部省合建高校要明确要求两院院士、国家“千人计划”“万人计划”专家、“长江学者奖励计划”入选者、国家杰出青年科学基金获得者等高层次人才建设名课、讲授基础课和专业基础课，建设一批中国特色、世界水平的一流本科课程。聚焦新工科、新医科、新农科、新文科建设，体现多学科思维融合、产业技术与学科理论融合、跨专业能力融合、多学科项目实践融合，建设一批培养创新型、复合型人才的一流本科课程。服务区域经济社会发展主战场，深化产教融合协同育人，建设一批培养应用型人才的一流本科课程。 1月前 北大袁萌一流本科数学课程不能培养“傻呆呆”

“提升高阶性。课程目标坚持知识、能力、素质有机融合，培养学生解决复杂问题的综合能力和高级思维。课程内容强调广度和深度，突破习惯性认知模式，培养学生深度分析、大胆质疑、勇于创新的精神和能力。” 