无穷小微积分再次放飞互联网，祝大家新春快乐！

        当前，在国内互联网上搜索“无穷小微积分”组合关键词，相关搜索结果稳定在6万篇左右。这是菲氏微积分徒子徒孙们抹杀不掉的事实。

袁萌  2月14日

附：（上传文件）
        Elementary Calculus: An Infinitesimal approach is a textbook by H.Jerome Keisler. The subtitle alludes to the infinitesimal numbers of thehyperreal number system of Abraham Robinson and is sometimes given as Anapproach using infinitesimals. The book is available freely online and iscurrently published by Dover.[1]
 
Contents
 
    1 Textbook
    2 Reception
    3 Transfer principle
    4 See also
    5 Notes
    6 References
    7 External links
 
Textbook
 
        Keisler's textbook is based on Robinson's construction of the hyperrealnumbers. Keisler also published a companion book, Foundations of InfinitesimalCalculus, for instructors which covers the foundational material in more depth.
 
            Keisler defines all basic notions of the calculus such as continuity,derivative, and integral using infinitesimals. The usual definitions in termsof ε-δ techniques are provided at the end of Chapter 5 to enable a transitionto a standard sequence.
 
           In his textbook, Keisler used the pedagogical technique of aninfinite-magnification microscope, so as to represent graphically, distincthyperreal numbers infinitely close to each other. Similarly, aninfinite-resolution telescope is used to represent infinite numbers.
 
        When one examines a curve, say the graph of ?, under a magnifyingglass, its curvature decreases proportionally to the magnification power of thelens. Similarly, an infinite-magnification microscope will transform aninfinitesimal arc of a graph of ?, into a straight line, up to an infinitesimalerror (only visible by applying a higher-magnification "microscope").The derivative of ? is then the (standard part of the) slope of that line (seefigure).
The standard part function "rounds off" a finite hyperreal tothe nearest real number. The "infinitesimal microscope" is used toview an infinitesimal neighborhood of a standard real.
 
Thus the microscope is used as a device in explaining the derivative.
Reception
 
The book was first reviewed by Errett Bishop, noted for his work inconstructive mathematics. Bishop's review was harshly critical; see Criticismof non-standard analysis. Shortly after, Martin Davis and Hausner published adetailed favorable review, as did Andreas Blass and Keith Stroyan.[2][3][4]Keisler's student K. Sullivan,[5] as part of her Ph.D. thesis, performed acontrolled experiment involving 5 schools which found Elementary Calculus tohave advantages over the standard method of teaching calculus.[1][6] Despitethe benefits described by Sullivan, the vast majority of mathematicians havenot adopted infinitesimal methods in their teaching.[7] Recently, Katz &Katz[8] give a positive account of a calculus course based on Keisler's book.O'Donovan also described his experience teaching calculus using infinitesimals.His initial point of view was positive, [9] but later he found pedagogicaldifficulties with approach to non-standard calculus taken by this text andothers.[10]
 
G. R. Blackley remarked in a letter to Prindle, Weber & Schmidt,concerning Elementary Calculus: An Approach Using Infinitesimals, "Suchproblems as might arise with the book will be political. It is revolutionary.Revolutions are seldom welcomed by the established party, althoughrevolutionaries often are."[11]
 
Hrbacek writes that the definitions of continuity, derivative, andintegral implicitly must be grounded in the ε-δ method in Robinson'stheoretical framework, in order to extend definitions to include non-standardvalues of the inputs, claiming that the hope that non-standard calculus couldbe done without ε-δ methods could not be realized in full.[12] B?aszczyk et al.detail the usefulness of microcontinuity in developing a transparent definitionof uniform continuity, and characterize Hrbacek's criticism as a "dubiouslament".[13]
Transfer principle
 
Between the first and second edition of the Elementary Calculus, muchof the theoretical material that was in the first chapter was moved to theepilogue at the end of the book, including the theoretical groundwork ofnon-standard analysis.
 
In the second edition Keisler introduces the extension principle andthe transfer principle in the following form:
 
    Every real statement thatholds for one or more particular real functions holds for the hyperreal naturalextensions of these functions.（这句话是无穷小微积分的关键！）
 
Keisler then gives a few examples of real statements to which theprinciple applies:
 
    Closure law for addition: forany x and y, the sum x + y is defined.
    Commutative law for addition:x + y = y + x.
    A rule for order: if 0 < x< y then 0 < 1/y < 1/x.
    Division by zero is neverallowed: x/0 is undefined.
    An algebraic identity: ( x ?y ) 2 = x 2 ? 2 x y + y 2 {\displaystyle (x-y)^{2}=x^{2}-2xy+y^{2}}(x-y)^{2}=x^{2}-2xy+y^{2}.
    A trigonometric identity: sin2 ? x + cos 2 ? x = 1 {\displaystyle \sin ^{2}x+\cos ^{2}x=1}\sin ^{2}x+\cos ^{2}x=1.
A rule for logarithms: If x> 0 and y > 0, then log 10 ? ( x y ) = log 10 ? x + log 10