Editing Nonstandard calculus

{{Short description|Modern application of infinitesimals}}
In [[mathematics]], '''nonstandard calculus''' is the modern application of [[infinitesimal]]s, in the sense of [[nonstandard analysis]], to infinitesimal [[calculus]]. It provides a rigorous justification for some arguments in calculus that were previously considered merely [[heuristic]].

Non-rigorous calculations with infinitesimals were widely used before [[Karl Weierstrass]] sought to replace them with the [[(ε, δ)-definition of limit]] starting in the 1870s. For almost one hundred years thereafter, mathematicians such as [[Richard Courant]] viewed infinitesimals as being naive and vague or meaningless.<ref>Courant described infinitesimals on page 81 of ''Differential and Integral Calculus, Vol I'', as "devoid of any clear meaning" and "naive befogging". Similarly on page 101, Courant described them as "incompatible with the clarity of ideas demanded in mathematics", "entirely meaningless", "fog which hung round the foundations", and a "hazy idea".</ref> <!--On this paragraph, see Talk, three sections that mention Courant and/or quotations-->

Contrary to such views, [[Abraham Robinson]] showed in 1960 that infinitesimals are precise, clear, and meaningful, building upon work by [[Edwin Hewitt]] and [[Jerzy Łoś]]. According to [[Howard Jerome Keisler|Howard Keisler]], "Robinson solved a three hundred year old problem by giving a precise treatment of infinitesimals.  Robinson's achievement will probably rank as one of the major mathematical advances of the twentieth century."<ref>[[Elementary Calculus: An Infinitesimal Approach]], p. iv.</ref>

== History ==
The history of nonstandard calculus began with the use of infinitely small quantities, called [[infinitesimal]]s in [[calculus]].  The use of infinitesimals can be found in the foundations of calculus independently developed by [[Gottfried Leibniz]] and [[Isaac Newton]] starting in the 1660s. [[John Wallis]] refined earlier techniques of [[indivisibles]] of [[Bonaventura Cavalieri|Cavalieri]] and others by exploiting an [[infinitesimal]] quantity he denoted <math>\tfrac{1}{\infty}</math> in area calculations, preparing the ground for integral [[calculus]].<ref>Scott, J.F. 1981. "The Mathematical Work of John Wallis, D.D., F.R.S. (1616&ndash;1703)". Chelsea Publishing Co. New York, NY. p. 18.</ref> They drew on the work of such mathematicians as [[Pierre de Fermat]], [[Isaac Barrow]] and [[René Descartes]].

In early calculus the use of [[infinitesimal]] quantities was criticized by a number of authors, most notably [[Michel Rolle]] and [[George Berkeley|Bishop Berkeley]] in his book ''[[The Analyst]]''.

Several mathematicians, including [[Colin Maclaurin|Maclaurin]] and [[Jean le Rond d'Alembert|d'Alembert]], advocated the use of limits. [[Augustin Louis Cauchy]] developed a versatile spectrum of foundational approaches, including a definition of [[continuous function|continuity]] in terms of infinitesimals and a (somewhat imprecise) prototype of an [[(ε, δ)-definition of limit|ε, δ argument]] in working with differentiation. [[Karl Weierstrass]] formalized the concept of [[Limit of a function|limit]]<!--correct link?--> in the context of a (real) number system without infinitesimals. Following the work of Weierstrass, it eventually became common to base calculus on ε, δ arguments instead of infinitesimals.

This approach formalized by Weierstrass came to be known as the ''standard'' calculus.  After many years of the infinitesimal approach to calculus having fallen into disuse other than as an introductory pedagogical tool, use of infinitesimal quantities was finally given a rigorous foundation by [[Abraham Robinson]] in the 1960s.  Robinson's approach is called [[nonstandard analysis]] to distinguish it from the standard use of limits.  This approach used technical machinery from [[mathematical logic]] to create a theory of [[hyperreal number]]s that interpret infinitesimals in a manner that allows a Leibniz-like development of the usual rules of calculus. An alternative approach, developed by [[Edward Nelson]], finds infinitesimals on the ordinary real line itself, and involves a modification of the foundational setting by extending [[ZFC]] through the introduction of a new unary predicate "standard".

==Motivation==

To calculate the derivative <math>f '</math> of the function <math> y =f(x)=x^2</math> at ''x'', both approaches agree on the algebraic manipulations:

: <math> \frac{\Delta y}{\Delta x} = \frac{(x + \Delta  x)^2 - x^2}{\Delta  x} = \frac{2 x \Delta x + (\Delta x)^2}{\Delta x} = 2 x  + \Delta x \approx 2 x</math>

This becomes a computation of the derivatives using the [[hyperreals]] if <math>\Delta x</math> is interpreted as an infinitesimal and the symbol "<math>\approx</math>" is the relation "is infinitely close to".

In order to make ''f ''' a real-valued function, the final term <math>\Delta x</math> is dispensed with. In the standard approach using only real numbers, that is done by taking the limit as <math>\Delta x</math> tends to zero. In the [[hyperreal number|hyperreal]] approach, the quantity <math>\Delta x</math> is taken to be an infinitesimal, a nonzero number that is closer to 0 than to any nonzero real. The manipulations displayed above then show that <math>\Delta y /\Delta x</math> is infinitely close to 2''x'', so the derivative of ''f'' at ''x'' is then 2''x''.

Discarding the "error term" is accomplished by an application of the [[standard part function]]. Dispensing with infinitesimal error terms was historically considered paradoxical by some writers, most notably [[George Berkeley]].

Once the hyperreal number system (an infinitesimal-enriched continuum) is in place, one has successfully incorporated a large part of the technical difficulties at the foundational level.  Thus, the [[epsilon, delta technique]]s that some believe to be the essence of analysis can be implemented once and for all at the foundational level, and the students needn't be "dressed to perform multiple-quantifier logical stunts on pretense of being taught [[infinitesimal calculus]]", to quote a recent study.<ref>{{citation
 | last1 = Katz | first1 = Mikhail
 | author1-link = Mikhail Katz
 | last2 = Tall | first2 = David
 | author2-link = David Tall
 | arxiv = 1110.5747 
 | doi = 
 | issue = 
 | publisher = [[Bharath Sriraman]], Editor. Crossroads in the History of Mathematics and Mathematics Education. [[The Montana Mathematics Enthusiast]] Monographs in Mathematics Education 12, Information Age Publishing, Inc., Charlotte, NC
 | pages = 
 | title = Tension between Intuitive Infinitesimals and Formal Mathematical Analysis
 | volume = 
 | year = 2011| bibcode =2011arXiv1110.5747K}}</ref> More specifically, the basic concepts of calculus such as continuity, derivative, and integral can be defined using infinitesimals without reference to epsilon, delta.

==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.

==Definition of derivative==

The [[Hyperreal number|hyperreals]] can be constructed in the framework of [[Zermelo–Fraenkel set theory]], the standard axiomatisation of set theory used elsewhere in mathematics.  To give an intuitive idea for the hyperreal approach, note that, naively speaking, nonstandard analysis postulates the existence of positive numbers ε ''which  are infinitely small'', meaning that ε is smaller than any standard positive real, yet greater than zero.  Every real number ''x'' is surrounded by an infinitesimal "cloud" of hyperreal numbers infinitely close to it.  To define the derivative of ''f'' at a standard real number ''x'' in this approach, one no longer needs an infinite limiting process as in standard calculus.  Instead, one sets

:<math> f'(x) = \mathrm{st} \left( \frac{f^*(x+\varepsilon)-f^*(x)}{\varepsilon} \right),</math>

where '''st''' is the [[standard part function]], yielding the real number infinitely close to the hyperreal argument of '''st''', and <math>f^*</math> is the natural extension of <math>f</math> to the hyperreals.

==Continuity==
A real function ''f'' is continuous at a standard real number ''x'' if for every hyperreal ''x' '' infinitely close to ''x'', the value ''f''(''x' '') is also infinitely close to ''f''(''x'').  This captures [[Cauchy]]'s definition of continuity as presented in his 1821 textbook [[Cours d'Analyse]], p.&nbsp;34.

Here to be precise, ''f'' would have to be replaced by its natural hyperreal extension usually denoted ''f''<sup>*</sup>.

Using the notation <math>\approx</math> for the relation of being infinitely close as above,
the definition can be extended to arbitrary (standard or nonstandard) points as follows:

A function ''f'' is ''[[microcontinuous]]'' at ''x'' if whenever <math>x'\approx x</math>, one has <math>f^*(x')\approx f^*(x)</math>

Here the point x' is assumed to be in the domain of (the natural extension of) ''f''.

The above requires fewer quantifiers than the [[(ε, δ)-definition of limit|(''ε'',&nbsp;''δ'')-definition]] familiar from  standard elementary calculus:

''f'' is continuous at ''x'' if for every ''ε''&nbsp;>&nbsp;0, there exists a ''δ''&nbsp;>&nbsp;0 such that for every ''x' '', whenever |''x''&nbsp;&minus;&nbsp;''x'&thinsp;''|&nbsp;<&nbsp;''δ'', one has |''f''(''x'')&nbsp;&minus;&nbsp;''f''(''x'&thinsp;'')|&nbsp;<&nbsp;''ε''.

==Uniform continuity==
A function ''f'' on an interval ''I'' is [[uniform continuity|uniformly continuous]] if its natural extension ''f''* in ''I''* has the following property:<ref>Keisler, Foundations of Infinitesimal Calculus ('07), p.&nbsp;45</ref>

for every pair of hyperreals ''x'' and ''y'' in ''I''*, if <math>x\approx y</math> then <math>f^*(x)\approx f^*(y)</math>.

In terms of microcontinuity defined in the previous section, this can be stated as follows: a real function is uniformly continuous if its natural extension f* is microcontinuous at every point of the domain of f*.

This definition has a reduced quantifier complexity when compared with the standard [[(ε, δ)-definition of limit|(ε,&nbsp;δ)-definition]].  Namely, the epsilon-delta definition of uniform continuity requires four quantifiers, while the infinitesimal definition requires only two quantifiers.  It has the same quantifier complexity as the definition of uniform continuity in terms of ''sequences'' in standard calculus, which however is not expressible in the [[first-order logic|first-order language]] of the real numbers.

The hyperreal definition can be illustrated by the following three examples.

Example 1: a function ''f'' is uniformly continuous on the semi-open interval (0,1], if and only if its natural extension f* is microcontinuous (in the sense of the formula above) at every positive infinitesimal, in addition to continuity at the standard points of the interval.

Example 2: a function ''f'' is uniformly continuous on the semi-open interval [0,∞) if and only if it is continuous at the standard points of the interval, and in addition, the natural extension ''f''* is microcontinuous at every positive infinite hyperreal point.

Example 3: similarly, the failure of uniform continuity for the squaring function

:<math>x^2</math>

is due to the absence of microcontinuity at a single infinite hyperreal point.

Concerning quantifier complexity, the following remarks were made by [[Kevin Houston (mathematician)|Kevin Houston]]:<ref>[[Kevin Houston (mathematician)|Kevin Houston]], How to Think Like a Mathematician, {{ISBN|978-0-521-71978-0}}</ref>

:The number of quantifiers in a mathematical statement gives a rough measure of the statement’s complexity. Statements involving three or more quantifiers can be difficult to understand. This is the main reason why it is hard to understand the rigorous definitions of limit, convergence, continuity and differentiability in analysis as they have many quantifiers.  In fact, it is the alternation of the <math>\forall</math> and <math>\exists</math> that causes the complexity.

[[Andreas Blass]] wrote as follows:
:Often ... the nonstandard definition of a concept is simpler than the standard definition (both intuitively simpler and simpler in a technical sense, such as quantifiers over lower types or fewer alternations of quantifiers).<ref>{{citation|first1=Andreas|last1=Blass
 |author1-link = Andreas Blass
 |title=Review: Martin Davis, Applied nonstandard analysis, and K. D. Stroyan and W. A. J. Luxemburg, Introduction to the theory of infinitesimals, and H. Jerome Keisler, Foundations of infinitesimal calculus|journal=Bull. Amer. Math. Soc.
 |volume=84
 |number=1
 |year=1978
 |pages=34–41|url=https://www.ams.org/journals/bull/1978-84-01/S0002-9904-1978-14401-2/home.html |doi=10.1090/S0002-9904-1978-14401-2|doi-access=free
 }}, p. 37.</ref>

==Compactness==

A set A is compact if and only if its natural extension A* has the following property: every point in A* is infinitely close to a point of A.  Thus, the open interval (0,1) is not compact because its natural extension contains positive infinitesimals which are not infinitely close to any positive real number.

==Heine–Cantor theorem==
The fact that a continuous function on a compact interval ''I'' is necessarily uniformly continuous (the [[Heine–Cantor theorem]]) admits a succinct hyperreal proof.  Let ''x'', ''y'' be hyperreals in the natural extension ''I*'' of ''I''.  Since ''I'' is compact, both st(''x'') and st(''y'') belong to ''I''.  If ''x'' and ''y'' were infinitely close, then by the triangle inequality, they would have the same standard part

:<math>c = \operatorname{st}(x) = \operatorname{st}(y).</math>

Since the function is assumed continuous at c,

:<math>f(x)\approx f(c)\approx f(y),</math>

and therefore ''f''(''x'') and ''f''(''y'') are infinitely close, proving uniform continuity of ''f''.

==Why is the squaring function not uniformly continuous?==

Let ''f''(''x'') = ''x''<sup>2</sup> defined on <math>\mathbb{R}</math>.  Let <math>N\in \mathbb{R}^*</math> be an infinite hyperreal.  The hyperreal number <math>N + \tfrac{1}{N}</math> is infinitely close to ''N''.  Meanwhile, the difference

:<math> f(N+\tfrac{1}{N}) - f(N) = N^2 + 2 + \tfrac{1}{N^2} - N^2 = 2 + \tfrac{1}{N^2}</math>

is not infinitesimal.   Therefore, ''f*'' fails to be microcontinuous at the hyperreal point ''N''.  Thus, the squaring function is not uniformly continuous, according to the definition in [[#Uniform continuity|uniform continuity]] above.

A similar proof may be given in the standard setting {{harv|Fitzpatrick|2006|loc=Example 3.15}}.

==Example: Dirichlet function==
Consider the [[Dirichlet function]]
:<math>I_Q(x):=\begin{cases} 1 & \text{ if }x \text{ is rational}, \\
0 & \text{ if } x \text{ is irrational}. \end{cases}</math>

It is well known that, under the [[continuous function|standard definition of continuity]], the function is discontinuous at every point.  Let us check this in terms of the hyperreal definition of continuity above, for instance let us show that the Dirichlet function is not continuous at π.  Consider the continued fraction approximation a<sub>n</sub> of π.  Now let the index n be an infinite [[hypernatural]] number.  By the [[transfer principle]], the natural extension of the Dirichlet function takes the value 1 at a<sub>n</sub>.  Note that the hyperrational point a<sub>n</sub> is infinitely close to π.  Thus the natural extension of the Dirichlet function takes different values (0 and 1) at these two infinitely close points, and therefore the Dirichlet function is not continuous at&nbsp;''π''.

==Limit==
While the thrust of Robinson's approach is that one can dispense with the approach using multiple quantifiers, the notion of limit can be easily recaptured in terms of the [[standard part function]] '''st''', namely

:<math>\lim_{x\to a} f(x) = L</math>

if and only if whenever the difference ''x''&nbsp;&minus;&nbsp;''a'' is infinitesimal, the difference ''f''(''x'')&nbsp;&minus;&nbsp;''L'' is infinitesimal, as well, or in formulas:

:if st(''x'') = ''a''&nbsp;  then st(''f''(''x'')) = L,

cf. [[(ε, δ)-definition of limit]].

==Limit of sequence==
Given a sequence of real numbers <math>\{x_n \mid n\in \N\}</math>, if <math>L\in \R</math> ''L'' is ''the limit'' of the sequence and

:<math> L = \lim_{n \to \infty} x_n </math>

if for every infinite [[hypernatural]] ''n'', st(''x''<sub>''n''</sub>)=''L'' (here the extension principle is used to define ''x''<sub>''n''</sub> for every hyperinteger ''n'').

This definition has no [[Quantifier (logic)|quantifier]] alternations. The standard [[(ε, δ)-definition of limit|(ε, δ)-style]] definition, on the other hand, does have quantifier alternations:

:<math>L = \lim_{n \to \infty} x_n \Longleftrightarrow \forall \varepsilon>0\;, \exists N \in \N\;,  \forall n \in \N : n >N \rightarrow |x_n - L| < \varepsilon.</math>

==Extreme value theorem==

To show that a real continuous function ''f'' on [0,1] has a maximum, let ''N'' be an infinite [[hyperinteger]].  The interval [0,&nbsp;1] has a natural hyperreal extension.  The function ''f'' is also naturally extended to hyperreals between 0 and 1.  Consider the partition of the hyperreal interval [0,1] into ''N'' subintervals of equal [[infinitesimal]] length 1/''N'', with partition points ''x''<sub>''i''</sub>&nbsp;= ''i''&nbsp;/''N'' as ''i'' "runs" from 0 to ''N''.  In the standard setting (when ''N'' is finite), a point with the maximal value of ''f'' can always be chosen among the ''N''+1 points ''x''<sub>''i''</sub>, by induction.  Hence, by the [[transfer principle]], there is a hyperinteger ''i''<sub>0</sub> such that 0&nbsp;≤ ''i''<sub>0</sub>&nbsp;≤ ''N'' and <math>f(x_{i_0})\geq f(x_i)</math> for all ''i''&nbsp;=&nbsp;0,&nbsp;…,&nbsp;''N'' (an alternative explanation is that every [[hyperfinite set]] admits a maximum).  Consider the real point

:<math>c= {\rm st}(x_{i_0})</math>

where '''st''' is the [[standard part function]].  An arbitrary real point ''x'' lies in a suitable sub-interval of the partition, namely <math>x\in [x_i,x_{i+1}]</math>, so that '''st'''(''x''<sub>''i''</sub>)&nbsp;= ''x''.  Applying '''st''' to the inequality <math>f(x_{i_0})\geq f(x_i)</math>, <math>{\rm st}(f(x_{i_0}))\geq {\rm st}(f(x_i))</math>.  By continuity of ''f'',
:<math>{\rm st}(f(x_{i_0}))= f({\rm st} (x_{i_0}))=f(c)</math>.
Hence ''f''(''c'')&nbsp;≥ ''f''(''x''), for all ''x'', proving ''c'' to be a maximum of the real function ''f''.<ref>{{harvtxt |Keisler|1986|p=164}}</ref>

==Intermediate value theorem==

As another illustration of the power of [[Abraham Robinson|Robinson]]'s approach, a short proof of the [[intermediate value theorem]] (Bolzano's theorem) using infinitesimals is done by the following.

Let ''f'' be a continuous function on [''a'',''b''] such that ''f''(''a'')<0 while ''f''(''b'')>0.  Then there exists a point ''c'' in [''a'',''b''] such that ''f''(''c'')=0.

The proof proceeds as follows.  Let ''N'' be an infinite [[hyperinteger]].  Consider a partition of [''a'',''b''] into ''N'' intervals of equal length, with partition points ''x<sub>i</sub>'' as ''i'' runs from 0 to ''N''.  Consider the collection ''I'' of indices such that ''f''(''x''<sub>''i''</sub>)>0.  Let ''i''<sub>0</sub> be the least element in ''I'' (such an element exists by the [[transfer principle]], as ''I'' is a [[hyperfinite set]]).  Then the real number
<math display="block">c=\mathrm{st}(x_{i_0})</math>
is the desired zero of ''f''.
Such a proof reduces the [[Predicate logic|quantifier]] complexity of a standard proof of the IVT.

== Basic theorems ==

If ''f'' is a real valued function defined on an interval [''a'', ''b''], then the transfer operator applied to ''f'', denoted by ''*f'',  is an ''internal'', hyperreal-valued function defined on the hyperreal interval [*''a'', *''b''].

''Theorem'':  Let ''f'' be a real-valued function defined on an
interval [''a'', ''b'']. Then ''f'' is differentiable at ''a < x < b'' if and only if
for every ''non-zero'' infinitesimal ''h'', the value

:<math> \Delta_h f := \operatorname{st} \frac{[{}^*\!f](x+h)-[{}^*\!f](x)}{h} </math>

is independent of ''h''.  In that case, the common value is the derivative of ''f'' at ''x''.

This fact follows from the [[transfer principle]] of nonstandard analysis and [[overspill]].

Note that a similar result holds for differentiability at the endpoints ''a'', ''b'' provided the sign of the infinitesimal ''h'' is suitably restricted.

For the second theorem, the Riemann integral is defined as the limit, if it exists, of a directed family of ''Riemann sums''; these are sums of the form

:<math> \sum_{k=0}^{n-1} f(\xi_k) (x_{k+1} - x_k) </math>

where

:<math>a = x_0 \leq \xi_0 \leq x_1 \leq \ldots x_{n-1} \leq \xi_{n-1} \leq x_n = b.</math>

Such a sequence of values is called a ''partition'' or ''mesh'' and

:<math> \sup_k (x_{k+1} - x_k) </math>

the width of the mesh.  In the definition of the Riemann integral, the limit of the Riemann sums is taken as the width of the mesh goes to 0.

''Theorem'':  Let ''f'' be a real-valued function defined on an
interval [''a'', ''b'']. Then ''f'' is Riemann-integrable on [''a'', ''b''] if and only if
for every internal mesh of infinitesimal width, the quantity

:<math> S_M = \operatorname{st} \sum_{k=0}^{n-1} [*f](\xi_k) (x_{k+1} - x_k) </math>

is independent of the mesh.  In this case, the common value is
the Riemann integral of ''f'' over [''a'', ''b''].

== Applications ==

One immediate application is an extension of the standard definitions of differentiation and integration to [[internal set|internal function]]s on intervals of hyperreal numbers.

An internal hyperreal-valued function ''f'' on [''a, b''] is ''S''-differentiable at ''x'', provided

:<math>  \Delta_h f = \operatorname{st} \frac{f(x+h)-f(x)}{h} </math>

exists and is independent of the infinitesimal ''h''. The value is the ''S'' derivative at ''x''.

''Theorem'':  Suppose ''f'' is ''S''-differentiable at every point of [''a, b''] where ''b'' &minus; ''a'' is a bounded hyperreal.  Suppose furthermore that

:<math> |f'(x)| \leq M \quad a \leq x \leq b. </math>

Then for some infinitesimal ε

:<math> |f(b) - f(a)| \leq M (b-a) +  \epsilon.</math>

To prove this, let ''N'' be a nonstandard natural number.  Divide the interval [''a'', ''b''] into ''N'' subintervals by placing ''N'' &minus; 1 equally spaced  intermediate points:

:<math>a = x_0 <  x_1< \cdots < x_{N-1} < x_N = b</math>

Then

:<math> |f(b) - f(a)|  \leq \sum_{k=1}^{N-1} |f(x_{k+1}) - f(x_{k})| \leq \sum_{k=1}^{N-1} \left\{|f'(x_k)| + \epsilon_k\right\}|x_{k+1} - x_{k}|.</math>

Now the maximum of any internal set of infinitesimals is infinitesimal.  Thus all the ε<sub>k</sub>'s are dominated by an infinitesimal ε. Therefore,

:<math> |f(b) - f(a)|  \leq \sum_{k=1}^{N-1} (M + \epsilon)(x_{k+1} - x_{k}) = M(b-a) + \epsilon (b-a)</math>

from which the result follows.

== See also ==
* [[Adequality]]
* [[Archimedes' use of infinitesimals]]
* [[Criticism of nonstandard analysis]]
* [[Differential_(mathematics)]]
* ''[[Elementary Calculus: An Infinitesimal Approach]]''
* [[Non-classical analysis]]
* [[History of calculus]]

==Notes==
{{Reflist}}

==References==

{{sfn whitelist|CITEREFKeisler1986}}
* {{citation | first = Patrick | last = Fitzpatrick| title= Advanced Calculus|publisher = Brooks/Cole| year=2006}}
*<cite id="CITEREFKeisler1986">H. Jerome Keisler: Elementary Calculus: An Approach Using Infinitesimals. First edition 1976; 2nd edition 1986. (This book is now out of print. The publisher has reverted the copyright to the author, who has made available the 2nd edition in .pdf format available for downloading at http://www.math.wisc.edu/~keisler/calc.html.)</cite>
*H. Jerome Keisler: Foundations of Infinitesimal Calculus, available for downloading at http://www.math.wisc.edu/~keisler/foundations.html (10 jan '07)
*{{citation|first1=Andreas|last1=Blass|title=Review: Martin Davis, Applied nonstandard analysis, and K. D. Stroyan and W. A. J. Luxemburg, Introduction to the theory of infinitesimals, and H. Jerome Keisler, Foundations of infinitesimal calculus|journal=Bull. Amer. Math. Soc.| volume=84|number=1|year=1978|pages=34–41|url=https://www.ams.org/journals/bull/1978-84-01/S0002-9904-1978-14401-2/home.html|doi=10.1090/S0002-9904-1978-14401-2|doi-access=free}}
* [[Margaret Baron|Baron, Margaret E.]]: The origins of the infinitesimal calculus. Pergamon Press, Oxford-Edinburgh-New York 1969. Dover Publications, Inc., New York, 1987. (A new edition of Baron's book appeared in 2004)
* {{springer|title=Infinitesimal calculus|id=p/i050950}}

== External links ==
* {{Cite book|title=Elementary Calculus: An Infinitesimal Approach|last1=Keisler|first1=H. Jerome|publisher=Dover Publications|year=2007|isbn=978-0-48-648452-5}}  [https://people.math.wisc.edu/~keisler/calc.html On-line version (2022)]

* {{Cite book|title=Infinitesimal Calculus|last1=Henle|first1=James M.|last2=Kleinberg|first2=Eugene M.|year=1979|publisher=Dover Publications|isbn=978-0-48-642886-4}} {{Internet Archive|id=infinitesimalcal0000henl_j1k6|name=Infinitesimal Calculus}}

* [http://www.lightandmatter.com/calc/ ''Brief Calculus''] (2005, rev. 2015) by Benjamin Crowel. This short text is designed more for self-study or review than for classroom use.  Infinitesimals are used when appropriate, and are treated more rigorously than in old books like Thompson's ''Calculus Made Easy'', but in less detail than in Keisler's ''Elementary Calculus: An Approach Using Infinitesimals''.

{{Infinitesimals}}

[[Category:Nonstandard analysis]]
[[Category:Calculus]]
[[Category:Infinity]]