Infinitesimals in mathematics teaching

I am considering using infinitesimals in my undergraduate teaching; my experience has been that limits serve only to confuse my students. Now, for students who will not become professional mathematicians, but more users of cheap jerseys mathematics, such as cheap jerseys engineers, I believe that the infinitesimal approach has a Ny lot to offer.

There are several approaches. The best known is non-standard analysis, where the real numbers are extended to the hyperreals by means of an ultraproduct construction, and these hyperreals contain non-zero positive infinitesimal numbers epsilon which are less than any real number:

0<epsilon<r for all real numbers r.

It can be shown that a function defined on the reals can be extended to a function on the hyprereals, and by use of the transfer principle, any statement which is true for real numbers is true over the hyperreals. Any bounded hyperreal r^ast will be infinitely close to a real number cheap jerseys China r, called that standard part of r^ast and denoted mbox{st}(r^ast). Then, for example, the derivative of a function f(x) can be defined as the value D for which


where D is assumed to be RetroCampus! independent of the choice wholesale jerseys China of epsilon.

However, there is another Media approach, called smooth infinitesimal analysis which in some ways is pedagogically easier; based on the notion of nilsquare infinitesimals. These are infinitesimals which have the property that their squares are zero. Whereas non-standard analysis has its roots in mathematical logic, and in Reach particular model theory, nilsquare infinitesimals arise from category theory, and in particular topos theory. Models for mathematics in football which nilsquare infinitesimals exist can be constructed by using a topos: the smooth topos, in which mathematics must be constructive (the law of the excluded middle fails), and all defined functions are continuous. The derivative can then be defined as the value D for which

f(x+epsilon)=f(x)+epsilon D

again assuming independence of epsilon. We can’t divide by a nilsquare infinitesimal, but we can cancel them out: the law of microcancellation claims that if

epsilon a=epsilon b

for all infinitesimals epsilon, then a=b.

Under this method, elementary calculus reduces to simple algebra. As far as I can find out, nobody has used precisely this method in their cheap jerseys teaching. A similar method, using differentials for which all higher powers and products reduce to zero, has been tried, but lacks the rigour of smooth infinitesimal analysis. Nobody has tried to write an elementary textbook from this perspective; the closest to a text is John Bell’s excellent “A Primer of Infinitesimal Analysis”, which is more of a gentle introduction for the interested mathematician, than a textbook for the beginning student.

Anyway, I intend to try it out with some students this year.

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