By Nigel J. Cutland, Mauro Di Nasso, David A. Ross
A convention on Nonstandard tools and functions in arithmetic (NS2002) used to be held in Pisa, Italy from June 12-16, 2002. Nonstandard research is without doubt one of the nice achievements of contemporary utilized mathematical good judgment. as well as the real philosophical success of offering a legitimate mathematical foundation for utilizing infinitesimals in research, the technique is now good verified as a device for either study and educating, and has turn into a fruitful box of research in its personal correct. This publication is a suite of peer-reviewed papers solicited from the various contributors of this convention with the purpose of supplying anything extra well timed than a textbook, yet much less ephemeral than a traditional lawsuits. It includes either survey papers and learn articles with unique attention for one, "Nonstandard research at pre-university point: naive value research" within which the writer discusses his event instructing calculus via an infinitesimal process.
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Additional info for Nonstandard Methods and Applications in Mathematics
Sample text
Seulem ent si, elle admet des dérivées partielles continues de tout ordre. 2 Montrer que toute appl ication mul tilinéaire co ntinue est de classe e 00 . (xi , , .. , x; J D ;"< i J .. (xi"< 'l' . . , x ;"<' ). Il en r és ulte, en particulier, que les dérivées parti e lles D~ f (a) sont des appli catio ns dan s F. 7 ) et par co nséquent il n' y a pas lie u de préciser dans quel ord re sont effectuées les déri vati ons. Nous noterons Œ; le nombre d ' indices ij égaux à i E [1, l] ; la dériv ée partie lle Di, .
J=O lorsq ue x , y E 0, llx - ail :S ô et l Y- ail :S ô [raisonner par réc urre nce sur k]. 2. 2) soit vérifiée dès que x , y E K et E > 0, il ex iste ô > 0 tel que l' inégalité llx - Yll :S ô. ~; J. le théorème précédent signifie que rk( h) = o(llhllk). 2. 2 Soit f: [a, b]--+ F une fonction de classe ek, k E N, (k + 1)f ois dérivable sur ]a, b[ telle que 1 1Dk+ 1 f(x) ll : : ; M pour a < x < b. On a alors ~ . 3) + rk(x), J. llrk(x) ll :S M (x - a)k+i (k + l )! 2. Supposons donc le théorème démontré jusqu'à l'ordre k - 1.
On peut également écrire ces relations sous la f orme Dii ... ), Di,f(a) o (p; 1 , • . ;, x ... x E i. i1 (xit ), . . )) E Ek et (Pi, , . . , Pi,. ) l 'application linéaire continue (x 1, . ; ,(x 1 ) , . (x"')) E E; , x . 5). On raisonne ensuite par réc urrence ; la propos itio n étant é ta blie j usqu 'à l'ordre k, é tabli ssons la à l'ord re k + 1. ;, ... , ,, ... ). La fon ctio n x r-+ Di 1 • •• Di. i,, ... , (Ei" ... , Eic ; F). La première application est différentiable au point a par hypothèse et la seconde esl 00 e n ta nt q u ' a pplication li néaire continue.