超实数 (非标准分析)

${\displaystyle \mathbb {N} \subseteq \mathbb {Z} \subseteq \mathbb {Q} \subseteq \mathbb {R} \subseteq \mathbb {C} }$

${\displaystyle 1+1+\cdots +1}$有限個）

${\displaystyle \forall x,y\in \mathbb {R} ,x+y=y+x}$

${\displaystyle \forall x,y\in ^{*}\mathbb {R} ,x+y=y+x}$

傳達原理

${\displaystyle 1<\omega ,\quad 1+1<\omega ,\quad 1+1+1<\omega ,\quad 1+1+1+1<\omega ,\ldots .}$

用於分析中

微分

${\displaystyle df(x,dx):=\operatorname {st} \left({\frac {f(x+dx)-f(x)}{dx}}\right)\ dx.}$

${\displaystyle {\frac {df(x,dx)}{dx}}=\operatorname {st} \left({\frac {f(x+dx)-f(x)}{dx}}\right)}$

 ${\displaystyle {\frac {df(x,dx)}{dx}}}$ ${\displaystyle =\operatorname {st} \left({\frac {f(x+dx)-f(x)}{dx}}\right)}$ ${\displaystyle =\operatorname {st} \left({\frac {x^{2}+2x\cdot dx+(dx)^{2}-x^{2}}{dx}}\right)}$ ${\displaystyle =\operatorname {st} \left({\frac {2x\cdot dx+(dx)^{2}}{dx}}\right)}$ ${\displaystyle =\operatorname {st} \left({\frac {2x\cdot dx}{dx}}+{\frac {(dx)^{2}}{dx}}\right)}$ ${\displaystyle =\operatorname {st} \left(2x+dx\right)}$ ${\displaystyle =2x}$

積分

${\displaystyle \int _{a}^{b}(\varepsilon ,dx):=\operatorname {st} \left(\sum _{n=0}^{N}\varepsilon (a+n\ dx)\right),}$

${\displaystyle \int _{a}^{b}(f\ dx,dx)}$

参考资料

1. ^ Ball, p. 31
2. ^ Fite, Isabelle. Total and Partial Differentials as Algebraically Manipulable Entities. 2022. .
3. ^ Kanovei, Vladimir; Shelah, Saharon, A definable nonstandard model of the reals (PDF), Journal of Symbolic Logic, 2004, 69: 159–164 [2004-10-13], S2CID 15104702, , doi:10.2178/jsl/1080938834, （原始内容 (PDF)存档于2004-08-05）
4. ^ Woodin, W. H.; Dales, H. G., Super-real fields: totally ordered fields with additional structure, Oxford: Clarendon Press, 1996, ISBN 978-0-19-853991-9
5. ^ Robinson, Abraham, Non-standard analysis, Princeton University Press, 1996, ISBN 978-0-691-04490-3. The classic introduction to nonstandard analysis.