So basically, in trying to make sense of a certain math aspect of a thermodynamic problem (how to manipulate differentials) I end up reading this http://www.tau.ac.il/~corry/teaching/toldot/download/Bos1974.pdf
After reading it, I now have some preliminary idea on manipulating differentials (at least for those back in the days of Lebniz) but it also raise some interesting thoughts
- Are there mathematical entities which has the following property?
a. Similar to differentials, they are in different orders such that if say an operator $a$ can turn x into an entity such that
$a(x)$ is infinitely smaller than $x$ and
$a^2(x)$ is infinitely infinitely smaller than $x$
b. Multiplication by a FINITE value $b$ can bring the $a$'s to other orders of infinity
e.g. $$b \times a(x)$$ has the same order as $x$ and $$b^2 \times a^3(x)$$ has the same order as $a(x)$.
Phrasing in this way, differentials are a special case of these class of objects when setting $b$ tends to infinity
In short, what is the field of maths that study objects that are sort of like a generalisation of differentials in that
objects can be made infinitely larger or smaller than another by multiplying by a certain finite number $b$?
I also noticed the idea of different scales reminds of the Hausdorff dimensions of fractals http://davis.wpi.edu/~matt/courses/fractals/intro.html Which basically showing how things from different scales are related to each other,
- is the mathematical class of objects I am trying to seek related to fractals?