Please Register or Log in to view the hidden image! 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 differnetials but it also raise an interesting throught Are there mathematical entities which has the following property? 1. Similar to differentials, they are in different orders such that if say an operator 'a' can turn x into such entity such that a(x) is infinitely smaller than x and a^2(x) is infinitely infinitely smaller than x 2. Multiplication by a finite value b can bring the a s to other orders of infinity e.g. ba(x) has the same order as x and b^2a^3(x) has the same order as a(x) In short, what is the field of maths that study objects that are sort of like a generalisation of differenetials in that objects can be made infinitely larger or smaller than another by multiplying by a certain finite number b? PS those itallic looking things are supposed to be words enclosed by round brackets...
That's the field of non-standard analysis where the Real Numbers of standard analysis are augmented by an unending procession of "transfinite" infinitely large numbers and their multiplicative inverses. The thing is, whenever non-standard analysis gives a Real Number as an answer, standard analysis also gives the same answer, so there's no obvious benefit to bringing in the machinery of non-standard analysis if Real Numbers suffice.
