I have been working on some physics problems, and have realised that I sometimes write a single infinitesimal (or delta) quantity that actually is a product of two independent delta quantities, e.g. $dA=r dr d\phi$ for an area, but equally we have an infinitesimal area even if only one of $r$ or $\phi$ is infinitesimal; as in you might have an arc subtending a non-infinitesimal angle $\phi$ with area $dA=r dr \phi$.

I realise that in physical situations it is generally clear whether the quantity is infinitesimal in one or two quantities, but I was wondering whether mathematicians distinguish between these quantities? It seems to me non-trivial to have an infinitesimal quantity that 'hides' several infintesimals. And could it be possible to break the problem up to have more infinitesimal? I.e. are infuntesimals fundamental in that any infinitesimal quantity is the product of a given number of infinitesimal and not more or less?


It depends on how you formalize the notion of infinitesimal. The standard answer is that infinitesimals are imprecise in nature and don't have a place in "real" mathematical proof, so the answer from that perspective is "no."

Nowadays though, we know that there are mathematically precise ways to use infinitesimals. The most popular way is using the hyperreal numbers, where we assume the existence of numbers that are positive but smaller than all other real numbers. Unfortunately, I have never seen these used in more than one dimension.

The standard way I have seen infinitesimals formalized in multiple dimensions is as differential forms. Here, we identify different infinitesimals with different linear functions from $\mathbb{R}^N$ to the reals. These have a definite dimension, which is the dimension of the domain of the linear function.

In your setting, my guess is that the answer to what you really want to know is "yes," mathematicians do distinguish between the dimensions of different infinitesimals, so that no infinitesimal can have an ambiguous dimension. The basic reason (which can be made precise in the language of differential forms) is that an $n$-dimensional infinitesimal can only be used to integrate over an $n$-dimensional set. I.e $\int_{\mathbb{R^3}}f dx dy$ does not make sense because the differential is only two-dimensional. So infinitesimals must have a definite dimension for the same reason that double and triple integrals are different.

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