Are infinitesimals, i.e. $dx = ...$, rigorous and correct notation?

Question

In many fields of physics and engineering, when we want to describe an infinitesimal, for example, the electric field, we could say $dE(x_i,y_i) = e^{jkr}...dx_0dy_0$

Since derivatives are not fractions, does it still make sense, in a mathematically rigorous way to talk about infinitesimals and would this be the correct notation of describing it? What would be the rigorous way of dealing with infinitesimals?

--Edit--

The history of this question went from "duplicate to an irrelevant question" but once reopened was closed for being "missing context" despite proper english, a clear yes or no question with already great answers (which would be impossible if the question was indecipherable). It seems like those with points have been abusing their powers an incorrectly closing questions that they simply don't like.

Answer 1

There are basically two rigorous ways to deal with differentials. One is to treat them as differential forms. This is kind of an algebraic way of doing things, it sets rules for how you can manipulate differentials without trying to describe them as, say, "limits of small differences".

The other way is nonstandard analysis, of which there are at least two incompatible types. One of those is the one from which that name originated, which originally used the idea of a nonstandard model (from model theory) to construct a self-consistent theory containing infinite and infinitesimal "hyperreal" numbers. This originated with Robinson. A different formalism with the same semantics (which is probably easier to understand for non-logicians) was made later by Nelson.

An entirely different semantics arises in smooth infinitesimal analysis. SIA is somewhat alien to "mainstream" mathematicians, because it works in a field which has nonzero nilpotent elements (e.g. $dx \neq 0$ but $(dx)^2=0$). Such a thing is a contradiction in terms in classical logic, so this subject requires a weaker logic called intuitionistic logic in order to function (and even then $dx \neq 0$ is really "it cannot be proven that $dx=0$", a weaker statement).

Honestly, most mathematicians, scientists, and engineers don't need either one. It is better to learn methods for manipulating differentials in formal (i.e. "regarding only form", which is sort of like "non-rigorous") calculations. Optionally you can also learn proofs in standard analysis (which use finite but arbitrarily small numbers). These never explicitly use differentials.

Answer 2

One can deal with infinitesimals properly, although it is far from trivial.

But, more importantly, equalities of the kind $$\tag1 dy=f(x)\,dx$$ are just notation. What I mean is that you never do any algebra nor any other operation with $(1)$: you don't add it, you don't multiply it, you don't take its square root.

So, whether $dy$ and $dx$ in $(1)$ have meaning as objects is irrelevant. One uses an expression as $(1)$ to indicate a substitution in an integral; and, in an integral, $dx$ and $dy$ are notation and not mathematical objects per se.

Answer 3

I join those who would rather call them differentials instead of infinitesimals. While there is a rigorous way of using infinitesimals as they were historically used (say, by Leibniz), I prefer to think in terms of differentials.

The differential of function $f=f(x)=y$ at point $x$, wlog, is the amount of change in (not the function $f$ itself) but in its best linear approximation when the independent variable is given a change $\Delta x$. This is written $df(x)=dy$ and is given by $$df(x)=dy=f'(x)\Delta x.$$ If $f$ is the identity function, then we obtain $df=dx=\Delta x$, so that we usually replace $\Delta x$ with $dx$ in the definition to have $$dy=f'(x)dx.$$ It turns out that this justifies manipulations with differentials as historically practised (and even today by physicists, engineers and differential geometers), so that one has $$dy=\frac{dy}{dx}dx.$$ This is one way to scrupulously practise this use of so-called "infinitesimals".