Physically, I certainly think it has meaning as it can contain. You'll usually get a split of folks on this matter. You'll see it in many derivations and you'll generally notice that you could redo the derivation with \Delta x instead and take limits. You might also find nonstandard analysis interesting in that it constructs infinitesimals (and infinite numbers) rigorously and has some poetic language in it (halos, shadows, etc)īut for a great deal of basic physics treating dx as a 'little change in x' doesn't get you into trouble. However there is nothing to say that the difference dx has to belong to this set.ĭifferential forms is definitely a good way to think about differentials. One advantage of the two point approach is that x 1 and x 2 can belong to the same set of points, X : x n is in X. I also think it is better to reserve the Greek lower case delta (\delta) for incremental (small) chages in something. Alternatively, if x 1 has no neighbourhood points in a certain direction that is the condition for the derivative not to exist. It allows us to use a limiting process/argument as there are always more neighberhood points between x 1 and x 2, no matter how close we get.
![calculus of infinitesimals calculus of infinitesimals](https://image.slidesharecdn.com/thehistoryofcalculus-140706021407-phpapp02/95/the-history-of-calculus-13-638.jpg)
If x 2 is a neighbourhood point for x 1 then this guarantees the conditions for differentiabilty.
![calculus of infinitesimals calculus of infinitesimals](https://infinityisreallybig.com/wp-content/uploads/2019/11/Calculus1Lecture16Thumb.png)
![calculus of infinitesimals calculus of infinitesimals](https://i.ytimg.com/vi/ywSQRggFjVg/maxresdefault.jpg)
The distinction is very subtle but consider this: Then dx = x 2 - x 1 and dy = f(x 2) - f(x 1) The traditional way out of this question is not to regard dx as change in x, small or otherwise, but to regard it as the difference between two points, say x 1 and x 2, where x 2 is in the neighbourhood of x 1