Sorry for my longtime absence on this platform. Was just thinking of you all the other day and thought I’d post. Instead of my normal ID content, I thought I’d post another interesting area for discussion – the Leibniz notation for the second derivative. My contention is that it is wrong in a deep way, though there are enough kludges available to make the numbers work in normal situations, which is why we get correct answers while using the wrong notation.
In short, the Leibniz notation is a ratio of differentials. However, if you actually treated the second derivative as a ratio of differentials you would get the wrong answer. The response is normally “well, it’s just notation, it’s not a real ratio”. I could understand that perspective EXCEPT that there is an ACTUAL RATIO of differentials that you CAN treat directly as a ratio. As such, since there exists a ratio of differentials that works, and the Leibniz notation is not it, that means that the standard Leibniz notation is actually an incorrect ratio of differentials.
