A paper published this week shows that elementary calculus contains a longstanding flaw that has been present for over a century. "Extending the Algebraic Manipulability of Differentials," by Jonathan Bartlett of The Blyth Institute and Asatur Zh. Khurshudyan of the National Academy of Sciences of Armenia, is published in the peer-reviewed journal Dynamics of Continuous, Discrete & Impulsive Systems, Series A: Mathematical Analysis. The journal has been published for a quarter of a century and many major universities across the United States subscribe to it.
The flaw they discovered is one of notation. Notation can be wrong when it implies untrue things, especially when notation exists that implies the correct things.
As for the technical details, the second derivative of y with respect to x has traditionally had the notation "d2 y/dx 2." While this notation is expressed as a fraction, the problem is that it doesn't actually work as a fraction. The problem is well-known but it has been generally assumed that there is no way to express the second derivative in fraction form. However, it turns out that, with minor modifications to the notation, the terms of the second derivative (and higher derivatives) can indeed be manipulated as an algebraic fraction. The revised notation for the second derivative is "(d 2 y/dx 2) – (dy/dx)(d 2x/dx 2)".
Correcting the notation will also likely open doors in fundamental calculus research. Better notation will improve the ability of mathematicians to do advanced work within calculus. Some of those fruits are already apparent, as the authors have already been using the new notation in published work with fruitful results.
From Mind Matters
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