Jonathan Bartlett, Asatur Zh. Khurshudyan
2018 · arXiv
Treating differentials as independent algebraic units have a long history of use and abuse. It is generally considered problematic to treat the derivative as a fraction of differentials rather than as a holistic unit acting as a limit, though for practical reasons it is often done for the first derivative. However, using a revised notation for the second and higher derivatives will allow for the ability to treat differentials as independent units for a much larger number of cases.
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There is a really good paper on better notations for differentiation https://arxiv.org/abs/1801.09553 in such a way that algebraic manipulations of the "dx" etc have meaning. The main proposition is to stop calling the second derivative of y along x d^2y/(dx)^2 since this is completely wrong if you want to manipulate things algebraically. In this notation the exponent is denoting function composition and not multiplication.
I've learned calculus a few times in and out of school. I learned it best from a home-schooling focused text book. The author published a paper describing his notation [0]; see section 3 which is very readable. The paper explains: > Most calculus students glaze over the notation for higher derivatives, and few, if any, books bother to give any reasons behind what the notation means. It's important to go back and consider why the notation is what it is, and what the pieces are supposed to represent. > In modern calculus, the derivative is always taken with respect to some variable. However, this is not strictly required, as the differential operation can be used in a…
context-free manner. The processes of taking a differential and solving for a derivative (i.e., some ratio of differentials) can be separated out into logically separate operations. > In such an operation, instead of doing d/dx (taking the derivative with respect to the variable x), one would separate out performing the differential and dividing by dx as separate steps. Originally, in the Leibnizian conception of the differential, one did not even bother solving for derivatives, as they made little sense from the original geometric construction of them. > For a simple example, the differential of x^3 can be found using a basic differential operator such that d(x^3) = 3x^2 dx. The derivative is simply the differential divided by dx. This would yield d(x^3)/dx = 3x^2. Is this significantly different than what is normally taught in schools? Using the notation described here is the first time it has felt like a tool rather than a formality to me, and it's quite different than the way I was thinking while taking calculus in high school. I'm not really sure how unusual this notation is though? [0]: https://arxiv.org/pdf/1801.09553.pdf
A somewhat relevant article, it explores an alternative to Leibniz notation: Extending the Algebraic Manipulability of Differentials (https://arxiv.org/pdf/1801.09553.pdf).
Math traditionally has had some bad notation from a formal point of view, because humans are good at coping with bad notations (or going back and forth between variants), unlike machines and formal systems. Computer science being a (more) formal science (vs math which is overwhelmingly not done formally), it has criticized some traditional math notation which are ad-hoc and not nicely formalizable (and put forward variants that are actually better behaved in terms of mathematical structures). For indices: indices are about referencing elements of finite ordered sets, say of size N. Hence the 'abstract' indexing set for N elements is the ordinal N. The most canonical way…
to represent it is to take the length-N prefix of the natural numbers (eg 0-based indexing, von neumann ordinals), which happen to have all sorts of additional structure (eg mod-N arithmetic). This is also consistent with the offset view (the i-th element is at offset i). The fact that people tend to start ordinal numbers at 1 doesn't change anything that mathematicians working with ordinal numbers take them to start at 0, for the same reason we start naturals at 0. See also: notation for higher derivatives https://arxiv.org/abs/1801.09553; a bit further but in the same vein: notations for free variables in programs as de-bruijn indices (or some variant thereof) (it's further because it's practical for doing proofs, but not for writing concrete terms). There are probably other instances.
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