the notation for differentiation is exemplary of the idiocy of math notations. Among which, the leibniz dy/dx is worst. Best is Euler's D[y]

https://en.wikipedia.org/wiki/Notation_for_differentiation

Bernard Bolzano, the guy who introduced limit δ ε into the ghost of departed souls dy/dx.

http://www-groups.dcs.st-and.ac.uk/history/Biographies/Bolzano.html

#calculus #math

https://en.wikipedia.org/wiki/Notation_for_differentiation

Bernard Bolzano, the guy who introduced limit δ ε into the ghost of departed souls dy/dx.

http://www-groups.dcs.st-and.ac.uk/history/Biographies/Bolzano.html

#calculus #math

View 17 previous comments

- Now, I am not a fan of the infinitesimal approach to calculus since an approach via (locally) uniform estimates is more direct, more elementary and does not rely on cumbersome technicalities like compactness or some other items of mathematical mythology like ultrafilters, it is closer to the original ideas of Newton https://archive.org/stream/methodoffluxions00newt It also allows to avoid the technicalities of limits and continuity and get right to the practical parts, like numerical methods and differential equations.Oct 14, 2017
- +michael livshits i've just read

https://en.m.wikipedia.org/wiki/Hyperreal_number#An_intuitive_approach_to_the_ultrapower_construction

ha, that intuitive approach is just as intuitive as the non-intuitive one in main text.

hyperreal to me is like, who would want to study it?

and as far as i know, not many mathematicians buy it; and its use in teaching is niche.

this

https://en.m.wikipedia.org/wiki/Influence_of_non-standard_analysis

showed how niche it is, even needs a pseudo justification.

but of course, still interesting as math, since pure math is made of niches.

just curious i went on to read

en.m.wikipedia.org - Criticism of non-standard analysis - Wikipedia

haha, gossip and drama.

though, it's interesting to note that non-standard analysis is considered “formal”, as Hilbert's “formalism”. Oh, that i like! and, i realized, that delta epison contained this nasty “there exist”, which i understand now why math has so much trouble as a programing lang.

i never liked those “constructivists” and strange argument about “excluded middle”, but now the constructism seems very attractive.

just my random rants!Oct 15, 2017 - +michael livshits archive.org - The method of fluxions and infinite series : with its application to the geometry of curve-lines

very interesting to read old book. I wish i have time to do so!

thanks for the link.Oct 15, 2017 - +Xah Lee, You may like so-called automatic differentiation if you want to get away from epsilons and deltas, check it out on wikipedia. Also differentiation can be viewed as factoring f(x)-f(a) through x-a in a certain class of functions, for polynomials it is explicit, the most immediate generalization is to assume that the functions involved are Lipschitz, it takes care of all functions with Lipschitz derivative, and the definition of the derivative becomes the inequality

|f(x)-f(a)-f'(a)(x-a)|≤K(x-a)^2 with some constant K, uniform in x and a. See Karcher's lectures and my article at http://www.mathfoolery.com/Article/simpcalc-v1.pdf for details. To prove differentiability in this context is to check this specific inequality. Likewise, to check continuity it's enough to check the inequality |f(x)-f(a)|≤Km(|x-a|), uniform in x and a, where m is a modulus of continuity and K is a constant, any continuous function on a closed bounded set has a modulus of continuity. These notions are widely used in PDEs, harmonic analysis and function theory, where we deal with spaces of functions with certain modulus of continuity (Lipscitz, Holder, etc.) instead of dealing with all the continuous functions at once.

The bottom line is that the elementary calculus can be greatly simplified and made more widely accessible and understandable if we focus on reasonable functions instead of slavishly following the orthodox classical way of defining pointwise continuity and differentiability by epsilon-delta and then using compactness to get any useful or applicable results.

Actually most of just continuous functions are highly pathological, the derivatives of pointwise differentiable functions are even more pathological. As Peter Lax said, no respectful analyst will talk about the space of pointwise differentiable functions. But then again, there is Henstock-Kurzweil integral that can integrate any pointwise derivative and get the fundamental theorem of calculus without any restrictions on the derivative, but it heavily relies on compactness and is hardly appropriate for elementary calculus.Oct 26, 2017 - +Xah Lee, You can learn more from reading Newton than from modern stuff that is dry and sterile in comparison. Make time if you enjoy it, we only live once :-)Oct 15, 2017
- Anyway, I would not worry too much about differentiation, it is a sort of an ideological part of calculus, integration and solving other differential equations is much more important. Look at it this way: differentiation of polynomials is easy, and all the functions we deal with can be approximated by polynomials in this or that metric, differentiation extends on these functions by continuity, the same way as the arithmetics of the rationals extends to the reals :-)Oct 15, 2017

Add a comment...