I'm reading through Concrete Math, and learning about finite calculus. I've never heard of it anywhere else, and a Google search found very few relevant sources. It seems to me an incredibly powerful tool for evaluating sums, essentially a systematization of the use of telescoping sums. Why isn't it more widely known and used? (Related question)
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2I think it has no version of the chain rule. Which is a pretty big disadvantage, to me. – Akiva Weinberger Feb 10 '16 at 03:43

1@AkivaWeinberger false, it does have a very obscure chain rule, but it is significantly trickier to use – Sidharth Ghoshal Feb 10 '16 at 03:49

1Let $D_{h,x}(f) = \frac{f(x+h)f(x)}{h}$ then $D_{h,x}(f(g)) = D_{h,x}(g) D_{h*D_{h,x}(g),g}(f(g))$ – Sidharth Ghoshal Feb 10 '16 at 03:50

1The challenge here is that unless $h=0$ or some function that is known to cancel out with the difference g, then you end up having to recursively work with subdifference equations, in order to say make a change variables (like a u subsitution) etc... that being said it is doable with enough machinery built up – Sidharth Ghoshal Feb 10 '16 at 03:52

2It's thriving as qcalculus or quantumcalculus. See https://mathematicalgarden.wordpress.com/2008/12/15/whatisqcalculus/, https://en.wikipedia.org/wiki/Quantum_calculus, and surveys by Ernst on the topic. Academia is incredibly conservative and much too rigid to introduce anything but thoroughly mainstream topics to students (yawn). See also the finite operator calculus (Rota, Roman, ...) and umbral calculus, as well. – Tom Copeland Feb 10 '16 at 04:27
4 Answers
I'm reading through Concrete Math, and learning about finite calculus. I've never heard of it anywhere else, and a Google search found very few relevant sources.
The name "finite calculus" is unusual. The traditional term is calculus of finite differences or variants such as difference calculus. A search for those terms will be more productive.
It seems to me an incredibly powerful tool for evaluating sums, essentially a systematization of the use of telescoping sums. Why isn't it more widely known and used?
It is widely known and used, most obviously in numerical analysis, but also in many other subjects.
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Thank you for ending my search term scavenger hunt! I was already suspecting it had a different name. – Melvin Roest Apr 25 '19 at 18:07
All formulas of discrete calculus will depend on $\Delta x$ (and $\Delta y,\Delta z$, etc.). After $\Delta x\to 0$, the formulas will inevitably become simpler. Those more cumbersome formulas may be the main turnoff, especially for the people who already know calculus well. On the other hand, the limits themselves seem to be an even bigger turnoff, especially for the people who don't know calculus yet.
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As much as finite differences are a beautiful subject, they are only useful once you need to understand discrete systems. Which in the digital age seems to be every problem these days.
That was NOT the case back in the 1600s when calculus really took off. In those days we didn't have a codified understanding of how forces and basic mechanics in physics worked. Being able to estimate the velocity of a moving object was a challenging problem. At this point in time the most important problems in math were "how to predict trajectories", "how to estimate the correct shape of a bridge", etc... Math, Physics, and Engineering were still highly intertwined and not separate subjects. Computer Science wouldn't even exist properly for another 400 years (which I'd argue is the primary consumer of finite differences).
So obviously we invented continuous calculus calculus first. Throughout the 1700s and 1800s continuous calculus made a big impact on society to the point that by the 1900s it was considered a critical cornerstone of a good math education. And thats how the story goes.
Finite Differences are in the grand scheme of things a cool mathematical curiosity which does have applications in computing, combinatorics, and more advanced functional analysis but (as much as I love this corner of math) they certainly aren't nearly as significant to our civilization as the Derivative.
Hence they aren't well known. And Concrete Math is still a pretty modern book by mathematical standards. Perhaps it will in a century's time become a cornerstone text but we still have a while to go for that to happen. And perhaps by then every schoolchild will also know about finite differences given the discrete and digital direction the world is headed.
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Using finite differences involves a considerable complication of the procedures of the calculus. Such an approach may be constructively more satisfactory but may not be appropriate for a first acquaintance with the subject. The technical complications of the epsilondelta approach are involved enough, and it is how certain things are defined (and it is way easier to do things rigorously with) and furthermore it is the received way of doing things.
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On the other side, finite calculus doesn't have to deal with technical complications as Jordan curve theorem etc... :) – Lehs Feb 18 '16 at 08:31

@Lehs, actually Kanovei has a combinatorial proof of this using infinitesimals. – Mikhail Katz Feb 18 '16 at 14:28
