I’ve always been curious about differentials and how to build a rigorous theory of what the fuck dx, dy, dy/dx, etc. are. For example, if you study Tao’s Analysis and Analysis 2, you will not see anything at all about differentials, and I think that maybe you won’t see the dy/dx notation at all. So, can anyone recommend a textbook about differentials?
Simplifying and Refactoring Introductory Calculus (2018)
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Re: Simplifying and Refactoring Introductory Calculus (2018)
#32That's an exercise under SICP (an infamous Scheme course) it works best with either
Racket with
#lang sicp
chicken-install srfi-203
chicken-install srfi-216
(import scheme)
(import (srfi 203))
(import (srfi 216))
Online, interactive SICP in the browser, you don't need to install anything:
Re: Simplifying and Refactoring Introductory Calculus (2018)
#33Got to the place where he says "As you can see, this is identical to the d/dx() operation except that the result is not divided by dx." What does it mean with his d() operator to "divide by dx"? All of a sudden it seems like he has changed dy/dx from unfortunate notation that sort of looks like a division into something that actually is dividing two meaningful things, dy and dx? And so what the hell are dy and dx?
dx and dy are the smallest change ever in a function. Think about the speed of something
(that's the reason of derivatives, you can't trace the speed of a stopped car in a track, you can't divide by zero, of course, there's no motion).
Re: Simplifying and Refactoring Introductory Calculus (2018)
#34My only experience is as a physics TA and teaching X-ray techs, so take this with a grain of salt. I think the author is trying to address a real problem, but he's not working on the right parts. First, limits are harder than derivatives. Historically, humans figured out the derivative in the late 1600s, but the modern rigorous definition of the limit didn't exist until the 1800s. Slow-walking the definition of the d…
SICP teaches you calculus in a really easy way, you are basically teaching the computer how to derivate and integrate in Lisp which a much easier notation. The functions almost define themselves.
Re: Simplifying and Refactoring Introductory Calculus (2018)
#35Eh. I think that the standard calculus approach is mostly fine, but just needs tweaking. The only major change that I'd like to make is the removal of sequences and all the associated theorems from the introductory calculus. Instead, start with limits of functions and the notion of continuity. It immediately leads to the notion of the derivative. And after that, it's just a lot of building blocks.
> the removal of sequences and all the associated theorems from the introductory calculus. Instead, start with limits of functions and the notion of continuity.
Strongly disagree.
Sequences(discrete) and Convergence are vital to understanding Calculus. Only then the idea of converging to a limit from left or right makes intuitive sense. Pair it with a graphical view of secants converging to a tangent(continuous) and you get the idea of instantaneous change however infinitesimal it might be.
You need both discrete and continuous ideas to build intuition before you introduce limits of functions and continuity.
Some books that i have found useful - https://news.ycombinator.com/item?id=49308281
Re: Simplifying and Refactoring Introductory Calculus (2018)
#36Earlier quoted context omitted.
Depends what you're looking for. Full Frontal Calculus[0], Intuitive Infinitesimal Calculus[1], and Elementary Calculus[2] are all textbooks on the calculus sequence using an infinitesimal pov. The basic approach is to extend the Real numbers to include infinitesimals (greater than zero but smaller than every positive real number) and transfinites (greater than every positive real number), collectively called the Hyp…
If we're going the hyperreal route, I quite like Goldblatt's GTM Lectures on the Hyperreals. You have to augment it with a paper or two if you want to work with other nonstandard objects, but when I was doing my graduate work it was the resource I kept going back to for clarity.
This looks like the most in-depth resource on the topic that I've seen so far; thanks for adding it! One of the reasons that I'm partial to the hyperreals is because it's such a natural thing, in the context of mathematical history, to extend the number system when that system isn't expressive enough to solve the problems we want to solve. The limit-based approach seems clumsy in comparison.
Re: Simplifying and Refactoring Introductory Calculus (2018)
#37Earlier quoted context omitted.
Actually Leibniz invented the modern dy and dx notation and did view the differentials as genuinely nonzero, which is generally speaking a safe assumption. In other words, dy/dx really was a quotient, albeit of really tiny values (at least we assume dx can become arbitrarily small while remaining nonzero). The calculation Leibniz would do looked something like this. First he would consider an equation y = x^2. Then h…
This is effectively how dual numbers work! https://en.wikipedia.org/wiki/Dual_number
Dual numbers are basically big/little O notation. They combine beautifully with Robinson's NSA to give the most 18th century-like approach to deriving integral/derivative formulas that I know. And it's fully rigorous!
Re: Simplifying and Refactoring Introductory Calculus (2018)
#38,, additionally, moving limits to the end of a first-year course allows students to develop intuitions around the derivative first before seeing the formal proof of their validity’’ Waiting a year to get from intuition to theorems is a perfect way to ruin math. Math is not supposed to be easy/simple, but it supposed to be a great way to understand systems based thinking. At the same time there could be more examples…
> Waiting a year to get from intuition to theorems is a perfect way to ruin math. It's interesting that you chose to make that point in a thread about calculus specifically. It had pretty shaky foundations for most of its history, and even today, there's a pretty significant disconnect between the mechanics of epsilon-delta and the meaning we assign to the result. > Math is not supposed to be easy/simple, but it supp…
> even today, there's a pretty significant disconnect between the mechanics of epsilon-delta and the meaning we assign to the result.
I think this is practically fixed by Robinson's NSA when it's combined with big/little O notation:
δy = f'(x) δx + o(δx)
Re: Simplifying and Refactoring Introductory Calculus (2018)
#39Does anyone know how to simplify the actually hard part of calculus: solving integrals? I refer to the million different substitutions and trig/hyperbolic formulas, along with the endless amount of other heuristics. I wonder if there's a way to bypass or simplify most of that.
Re: Simplifying and Refactoring Introductory Calculus (2018)
#40Eh. I think that the standard calculus approach is mostly fine, but just needs tweaking. The only major change that I'd like to make is the removal of sequences and all the associated theorems from the introductory calculus. Instead, start with limits of functions and the notion of continuity. It immediately leads to the notion of the derivative. And after that, it's just a lot of building blocks.
> the removal of sequences and all the associated theorems from the introductory calculus. Instead, start with limits of functions and the notion of continuity. Strongly disagree. Sequences(discrete) and Convergence are vital to understanding Calculus. Only then the idea of converging to a limit from left or right makes intuitive sense. Pair it with a graphical view of secants converging to a tangent(continuous) and…
I don't disagree. Sequences are important, and the bridge between sequences and functions (Bolzano–Weierstrass theorem, mean value theorem, etc.) is crucial.
But they are not immediately needed to understand the limits.
Try to see how far you can get just with the epsilon-delta formulation of limits of functions.