[HN Gopher] Calculus Made Easy by Silvanus P. Thompson (1910) ___________________________________________________________________ Calculus Made Easy by Silvanus P. Thompson (1910) Author : avinassh Score : 160 points Date : 2023-10-29 12:15 UTC (10 hours ago) (HTM) web link (calculusmadeeasy.org) (TXT) w3m dump (calculusmadeeasy.org) | 2OEH8eoCRo0 wrote: | My biggest mistake as a SWE (now in my 30s) was not learning | higher level mathematics and allowing what knowledge I did | possess to wither on the vine. | fiforpg wrote: | What better hobby to pick for those cold winter evenings, than | to do some integrals! | | Edit: I can recommend this book for a self-guided study | | https://archive.org/details/zeldovich-higher-mathematics-for... | | The author was a Soviet nuclear physicist (who participated in | the creation of the H-bomb), so his main point isn't rigor. It | can be a nice change of perspective from standard American | texts. | srvmshr wrote: | Although Soviet-era books were notoriously terse & difficult | to digest, they had some very aesthetic typesetting. I have | owned a few (Problems in Physics by Irodov, & another by | Krotov) and they all share similar design aesthetics. | hotnfresh wrote: | Late 30s here. I keep feeling like I should learn math better, | but damn, I just never need it. It's much easier to learn stuff | I need. As it is, I've lost everything back to about 8th grade | math because I've never used any of it, so it's just as gone as | all the French I used to know but never found an excuse to use. | | [edit] and I'm dreading my kids getting past elementary school | math because they're gonna be like "why the hell am I spending | months of my life on quadratic equations?" and I'm not gonna | have an answer, because IDK why we did that either. At least I | have answers for calculus, even if they're not much good ("so | you can do physics stuff", "right, but will I ever need to do | physics stuff?", "uhhh... unless you really want to, no.") | mlyle wrote: | Quadratics are useful-- finding dimensions of things in the | plane; relating area and constrained side lengths, etc. They | come up a lot if you want to solve problems. | | And good luck taking on calculus without being super solid in | the mathematical tools you use against quadratics -- | factoring, completing squares, manipulation of binomials, | pairing up like terms, etc. | JadeNB wrote: | The quadratic equation is completing the square while | inexplicably avoiding all the intuition of completing the | square. For example, to solve x^2 + 6x + 5 = 0, you would re- | write it as (x^2 + 6x + 9) - 4 = 0, which is (x + 3)^2 - 4 = | 0 and hence equivalent to (x + 3)^2 = 4, so that x + 3 = +-2 | and hence x = 3 +- 2 is 1 or 5. Euclid thought of things this | way, though his language is, of course, very different to | modern language; see, for example, Proposition 6 of Book II ( | http://aleph0.clarku.edu/~djoyce/elements/bookII/propII6.htm. | ..). | | That's the same answer as the quadratic formula, but makes a | lot more sense to me! Of course I've cooked the numbers so | that you don't wind up with surds in the answer, but those | are just complications in bookkeeping, not in concept. | Qem wrote: | Silvanus' book makes calculus simple by adopting an infinitesimal | approach, like Newton and Leibniz did when they invented | calculus. But that approach was shunned by mathematicians for a | long time, because it was only made rigorous in the 60s. After | Silvanus' book, I also recommend Elementary Calculus: An | Infinitesimal Approach, by professor Jerome Keisler, for those | interested in this alternative pathway to calculus. It can be | freely downloaded at | https://people.math.wisc.edu/~hkeisler/calc.html | | Relevant wikipedia entry: | https://en.m.wikipedia.org/wiki/Nonstandard_analysis | btilly wrote: | I have mixed feelings about this. I've been through nonstandard | analysis. My response was, "We shouldn't need the axiom of | choice to define the derivative." | | But I think that it is extremely important to understand that | the infinitesimal notation really MEANS something. Here is some | Python to demonstrate. # d is a functor. It | takes a function and returns a second function. # The | second function captures the change in f over a small distance. | # The dx/2 business reduces artefacts of it being a finite | distance. def d (f, dx=0.001): return | lambda t: (f(t + dx/2) - f(t - dx/2)) # d2x / dx2 | def second_derivative (f): return lambda t: | d(d(f))(t) / (d(x)(t) * d(x)(t)) def x (t): | return t def cubed (t): return t*t*t | print("The second derivative of cubed at 1 is near", | second_derivative(cubed)(1)) | btilly wrote: | Dang it! I hate re-reading after the time for editing is over | and finding a stupid mistake. | | The Python is correct, but the comment is not. The formula | for the second derivative should be, of course: | # d2y / dx2 | finite_depth wrote: | Most calculus students don't need the full formal power of | rigorous analysis. Calculus, taken alone and with the | elementary properties of the real numbers assumed and a few | elementary properties of infinitesimals (0 <<< | infinitesimal^2 <<< infinitesimal <<< any positive real), can | get you a lot of power for very little formal work. | btilly wrote: | Absolutely true. However this comes at the cost of having | to not think too hard about issues like "what is a | function". | | You generally don't run into trouble with 1, x, 1/x, sin(x) | and the like. But when you push past the analytic | functions, you wind up having to unlearn a lot of ideas so | that you can learn an entirely different foundation. | ezekiel68 wrote: | You're right. But then again, lots of scaffolding gets | discarded when an arch gets constructed also. | ZoltanAK2 wrote: | You write, "we shouldn't need the axiom of choice to define | the derivative." | | The good news is that we don't! | | Only model-theoretic approaches, which justify the | infinitesimal methods by constructing a hyperreal field, | require (a weak form of) the axiom of choice [2]. | | However, there are axioms for nonstandard analysis which are | conservative over the usual choice-free set theory ZF. The | three axioms of Hrbacek and Katz presented in the article | "Infinitesimal analysis without the Axiom of Choice" [1] are | the best recent example: these axioms allow you to do | everything that is done in Keisler's book and more (including | defining the derivative), and you never need to invoke the | axiom of choice to justify them. | | [1] https://arxiv.org/abs/2009.04980 | | [2] Essentially, the set of properties satisfied by a fixed | nonstandard hypernatural gives rise to a non-principal | ultrafilter over the naturals. The axiom of choice is | necessary to prove the existence of non-principal | ultrafilters in (choice-free) set theory, but the existence | of non-principal ultrafilters is not sufficient to prove the | axiom of choice. | btilly wrote: | Yeah, yeah. My comment was my reaction 30 years ago. | | I find it a mildly interesting intellectual exercise that | you can do NSA with weaker axioms than choice. But for all | cases I care about, I can already prove it with NSA without | ANY additional axioms! | | How is this possible? From Shoenfield's absoluteness | theorem, you can prove that all statements that an be made | in the Peano Axioms that can be proven in ZFC, are also | true in ZF. (Note, they must be statable in PA, but not | necessarily provable there.) But PA can encode any | statement we can make about computation. So take any | calculation we can talk about that can be approximated on a | computer. We can rewrite it in PA. We can prove it using | NSA. We then know that it is true in ZF. And we know that | it is true without any additional axioms beyond ZF! | | That which we can actually calculate in any useful way can | all be calculated on a computer. And therefore NSA can | prove anything about Calculus that I care about without | needing any axiom beyond ZF. | | But in the end this is using a mathematical sledgehammer to | drive in a thumb tack. Many approaches to Calculus do not | require assertions about the existence of sets that we | cannot construct, even in principle. Even though I | understand how NSA works, I'd prefer to use any of those. | dvt wrote: | I think I'd also prefer the infinitesimal version of calculus, | but the idea of limits is applicable to many other areas of | math, not just calculus (evaluating infinite series, for | example). So learning limits is probably a better pathway to | higher mathematics. | JadeNB wrote: | > I think I'd also prefer the infinitesimal version of | calculus, but the idea of limits is applicable to many other | areas of math, not just calculus (evaluating infinite series, | for example). So learning limits is probably a better pathway | to higher mathematics. | | I'd say that limits in the sense that you mean (as opposed to | category-theoretic limits) are precisely the domain of | calculus or, if one wishes so to call it (because one is | proving things!), analysis. For example, many US | universities, mine included, regard the computation of | infinite series as part of Calculus II. | pfdietz wrote: | And non-standard analysis is useful elsewhere in math as | well. Here's an example: | | https://discreteanalysisjournal.com/article/87772-a-simple-c. | .. | ykonstant wrote: | More importantly for this audience, the idea of limits lies | at the heart of numerical analysis. Explicating the | quantifiers of the definition of a limit is the first step in | obtaining control over any estimator of your data. This, | among other things, is why I am perpetually baffled at the | people _in this audience_ who say that limits is something | "they will never need". Limits, their algebra (which subsumes | all of high school algebra and inequalities) and their | delicate analysis is what makes a ton of numerical algorithms | work; Higham's classic has it all, and is perfectly clear | about it all. | btilly wrote: | I think that Big O / little o are both more approachable and | provide a richer understanding than limits. See, for example, | https://micromath.wordpress.com/2008/04/14/donald-knuth- | calc... to see Donald Knuth agreeing. | | As an example of the conceptual richness, pick up a Calculus | book and flip to the problem section for L'Hopital's rule. | Without using any special rules at all, attempt to write them | out in o-notation and observe that you generally don't need | L'Hopital's rule to work them out. It is possible to produce | examples that can be calculated by L'Hopital's rule, but not | by simply understanding o-notation. But it isn't easy, and | you're unlikely to find them in textbooks. | | It is probably true that as you go on, limits are more useful | in higher mathematics than o-notation. But o-notation is far | more useful in most subjects that use mathematics. Given how | easy it is to master limits if you know o-notation, why not | teach o-notation first? | analognoise wrote: | I think it would be fun to go through all of these books in a | group, but to also do worksheets with something like XCas, | Maxima, Fricas, etc. | threatofrain wrote: | Consider Analysis 1 by Terry Tao. It walks you from the very | beginning and also uses a conversational style. It's one of the | most pedagogically smooth or gentle books I've ever read. | wallflower wrote: | My calculus teacher tried to talk me out of taking the AP | Calculus AB exam because I wasn't doing that well in his class. | "Calculus the Easy Way" which teaches in the setting of a | mythical fantasy world where applied problems are solved with | calculus unlocked the secret of calculus for me. I ended up | scoring a "4" which was a big triumph for me, as it placed me out | of taking calculus with all the people who did not want to be | taking calculus. | | https://www.thriftbooks.com/w/calculus-the-easy-way-easy-way... | marai2 wrote: | Douglas Downing's Trigonometry Made Easy is also a really great | book for non-maths people. Same approach of situating | Trigonometry in a mythical fantasy land. I remember | Trigonometry as a chore of trying to memorize double-angle | formulas and such, but this book really helped connect trig in | an intuitive way. | | I didn't know he had a calculus book, I'll have to check it out | now. | del_operator wrote: | I picked this book up when I was learning Algebra to see what the | hype was about as a kid. It was dull. I went back to the library | stacks after some weeks and found Calculus and Pizza to be even | easier to swallow. | | Calculus Made Easy is a good book. It made me appreciate even | easier books when you need them and have enthusiasm for learning | a topic. | gramie wrote: | I graduated from engineering school more than 30 years ago, and | learned how to apply calculus without understanding _why_ it | worked. This book helped me to finally understand. | jencuduwvd wrote: | Does anyone know any other good resources for learning calculus | at home? Preferably ones that show how _and why_. I did some in | college but I 've forgotten just about everything, and now I'm | finding myself needing it again. | mindcrime wrote: | > Does anyone know any other good resources for learning | calculus at home? | | Professor Leonard: | | Calculus I - | https://www.youtube.com/playlist?list=PLF797E961509B4EB5 | | Calculus II - | https://www.youtube.com/playlist?list=PLDesaqWTN6EQ2J4vgsN1H... | | Calculus III - | https://www.youtube.com/playlist?list=PLDesaqWTN6ESk16YRmzuJ... | | I consider him one of the best lecturers in math education, at | least for these subjects. And in particular because he is | _very_ detailed in his explanations. He points out that most | students who struggle with Calculus struggle because they | (never mastered | forgot | whatever) their basic _Algebra_. So | he does a very thorough job of explaining all of the subtle | algebraic manipulations that go on as he works through | derivatives, integrals, etc. | | TBH, I think a person who wanted to learn the equivalent of | high-school algebra could just about doing it by watching his | Calc I series... and treat any Calculus they learn as "found | money." But assuming you remember at least a little algebra and | really want to learn Calculus, I think he's one of the best at | teaching it. | | Note that most of his lectures are live lectures to an actual | class, so IMO the best way to approach it is to pretend you're | right there in class. Listen, take notes, and then when he puts | an example on the board pause the video and work through the | example. Just restart the video when you finish the problem or | if you get stuck. | | If you want to work additional problems, go on Amazon or | Alibris or whatever and buy a cheap used copy of one of the | enormous Calculus books, and/or a Shaum's Outlines book on | Calculus, or one of those "1001 solved problems in $SUBJECT" | books... or some combination of all of the above. | | Also as a side-note, speaking for myself, I find that I can | follow his material find at 1.25x speed, so I pretty much | always watch on 1.25x. I could probably manage 1.5x if I really | tried, but the time savings from just doing 1.25x is enough to | make me happy. YMMV, of course. | nerdponx wrote: | [delayed] | xcjs wrote: | I love projects like this, but I really wish instead of | converting text resources to web sites that these projects would | produce epub outputs. It's great for distribution, offline | reading, and scaling to different display sizes, aspect ratios, | and resolutions. | MrBlueIncognito wrote: | I wonder if there will be a time when textbooks will be created | in digital-first format, instead of being mere replicas of what | print books are. It doesn't have to be static text and images | on A4 pages. | marcusverus wrote: | The page links to a pdf version[0], which can easily be | converted to epub using Calibre[0], which is free and open- | source. | | [0] https://www.gutenberg.org/ebooks/33283 | | [1] https://calibre-ebook.com/ | Almondsetat wrote: | Epubs are horrible for technical documents | tarkin2 wrote: | Has anyone been through the calculus courses on khan academy? | What did you think? | MrBlueIncognito wrote: | I went through them a long time ago. It's not the most in-depth | resource, but you will learn enough calculus for when you need | to actually apply it or even just pass tests. Also learning | math on KA is really fun, there's something they just get | right. I'd definitely recommend giving it a try. | | If you don't feel satisfied after going through the courses, | you can always pick up a book afterwards to dig deeper. | WillAdams wrote: | A newer text is: | | _Make: Calculus: Build models to learn, visualize, and explore_ | by Joan Horvath and Rich Cameron | | https://www.goodreads.com/book/show/61739368-make | | It's part of a series with matching books on Geometry and | Trigonometry. | arbuge wrote: | If you don't read anything else in this, read the "Epilogue and | Apologue". | lesona wrote: | Would anyone have any recommendations for books/textbooks of this | style and the comment's Elementary Calculus: An Infinitesimal | Approach, by professor Jerome Keisler on Algebra/PreCalc & Trig? | | I've always wanted to learn math but my teachers could never | explain it to me in a way that clicked and any textbook I've read | couldn't either. These two above really seem to be in my | wheelhouse. ___________________________________________________________________ (page generated 2023-10-29 23:00 UTC)