[HN Gopher] Calculus Made Easy by Silvanus P. Thompson (1910)
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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.
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