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Calculus Made Easy

698 points by susam 3y ago

theophrastus 3y ago
Shouldn't this make some initial direct reference to the author: Silvanus P. Thompson[1]? [1] https://en.wikipedia.org/wiki/Calculus_Made_Easy https://en.wikipedia.org/wiki/Calculus_Made_Easy
- Jtsummers 3y ago
  https://calculusmadeeasy.org/ https://calculusmadeeasy.org/ The link isn’t to the front page with his name.
- aaronbrethorst 3y ago
  Ohh, this really is from 1910. And here I just thought the author was being obnoxiously cutesy with their language.
willmeyers 3y ago
Another read on calculus is The Calculus: A Generic Approach by Otto Toeplitz. It goes through a similar process that I enjoyed. https://press.uchicago.edu/ucp/books/book/chicago/C/bo5485725.html https://press.uchicago.edu/ucp/books/book/chicago/C/bo548572...
- maroonblazer 3y ago
  >Generic Genetic
ineedaj0b 3y ago
This taught me Calculus! It’s fantastic and if you need to learn calculus this was better than any teacher I ever had.
yazzku 3y ago
Looks great, thank you.
jsmm 3y ago
Wow! this taught me more than 2 semesters of calculus. Thanks for sharing.
lupire 3y ago
I always conflate this with the book Feynman studied, Calculus for the Practical Man https://archive.org/details/calulusforthepra000526mbp https://archive.org/details/calulusforthepra000526mbp
- xNeil 3y ago
  Feynman started with Calculus Made Easy, and only did Calculus for the Practical Man after he finished it.
mettamage 3y ago
After years of wanting to but never really being disciplined or motivated enough, I have finally been able to tackle math. My math skills stagnated in high school and then I went on studying computer science (avoiding almost all of the math) and came out pretty good in programming but still bad in math. I tried to read books like calculus made easy but it was still too dry and too hard. How I fixed it: 1. I learned to become bored and be okay with it. I sometimes lie on the ground for 15 minutes to an hour and do nothing (and that's different than meditation!). I feel incredibly bored, bored to tears actually. I have noticed when I'm in this state, I'm in a much better position to do math or anything else that I slightly find boring but also really interesting. 2. I went to the root of my problem. I'm Dutch. I failed a course in high school called wiskunde B (math B). It teaches calculus, vectors and trig. I'm currently doing that [1]. It's been a few months, but it's going well. Results and observations: I think I'm almost at the level to do actual college level math. It helps that I studied CS, because I know a ton of math channels on YouTube and they are now immensely helpful. The thing is the course I'm following isn't explaining the theory well. It is amazing in making me practice, so fortunately I just need to go to good YouTube channels such as 3Blue1Brown or Khan Academy for an adequate explanation on theory. Another observation is that it has made me a better programmer. My debugging style has always been "turn on the debugger". Now my debugging style is: think deeply about code execution first. Math makes me comfortable with "code execution" because with math you have no choice! __Looking for a math tutor (email in profile)__ I'm looking for a math tutor. I'd love to ask questions during the week that I don't need an immediate answer to. I think it'll take about 1 hour per week for you. I have noticed I'll need one after the wiskunde B course because I have asked about 100 questions to them in total over the duration of 4 months. Edit: To add. I think for me what made calculus so hard was not having a strong understanding of the fundamentals. The gist of calculus of slope and area is easy. Manipulating the equations perfectly 100% of the time. That used to be really hard and it is getting easier the more I practice. I just happen to watch a YouTube video where a math professor mentions the same thing [2]. [1] The math course I've followed (400 euro - includes exam): https://kdvi.uva.nl/nl/onderwijs/wiskundecursussen/e-wintercursus-wiskunde-b/e-wintercursus-wiskunde-b-copy.html https://kdvi.uva.nl/nl/onderwijs/wiskundecursussen/e-winterc... It's in Dutch, but they have an English version in the language settings. [2] https://www.youtube.com/watch?v=M7febmLhS6E&ab_channel=BigThink https://www.youtube.com/watch?v=M7febmLhS6E&ab_channel=BigTh...
whatisthis9 3y ago
man this is awesome. Thank you! I took my last calculus course in undergrad over 10 years ago and honestly forgot a lot of math including calculus. I am currently back in school doing my masters and struggling at the moment after forgetting this and a lot of other undergrad math topics. Just out of curiosity, does anyone know of similar sites for linear algebra, discrete mathematics, statistics, etc?
- contact9879 3y ago
  Gilbert Strang's "Intro to Linear Algebra"[1] is widely recommended and I enjoyed it as a supplement to Friedberg, Insel, and Spence's much more formal "Linear Algebra"[2]. [1] https://math.mit.edu/~gs/linearalgebra/ila6/indexila6.html https://math.mit.edu/~gs/linearalgebra/ila6/indexila6.html [2] https://www.pearson.com/en-us/subject-catalog/p/linear-algebra/P200000006185/9780134876979 https://www.pearson.com/en-us/subject-catalog/p/linear-algeb...
  - nerdponx 3y ago
    I learned right from FIS without any "warmup" from Strang-level material and it was rough to say the least. I came away with a good final grade and a very poor understanding of most of what I was doing, I was just grinding through proofs with little practical intuition. Good reference book, but very difficult to learn from, especially with a professor who seemed to love abstract math and didn't place high value on geometric intuition.
  - threatofrain 3y ago
    In general, proof-oriented undergrad classes for Linear Algebra would be either the second class they take, or it's a math major going a pathway specifically for math people. Similar to Sheldon Axler's famous book.
- j_bum 3y ago
  +1 for Gilbert Strang’s linear algebra textbookcourse. Im working through his textbook now as I’m diving deeper into ML/DL methods. Here is an additional link to the Spring 2023 course materials that follow along the 6th ed. Of his textbook [0]. [0] https://github.com/mitmath/1806 https://github.com/mitmath/1806
wackget 3y ago
Great if this works for you but honestly I don't find this "easy" at all. The writing style is annoyingly formal and already on page 2 it jumps into "draw the rest of the fucking owl" territory: >> A very simple example will serve as illustration. >> Let us think of xx as a quantity that can grow by a small amount so as to become x+dxx+dx, where dxdx is the small increment added by growth. The square of this is x2+2x⋅dx+(dx)2x2+2x·dx+(dx)2. The second term is not negligible because it is a first-order quantity; while the third term is of the second order of smallness, being a bit of, a bit of x2x2.
- nextaccountic 3y ago
  I actually love the intuition built with geometric arguments using infinitesimals. You need something like smooth infinitesimal analysis [0] to make this rigorous, but it's much better than anything involving limits. [0] https://publish.uwo.ca/~jbell/invitation%20to%20SIA.pdf https://publish.uwo.ca/~jbell/invitation%20to%20SIA.pdf
  - gmane 3y ago
    I'm of the opinion that there's a reason why a subset (myself included) of people who when initially exposed to infinitesimals, and specifically the part where you start just disregarding terms, reject them (it's one of the oldest arguments related to calculus! [0]). Those geometric arguments are essentially less rigorous versions of limits! And that lack of rigor hurts those arguments until you have a rigorous justification for them (that didn't appear until the 1960's if my memory is right). I've come around to infinitesimals, but mostly through exposure to the large hyper-reals. (for context for someone who doesn't know, the idea is to define a number, k which is greater than all real numbers. If you take 1/k, you have a very small number and you can fit an infinite number of 1/k's between 0 and the "next" real number. This concept is what sold me on infinitesimals.) [0] https://en.wikipedia.org/wiki/The_Analyst https://en.wikipedia.org/wiki/The_Analyst
    - akira2501 3y ago
      I wonder if it's working with floating point numbers that made me less uncomfortable when first discovering infinitesimals. The idea that something just falls out of our current representable scope under certain operations seemed fine to me. I've always had a soft spot for infinitesimals and a slight dislike for epsilon-delta limits.
barfard 3y ago
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causality0 3y ago
I liked math up until my pre-calc teacher told me I couldn't go any further until I memorized two dozen different trigonometric identities and was able to immediately identify which to use.
- BenFranklin100 3y ago
  This is actually not a bad way to go. Understanding which is then followed by rote memorization of basic identities and formulas frees up your brain to focus on higher level concepts. Like the tennis amateur who didn’t make pro because they never bothered to drill on the fundamentals, many people get bogged down in higher level math and physics courses because they’ve long lost or never developed a fluency with the basics needed to derive and understand the advanced concepts.
  - raincole 3y ago
    I would have downvoted this comment one year ago. But my perspective changed. "Memorizing to free up brain space" is very real.
    - pictureofabear 3y ago
      This is a good example of why upvotes and downvotes don't always work well when you have class imbalances (more young people than old people or vice versa). We upvote or downvote mostly based on intuition, but wisdom is not always intuitive--we don't always recognize it when we see it.
  - bikenaga 3y ago
    "It is a profoundly erroneous truism, repeated by all copy-books and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them. Operations of thought are like cavalry charges in a battle - they are strictly limited in number, they require fresh horses, and must only be made at decisive moments." - Alfred North Whitehead, "An Introduction to Mathematics", pp. 45–46
canjobear 3y ago
(1910)
math_dandy 3y ago
If this book has exercises at the end of each chapter, I’d assign it to my calculus classes.
DoreenMichele 3y ago
I will recommend the book A tour of the calculus as another potential resource for those interested.
Razengan 3y ago
I skipped school and still trying to "catch up" to the all the stuff I was supposed to learn before gamedev, and I can't recommend these resources enough for actually grokking math stuff: https://www.youtube.com/c/3blue1brown https://www.youtube.com/c/3blue1brown https://www.mathsisfun.com https://www.mathsisfun.com
threatofrain 3y ago
If people here are looking to catchup or refresh their old calculus knowledge, I'd recommend Terry Tao's Analysis 1. It's pedagogically friendly, conversational, but also rigorous.
graycat 3y ago
I gave my 6th grade nephew a 90 second "Calculus Made Easy": First, credibility: Ah, I never took the first year of college calculus -- to make faster progress in college math, got a good book and taught myself. After an oral exam at a black board, was admitted to the second year. Majored in math. Took (so called) advanced calculus from Rudin's Principles .... Took applied advanced calculus from a famous MIT book from an MIT Ph.D. Taught calculus at Indiana University. Studied Fleming's Functions of Several Variables, right, through the inverse and implicit function theorems, Stokes formula, exterior algebra, etc. Published some advanced math, essentially advanced calculus. Once, with some calculus, at FedEx pleased the most serious investors on the Board, had them return their airline tickets back to Texas, stay after all, and saved the company. Okay, the 90 seconds: Consider a car, its speedometer and odometer. Calculus has two parts. For the first part, you read the data from the odometer and reconstruct the speedometer readings. Doing this, you take changes in (increments of, differences in) the odometer readings, say, every second. This is called differentiation. For the second part, you read the data from the speedometer and reconstruct the odometer readings. Doing this, you add the speedometer readings, say, every second. This is called integration. Starting with the odometer readings and differentiating to get the speedometer readings and then integrating the speedometer readings will give back the odometer readings -- this is the "fundamental theorem of calculus". ~90 seconds. For more, instead of the 1 second steps could use 0.1 seconds, .... 0.0001 seconds, etc. With really small steps, making them smaller will make no or nearly no difference. So, the reconstructions will have converged, reached a limit. Reaching this limit is mostly what was novel when Newton, Leibnitz, etc. invented calculus. It is fair to say that the first big application was to Newton's law F = ma where have some object -- baseball, airplane, rocket -- with mass m and are applying to the object force F. Then a is the acceleration of the object. Integrate the acceleration and get the velocity v. Integrate v and get distance d. So, can find where the rocket is after, say, 10 seconds. Other early applications were to planetary motion. There are applications to areas, volumes, classical mechanics. fluid flow, mechanical engineering, electricity and magnetism, quantum mechanics, relativity, electrical and electronic engineering, e.g., Fourier theory. Physics and engineering are big users. And there are applications in economics, e.g., work of Arrow, Hurwicz, and Uzawa on the Kuhn-Tucker conditions. By the early 20th century, calculus was refined, e.g., presented with careful assumptions, definitions, theorems, and proofs, e.g., B. Riemann and, soon, H. Lebesgue. By then there was the idea of the highly irregular Brownian motion and the observation that differentiation wouldn't work there -- Brownian motion was differentiable nowhere! Calculus? A pillar of science, technology, and, thus, civilization.
conwy 3y ago
As someone who's taking a university entrance course in Calculus I find these kind of "calculus made easy" pamphlets irritatingly trite. The hard part isn't the highest level concepts, which are actually fairly easy to grasp and somewhat intuitive. The hard part is all the foundational knowledge required to solve actual math problems with Calculus. The most difficult parts of Calculus (for me at least) are: 1. Having a very thorough grasp of the groundwork / assumed knowledge. Good enough that you can correctly solve an unexpected problem, from completing the square to long division of polynomials to an equation involving differentials. 2. Understanding and correctly applying the notation and graphing techniques, from Leibniz notation to sketching curves. This is why large books and courses exist covering only introductory Calculus, not even beginning to scrape the surface of more advanced math.
- getcrunk 3y ago
  Most of your first point is … algebra? Yes if your algebra is weak you will not be able to cope with solving calc equations. The solution to that problem is not to be found in a calculus made easy. It would be found in algebra made easy.
  - sieabahlpark 3y ago
    [dead]
  - conwy 3y ago
    Yes, it's algebra. Math isn't like programming. In programming you can often solve a problem using a library, framework, language facility, etc. without entirely understanding why it works all the way down to the binary level. In math you can't often solve a more advanced problem such as Calculus problem without understanding the more foundational math such as algebra, fractions, etc. If "information hiding" / layers of abstraction was possible in math, I would have completed my university entrance course months ago, but here I am still struggling. Sure, we could have Algebra made easy and also Trigonometry made easy, Fractions made easy, Functions made easy, etc. etc. I just find it personally irritating that all this foundational knowledge is brushed aside when it's really core to someone's actual competence dealing with actual math problems. Maybe it's just assumed that people went to a good high school or had a private math tutor and already learned the foundations very well, but I think at least that assumption would be coming from a place of privilege. It's similar to telling someone to take a Bootcamp in React and that will be enough for them to succeed as a software engineer. But to solve the kind of problems they are going to face in reality they will eventually have to learn at least some foundational Javascript and maybe a little about algorithms and data structures.
a-dub 3y ago
the hardest part of doing calculus is fluency in arithmetic, algebra and trigonometry. the most important idea in calculus is often glossed over, weakly presented or omitted entirely: the continuity of the reals. i feel that once this is fully understood, most of the ideas in calculus become intuitive.
beckthompson 3y ago
Whenever I see integrals I think "area under the curve" and when I see derivatives I think "slope of a function"!
- kekebo 3y ago
  https://x.com/afterveil/status/1746168116546093291?s=46 https://x.com/afterveil/status/1746168116546093291?s=46
  - beckthompson 3y ago
    Haha that's a funny video!
_virtu 3y ago
I’ve been craving some Physics courses since it’s been about a decade since I was in school. I picked up a Classical Mechanics book to get back into the swing of things and of course it went through some basic linear algebra. It’s been a while since I’ve thought about the dot product of two vectors. You know what blew me away though? Not one textbook I looked in mentioned “why” the dot product is important; that is it’s useful for determining the similarity of two vectors. They all focused on the mechanical details of computing the dot product, but never spelled out the reason it can be useful. I went through a few other resources before I broke down and had a little chat with ChatGPT to discuss the meaning behind it and it makes perfect sense after that. In comparison to when I was in college, things are much slower paced so I can take the time I need to ensure I have a full grasp of a concept before moving forward. I guess all of this is to say that as I’ve continued forward through more concepts I keep finding that the books I’m reading offer a mechanical view instead of a holistic view of the material. This feels like the biggest issue with most math books I’ve read and it makes me wonder where books that offer more semantic meaning of concepts instead of recipes exist.
- adhamsalama 3y ago
  Try 3blue1brown. You'll love it.
- liammclennan 3y ago
  It's not just the books it is the whole method of teaching. I remember learning the steps to calculate an eigenvector without a single comment on why one would ever want to do that. I think it is done so that the educator can claim "this course teaches all of calculus and linear algebra and quantum mechanics". To actually explain things properly would require more modest course goals.
  - aerhardt 3y ago
    I was taught linear algebra and multivariate calculus as a business major. They could hardly justify why they were teaching it in that context - they were weeder courses - but I always wished they had at least tried to give us a hint of applications. Nothing, it was all algebra for the sake of algebra. Atrocious.
noduerme 3y ago
Hand up, I failed precalc twice and never made it past high school. But give me a blank SQL query where I can use window functions on a large dataset, I will tease out the differentiating variables that matter to a business. I think I missed the point of precalc. I never had to solve any real world problems, and the jargon was too impenetrable. Grouping and summarizing sets within sets is like, piece of cake. Just as is formal logic. The way it's taught is completely upside down.
- woopsn 3y ago
  The lead up to calculus is brutal. I did poorly in high school math, got my requirements finished by the grace of BYU online. Never took precalc. Then in college I crammed for the placement exam and got directly into the upper level calculus track - and from then on I loved math. Unfortunately in the traditional track, calculus is the first and only class that "makes sense", as in someone would want to take it. But it takes years of preparation (supposedly). In the meantime like you I took some interest in programming.
- PennRobotics 3y ago
  My brother had a strong dislike for math that wasn't immediately practical. He ended up understanding integrals (at least numerical approximation) after building a small boat and needing to calculate how much air it would hold: obviously less than the containing rectangle, more than the contained rectangle; and then he figured out you can split the boat into sections and get a good guess of each section's volume, move from rectangles to trapezoids, etc.
  - noduerme 3y ago
    I'm a bit like that. I scored surprisingly high on the math portion of the SAT, but couldn't remember which thing was sin and which was cosine. In coding for 2D game platforms I think I've needed to reinvent parts of trigonometry and spatial geometry every time I encounter a new version of the problem. What your brother did makes perfect sense to me; not that I dislike math per se, but his way is probably how I would approach that problem too. I need to be able to visualize it in space. By breaking it up into discreet buckets if necessary. Once I do, I can figure out an iterative logic function that approximately describes it... although it may take me much longer if I'm trying to figure out an actual equation.
vixen99 3y ago
Just a reminder that this text was written in 1910 by https://en.wikipedia.org/wiki/Silvanus_P._Thompson https://en.wikipedia.org/wiki/Silvanus_P._Thompson
coolThingsFirst 3y ago
it's already easy -eastern european
- coolThingsFirst 3y ago
  mericans jealous
- emeril 3y ago
  some americans find it easy too!
cess11 3y ago
When I was in school calculus was the first time we encountered sleight-of-hand tricks as a technique in mathematical reasoning, along with math basically being a symbolic form of fiction it made it quite hard. While I enjoy Silvanus book, I don't think calculus can be made easy without preparing the students to accept the handwaving and trickery involved in the jump from an equation to its 'area' or 'velocity'. Compared to the solemn majesty of euclidic trigonometry or relatively straightforward step-by-step solving of quadratic equations the techniques foundational to calculus are rather devious (as are those that bring in complex numbers). In high school the combination with physics made it harder for some students, they had the impression that learning math amounted to learning about nature, rather than a language for expressing fictions about a view of nature. In turn the approximative nature of the problem solving and calculus didn't fit very well.
alberto_ol 3y ago
Here some of the previous submissions, with lots of comments. https://hn.algolia.com/?q=Calculus+Made+Easy https://hn.algolia.com/?q=Calculus+Made+Easy
melenaboija 3y ago
After 20 years dealing with calculus in school, professional career and spare time there is always joy and inevitably a smile when I see writings like this. The feeling is that the intuition that took me years to build it is a matter of minutes when properly developed. Things like: > Then (dx)^2 will mean a little bit of a little bit of x; is one of the pillars for one of my last struggles that has taken tens of my hours just to have a basic understanding of stochastic calculus and why this actually matters in this specific case. When I see things like this makes me think that humanity is progressing as new generations having access to this information will make them learn faster. Thanks :)
- smatija 3y ago
  The problem isn't that resources like this didn't exist in the past (Calculus Made Easy was written in 1910 after all), problem is that they aren't wellknown - and that is unlikely to change even today.
  - melenaboija 3y ago
    My opinion is that the problem (difference) is that digital format and internet is not the same as paper and libraries.
    - smatija 3y ago
      I would argue that internet makes good resources even harder to find - there is so much of everything, with no curation, that any resource's chance of gaining long term recognition is practically zilch.
      - melenaboija 3y ago
        For myself (not a mathematician) I would find hard to argue that in 1920 would have been easier to have access to this specific information. I was sitting in my sofa when I found this. Also, it should be easy to ask people attending school now if they would prefer books to internet. And to me if the justification is that youth doesn’t know what they are talking about is like denying progress.
anthk 3y ago
Re-edit that with metrical units and I'd recommend that.
spinlock_ 3y ago
The last couple of months, I have been studying the fundamentals of algebra using Professor Leonard's YT Channel[0]. My goal is to fill in the gaps in my knowledge before I refresh my Calculus. It takes a while to go through all this stuff, if you do it right. But man, I have so much more confidence in my skills now than I had before, which to me is in itself rewarding and motivating. I had no idea how big my knowledge gaps in algebra were before I started going through his playlists. My end goal is to be able to follow Andrej Karpathy's "Neural Networks: Zero to Hero"[1] without any big problems So starting basically from "zero" in order to learn the prerequisites before learning what you actually want to learn on your own can feel daunting at times. But I think taking shortcuts will result in frustration. So, here I am taking algebra courses on YT with 38 years. [0] https://youtube.com/@ProfessorLeonard?si=0kiGvmbZv4b9Sgf9 https://youtube.com/@ProfessorLeonard?si=0kiGvmbZv4b9Sgf9 [1] https://youtube.com/playlist?list=PLAqhIrjkxbuWI23v9cThsA9GvCAUhRvKZ&si=jKv7dUgvpQmwFnJg https://youtube.com/playlist?list=PLAqhIrjkxbuWI23v9cThsA9Gv...
raverbashing 3y ago
> The fools who write the textbooks of advanced mathematics — and they are mostly clever fools — seldom take the trouble to show you how easy the easy calculations are. Yes this hits the nail on the head The explanations overall look simple, maybe too long sometimes, but that's a killer prelude
WhitneyLand 3y ago
How much UX design / usability tests do the authors of math textbooks do?
- dagw 3y ago
  When I was at university I had probably 2 or 3 courses where we were beta testing textbooks that the lecturer was in the process of writing. On the downside the book had some rough edges, on the plus side the books where free
  - WhitneyLand 3y ago
    Interesting, did they ask you to look for mistakes, or also validate if things were easy to understand and learning efficient?
russfink 3y ago
WRITTEN IN 1910! Why oh why has every text written after this failed to include such a clear understanding of the concepts. There is no excuse for this, academia! Shame, shame! I should engage thee in a bout of fisticuffs were thou a being!
- theonemind 3y ago
  The 11th edition of the Encyclopedia Britannica is famous for being the last "opionated" edition, also from around 1910. It seems like there was a shift toward trying to be objective in academia around 1910 or so.
HarHarVeryFunny 3y ago
Easy derivatives are indeed easy - easier than that site makes them seem! We can approximate the slope (derivative) of any function y = f(x), at (x,y) as the slope of a straight line from (x,y) to another nearby point (x',y') where y' = f(x'). The slope of this line is (y'-y)/(x'-x). Since we need x' close to x to make this accurate, we use x' = x + dx where dx is small, and we can also represent y' as y' = y + dy. So, then our slope is (y'-y)/(x'-x) = dy/dx = (f(x+dx)-f(x))/dx And that's all there is to it. e.g. For f(x) = x^2 dy/dx = ((x+dx)^2-x^2)/dx = 2x + dx We get the true value of the slope at (x,y) in the limit of dx -> 0, giving us the derivative of f(x) = x^2 as f'(x) = 2x.
- HDThoreaun 3y ago
  Probably why derivatives are the first semester of calculus,. they are indeed the easiest part and a good way to introduce limits.
barfbagginus 3y ago
The title claims it makes calculus easy, and yet there is no category theory. "How can this be possible?!" I hear you ask. Well, the book was written in 1910, 50 years before category theory appeared. But do not worry! There is a book that uses categories to develop ordinary differential and integral calculus! What could be easier than that? I don't know! But if I find it, I'll let you know! Enjoy! https://books.google.com/books?id=gaE5EAAAQBAJ&newbks=1&newbks_redir=0&printsec=frontcover&pg=PP1&dq=info:e9n__jWuz-IJ:scholar.google.com/&hl=en&source=gb_mobile_entity&ovdme=1#v=onepage&q&f=false https://books.google.com/books?id=gaE5EAAAQBAJ&newbks=1&newb...
gen220 3y ago
I "knew" enough calculus to get good grades in math courses through high school and college as an engineer. But I didn't really "know" calculus until I read a book not too dissimilar from this one, "A Course of Pure Mathematics" from 1908 (!), which constructs calculus up from number theory (I think the fundamental theorem of calculus comes halfway through the book?). From that point it's impossible to forget. As for why it's not taught this way today, I'd blame our testing regime and large classrooms, which incentivizes temporary memorization of key formulae and knowing where to mindlessly apply them, over deep/lasting/semantic understanding. I'd also blame the fact that we have a different maths teacher every year, so students come in with heterogenous understandings of the pre-requisites for the next year's material, so the first part of a section is spent reviewing + consolidating. It takes maybe 10-20% more time to get a rich understanding of the subject that lasts a lifetime. But we value compression and instantly-measurable results more than actual learning. :/ Encountering material like this makes you really happy, but it's kind of bittersweet because it makes you realize that the modal state of modern pedagogy is pretty abysmal.
zackmorris 3y ago
This is great! For anyone reading this, I highly recommend attending the best college you can. The reason why is: the caliber of professor. I was lucky to receive (IMHO) a world-class engineering education at a university in Illinois. It would have been even better at an Ivy League school. But I have heard various accounts from students at other state schools and community colleges, and there's a distinct lack of nuance in their coursework. For example, my statistics and probability professor explained everything by counting apples and bananas on his fingers. It sounds hilarious, but that's all probability is - counting the number of certain outcome(s) out of all possible outcomes. He broke down even the most complex problems into a series of basic counting operations. A less dedicated professor might have just regurgitated the course material verbatim. Where this really matters is, I don't believe in complexity. Learning to learn is about breaking down problems into smaller pieces that are at most the maximum size that can fit in a human brain, and then working each problem until reaching the final answer. Once you can do that, you can tackle any challenge, like running hurdles, with full confidence that you will succeed in time and with enough resources. Which is why I'm so hard on fields which I don't feel are mature yet, due to their byzantine terminology and barriers to entry. Some low hanging fruit might be: AI's obsessions with tensors, CUDA and the entire concept of a GPU (instead of just using symmetric multiprocessing), quantum physics, organic chemistry, biology, medicine, genetics, law, accounting (the CPA test for example), imperative programming (vs functional programming and fundamentals like Lisp and spreadsheets) and design/repair of internal combustion engines. Now, paying for college is a whole different story. It stopped being affordable in the late 2000s, by design to enrich private interests and create a subjugated class that needs a job to pay off student loans, much like the scam of attaching health insurance to employment. The only feasible answer there is systems thinking. Which looks like working the problem backwards by enumerating the core problems and then deriving a general solution which allows everyone to attend for free, funded by taxes paid by automated robots and billionaires. I mean, that's how Star Trek works, why settle for less. Getting rid of billionaires is just a bonus.
bikenaga 3y ago
I got "Calculus Made Easy" back in high school, and it helped when I first took calc in 10th grade, and then BC Calc in 12th grade. It gave me some confidence to have seen the basic ideas presented in a gentle way. Actually, once you get past the old-fashioned language and layout many math books from the early 1900's are pretty good. They don't have a lot of the abstraction that came later, some of which makes you feel like you're trying to do surgery while wearing boxing gloves. Anyway, there are other calc books which have similar intents, though different styles. Here are two: Daniel Kleppner and Norman Ramsey, "Quick Calculus" - We used this for math background when I was taking AP Physics in high school. The physics textbook was Resnick and Halliday, which uses calc throughout. It is not particularly intuitive, but the "programmed learning" approach makes it very easy to pick up enough basic computational ability to do applications. (The phrase "programmed learning" is so old that maybe I should clarify that it has nothing to do with computers.) H. M. Schey, "Div, Grad, Curl, and All That" - This might be helpful for people taking vector calculus, or using it in an electromagnetism course in physics. In fact, a lot of his examples are from E-M. Very good at providing intuition without too many technicalities. I think the best uses for books like these are: (a) You want to get the "big ideas" without worrying too much about technical facility (maybe because you won't ever need it - e.g. you're just learning for your own interest); (b) To provide a fair amount of the math you'll need in (say) a physics course; (c) As prep before taking a standard calc course, or maybe as a refresher. Books like Thompson, Kleppner and Ramsey, or Schey will help with the big picture and to boost confidence but it's best to read them before taking the "real" calc course - since once the "real" course is running you'll be too busy with the work in the course to read something else on the side. But do whatever works for you. Calculus really is wonderful - have fun!
krick 3y ago
I appreciate the effort, but after a couple of pages I can already tell, that it wouldn't be something I wanted if it was the first time I'm learning calculus. And even for a refresher it isn't exactly perfect. IMO, the author is right about what's wrong with the most "proper" books, but overcompensates, and thus makes it unnecessarily complicated to understand, maybe even more so, than some "proper" textbooks. It is too wordy, too informal and pretty hard to read and follow. I don't need any Dean Swift poetry and references to "Queen Elizabeth's days", I just want to know what is calculus, why do we need it, and how do I actually do it, goddammit… That's if we follow the seemingly easier "engineering" approach. And we need it to be a bit more formal still if we follow even slightly more mathematical approach, which I think is actually necessary if you want even a bit of real understanding, because math is this thing where it's very easy to fool yourself into thinking you understand something, when you actually don't, and then get all flabbergasted and helpless against paradoxical "fake proofs" or being asked to prove anything yourself. And to tell valid derivation from invalid, yes, you do need some formal definitions. And in fact, the (formal) basics really aren't difficult to understand at all. Everyone can readily agree that x² grows faster than x, and being introduced into concept of limits to see why (dx)² can be considered negligible compared to dx. You don't need to be comparing weeks to minutes for that. If anything, the latter only throws you off. And I feel like it takes way less patience to read a handful of formal definitions than pages of these "old British"-stylized ramblings. Edit: oh, it isn't "stylized". Still, doesn't change the main point, that there are far better more recent resources to learn calculus.

698 points by susam 3y ago

Shouldn't this make some initial direct reference to the author: Silvanus P. Thompson[1]? [1] https://en.wikipedia.org/wiki/Calculus_Made_Easy https://en.wikipedia.org/wiki/Calculus_Made_Easy

Jtsummers 3y ago
https://calculusmadeeasy.org/ https://calculusmadeeasy.org/ The link isn’t to the front page with his name.
aaronbrethorst 3y ago
Ohh, this really is from 1910. And here I just thought the author was being obnoxiously cutesy with their language.

willmeyers 3y ago

Another read on calculus is The Calculus: A Generic Approach by Otto Toeplitz. It goes through a similar process that I enjoyed. https://press.uchicago.edu/ucp/books/book/chicago/C/bo5485725.html https://press.uchicago.edu/ucp/books/book/chicago/C/bo548572...

maroonblazer 3y ago
>Generic Genetic

ineedaj0b 3y ago

This taught me Calculus! It’s fantastic and if you need to learn calculus this was better than any teacher I ever had.

yazzku 3y ago

Looks great, thank you.

jsmm 3y ago

Wow! this taught me more than 2 semesters of calculus. Thanks for sharing.

lupire 3y ago

I always conflate this with the book Feynman studied, Calculus for the Practical Man https://archive.org/details/calulusforthepra000526mbp https://archive.org/details/calulusforthepra000526mbp

xNeil 3y ago
Feynman started with Calculus Made Easy, and only did Calculus for the Practical Man after he finished it.

mettamage 3y ago

After years of wanting to but never really being disciplined or motivated enough, I have finally been able to tackle math. My math skills stagnated in high school and then I went on studying computer science (avoiding almost all of the math) and came out pretty good in programming but still bad in math. I tried to read books like calculus made easy but it was still too dry and too hard. How I fixed it: 1. I learned to become bored and be okay with it. I sometimes lie on the ground for 15 minutes to an hour and do nothing (and that's different than meditation!). I feel incredibly bored, bored to tears actually. I have noticed when I'm in this state, I'm in a much better position to do math or anything else that I slightly find boring but also really interesting. 2. I went to the root of my problem. I'm Dutch. I failed a course in high school called wiskunde B (math B). It teaches calculus, vectors and trig. I'm currently doing that [1]. It's been a few months, but it's going well. Results and observations: I think I'm almost at the level to do actual college level math. It helps that I studied CS, because I know a ton of math channels on YouTube and they are now immensely helpful. The thing is the course I'm following isn't explaining the theory well. It is amazing in making me practice, so fortunately I just need to go to good YouTube channels such as 3Blue1Brown or Khan Academy for an adequate explanation on theory. Another observation is that it has made me a better programmer. My debugging style has always been "turn on the debugger". Now my debugging style is: think deeply about code execution first. Math makes me comfortable with "code execution" because with math you have no choice! __Looking for a math tutor (email in profile)__ I'm looking for a math tutor. I'd love to ask questions during the week that I don't need an immediate answer to. I think it'll take about 1 hour per week for you. I have noticed I'll need one after the wiskunde B course because I have asked about 100 questions to them in total over the duration of 4 months. Edit: To add. I think for me what made calculus so hard was not having a strong understanding of the fundamentals. The gist of calculus of slope and area is easy. Manipulating the equations perfectly 100% of the time. That used to be really hard and it is getting easier the more I practice. I just happen to watch a YouTube video where a math professor mentions the same thing [2]. [1] The math course I've followed (400 euro - includes exam): https://kdvi.uva.nl/nl/onderwijs/wiskundecursussen/e-wintercursus-wiskunde-b/e-wintercursus-wiskunde-b-copy.html https://kdvi.uva.nl/nl/onderwijs/wiskundecursussen/e-winterc... It's in Dutch, but they have an English version in the language settings. [2] https://www.youtube.com/watch?v=M7febmLhS6E&ab_channel=BigThink https://www.youtube.com/watch?v=M7febmLhS6E&ab_channel=BigTh...

whatisthis9 3y ago

man this is awesome. Thank you! I took my last calculus course in undergrad over 10 years ago and honestly forgot a lot of math including calculus. I am currently back in school doing my masters and struggling at the moment after forgetting this and a lot of other undergrad math topics. Just out of curiosity, does anyone know of similar sites for linear algebra, discrete mathematics, statistics, etc?

contact9879 3y ago
Gilbert Strang's "Intro to Linear Algebra"[1] is widely recommended and I enjoyed it as a supplement to Friedberg, Insel, and Spence's much more formal "Linear Algebra"[2]. [1] https://math.mit.edu/~gs/linearalgebra/ila6/indexila6.html https://math.mit.edu/~gs/linearalgebra/ila6/indexila6.html [2] https://www.pearson.com/en-us/subject-catalog/p/linear-algebra/P200000006185/9780134876979 https://www.pearson.com/en-us/subject-catalog/p/linear-algeb...
- nerdponx 3y ago
  I learned right from FIS without any "warmup" from Strang-level material and it was rough to say the least. I came away with a good final grade and a very poor understanding of most of what I was doing, I was just grinding through proofs with little practical intuition. Good reference book, but very difficult to learn from, especially with a professor who seemed to love abstract math and didn't place high value on geometric intuition.
- threatofrain 3y ago
  In general, proof-oriented undergrad classes for Linear Algebra would be either the second class they take, or it's a math major going a pathway specifically for math people. Similar to Sheldon Axler's famous book.
j_bum 3y ago
+1 for Gilbert Strang’s linear algebra textbookcourse. Im working through his textbook now as I’m diving deeper into ML/DL methods. Here is an additional link to the Spring 2023 course materials that follow along the 6th ed. Of his textbook [0]. [0] https://github.com/mitmath/1806 https://github.com/mitmath/1806

wackget 3y ago

Great if this works for you but honestly I don't find this "easy" at all. The writing style is annoyingly formal and already on page 2 it jumps into "draw the rest of the fucking owl" territory: >> A very simple example will serve as illustration. >> Let us think of xx as a quantity that can grow by a small amount so as to become x+dxx+dx, where dxdx is the small increment added by growth. The square of this is x2+2x⋅dx+(dx)2x2+2x·dx+(dx)2. The second term is not negligible because it is a first-order quantity; while the third term is of the second order of smallness, being a bit of, a bit of x2x2.

nextaccountic 3y ago
I actually love the intuition built with geometric arguments using infinitesimals. You need something like smooth infinitesimal analysis [0] to make this rigorous, but it's much better than anything involving limits. [0] https://publish.uwo.ca/~jbell/invitation%20to%20SIA.pdf https://publish.uwo.ca/~jbell/invitation%20to%20SIA.pdf
- gmane 3y ago
  I'm of the opinion that there's a reason why a subset (myself included) of people who when initially exposed to infinitesimals, and specifically the part where you start just disregarding terms, reject them (it's one of the oldest arguments related to calculus! [0]). Those geometric arguments are essentially less rigorous versions of limits! And that lack of rigor hurts those arguments until you have a rigorous justification for them (that didn't appear until the 1960's if my memory is right). I've come around to infinitesimals, but mostly through exposure to the large hyper-reals. (for context for someone who doesn't know, the idea is to define a number, k which is greater than all real numbers. If you take 1/k, you have a very small number and you can fit an infinite number of 1/k's between 0 and the "next" real number. This concept is what sold me on infinitesimals.) [0] https://en.wikipedia.org/wiki/The_Analyst https://en.wikipedia.org/wiki/The_Analyst
  - akira2501 3y ago
    I wonder if it's working with floating point numbers that made me less uncomfortable when first discovering infinitesimals. The idea that something just falls out of our current representable scope under certain operations seemed fine to me. I've always had a soft spot for infinitesimals and a slight dislike for epsilon-delta limits.

barfard 3y ago

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causality0 3y ago

I liked math up until my pre-calc teacher told me I couldn't go any further until I memorized two dozen different trigonometric identities and was able to immediately identify which to use.

BenFranklin100 3y ago
This is actually not a bad way to go. Understanding which is then followed by rote memorization of basic identities and formulas frees up your brain to focus on higher level concepts. Like the tennis amateur who didn’t make pro because they never bothered to drill on the fundamentals, many people get bogged down in higher level math and physics courses because they’ve long lost or never developed a fluency with the basics needed to derive and understand the advanced concepts.
- raincole 3y ago
  I would have downvoted this comment one year ago. But my perspective changed. "Memorizing to free up brain space" is very real.
  - pictureofabear 3y ago
    This is a good example of why upvotes and downvotes don't always work well when you have class imbalances (more young people than old people or vice versa). We upvote or downvote mostly based on intuition, but wisdom is not always intuitive--we don't always recognize it when we see it.
- bikenaga 3y ago
  "It is a profoundly erroneous truism, repeated by all copy-books and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them. Operations of thought are like cavalry charges in a battle - they are strictly limited in number, they require fresh horses, and must only be made at decisive moments." - Alfred North Whitehead, "An Introduction to Mathematics", pp. 45–46

canjobear 3y ago

(1910)

math_dandy 3y ago

If this book has exercises at the end of each chapter, I’d assign it to my calculus classes.

DoreenMichele 3y ago

I will recommend the book A tour of the calculus as another potential resource for those interested.

Razengan 3y ago

I skipped school and still trying to "catch up" to the all the stuff I was supposed to learn before gamedev, and I can't recommend these resources enough for actually grokking math stuff: https://www.youtube.com/c/3blue1brown https://www.youtube.com/c/3blue1brown https://www.mathsisfun.com https://www.mathsisfun.com

threatofrain 3y ago

If people here are looking to catchup or refresh their old calculus knowledge, I'd recommend Terry Tao's Analysis 1. It's pedagogically friendly, conversational, but also rigorous.

graycat 3y ago

I gave my 6th grade nephew a 90 second "Calculus Made Easy": First, credibility: Ah, I never took the first year of college calculus -- to make faster progress in college math, got a good book and taught myself. After an oral exam at a black board, was admitted to the second year. Majored in math. Took (so called) advanced calculus from Rudin's Principles .... Took applied advanced calculus from a famous MIT book from an MIT Ph.D. Taught calculus at Indiana University. Studied Fleming's Functions of Several Variables, right, through the inverse and implicit function theorems, Stokes formula, exterior algebra, etc. Published some advanced math, essentially advanced calculus. Once, with some calculus, at FedEx pleased the most serious investors on the Board, had them return their airline tickets back to Texas, stay after all, and saved the company. Okay, the 90 seconds: Consider a car, its speedometer and odometer. Calculus has two parts. For the first part, you read the data from the odometer and reconstruct the speedometer readings. Doing this, you take changes in (increments of, differences in) the odometer readings, say, every second. This is called differentiation. For the second part, you read the data from the speedometer and reconstruct the odometer readings. Doing this, you add the speedometer readings, say, every second. This is called integration. Starting with the odometer readings and differentiating to get the speedometer readings and then integrating the speedometer readings will give back the odometer readings -- this is the "fundamental theorem of calculus". ~90 seconds. For more, instead of the 1 second steps could use 0.1 seconds, .... 0.0001 seconds, etc. With really small steps, making them smaller will make no or nearly no difference. So, the reconstructions will have converged, reached a limit. Reaching this limit is mostly what was novel when Newton, Leibnitz, etc. invented calculus. It is fair to say that the first big application was to Newton's law F = ma where have some object -- baseball, airplane, rocket -- with mass m and are applying to the object force F. Then a is the acceleration of the object. Integrate the acceleration and get the velocity v. Integrate v and get distance d. So, can find where the rocket is after, say, 10 seconds. Other early applications were to planetary motion. There are applications to areas, volumes, classical mechanics. fluid flow, mechanical engineering, electricity and magnetism, quantum mechanics, relativity, electrical and electronic engineering, e.g., Fourier theory. Physics and engineering are big users. And there are applications in economics, e.g., work of Arrow, Hurwicz, and Uzawa on the Kuhn-Tucker conditions. By the early 20th century, calculus was refined, e.g., presented with careful assumptions, definitions, theorems, and proofs, e.g., B. Riemann and, soon, H. Lebesgue. By then there was the idea of the highly irregular Brownian motion and the observation that differentiation wouldn't work there -- Brownian motion was differentiable nowhere! Calculus? A pillar of science, technology, and, thus, civilization.

conwy 3y ago

As someone who's taking a university entrance course in Calculus I find these kind of "calculus made easy" pamphlets irritatingly trite. The hard part isn't the highest level concepts, which are actually fairly easy to grasp and somewhat intuitive. The hard part is all the foundational knowledge required to solve actual math problems with Calculus. The most difficult parts of Calculus (for me at least) are: 1. Having a very thorough grasp of the groundwork / assumed knowledge. Good enough that you can correctly solve an unexpected problem, from completing the square to long division of polynomials to an equation involving differentials. 2. Understanding and correctly applying the notation and graphing techniques, from Leibniz notation to sketching curves. This is why large books and courses exist covering only introductory Calculus, not even beginning to scrape the surface of more advanced math.

getcrunk 3y ago
Most of your first point is … algebra? Yes if your algebra is weak you will not be able to cope with solving calc equations. The solution to that problem is not to be found in a calculus made easy. It would be found in algebra made easy.
- sieabahlpark 3y ago
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- conwy 3y ago
  Yes, it's algebra. Math isn't like programming. In programming you can often solve a problem using a library, framework, language facility, etc. without entirely understanding why it works all the way down to the binary level. In math you can't often solve a more advanced problem such as Calculus problem without understanding the more foundational math such as algebra, fractions, etc. If "information hiding" / layers of abstraction was possible in math, I would have completed my university entrance course months ago, but here I am still struggling. Sure, we could have Algebra made easy and also Trigonometry made easy, Fractions made easy, Functions made easy, etc. etc. I just find it personally irritating that all this foundational knowledge is brushed aside when it's really core to someone's actual competence dealing with actual math problems. Maybe it's just assumed that people went to a good high school or had a private math tutor and already learned the foundations very well, but I think at least that assumption would be coming from a place of privilege. It's similar to telling someone to take a Bootcamp in React and that will be enough for them to succeed as a software engineer. But to solve the kind of problems they are going to face in reality they will eventually have to learn at least some foundational Javascript and maybe a little about algorithms and data structures.

a-dub 3y ago

the hardest part of doing calculus is fluency in arithmetic, algebra and trigonometry. the most important idea in calculus is often glossed over, weakly presented or omitted entirely: the continuity of the reals. i feel that once this is fully understood, most of the ideas in calculus become intuitive.

beckthompson 3y ago

Whenever I see integrals I think "area under the curve" and when I see derivatives I think "slope of a function"!

kekebo 3y ago
https://x.com/afterveil/status/1746168116546093291?s=46 https://x.com/afterveil/status/1746168116546093291?s=46
- beckthompson 3y ago
  Haha that's a funny video!

_virtu 3y ago

I’ve been craving some Physics courses since it’s been about a decade since I was in school. I picked up a Classical Mechanics book to get back into the swing of things and of course it went through some basic linear algebra. It’s been a while since I’ve thought about the dot product of two vectors. You know what blew me away though? Not one textbook I looked in mentioned “why” the dot product is important; that is it’s useful for determining the similarity of two vectors. They all focused on the mechanical details of computing the dot product, but never spelled out the reason it can be useful. I went through a few other resources before I broke down and had a little chat with ChatGPT to discuss the meaning behind it and it makes perfect sense after that. In comparison to when I was in college, things are much slower paced so I can take the time I need to ensure I have a full grasp of a concept before moving forward. I guess all of this is to say that as I’ve continued forward through more concepts I keep finding that the books I’m reading offer a mechanical view instead of a holistic view of the material. This feels like the biggest issue with most math books I’ve read and it makes me wonder where books that offer more semantic meaning of concepts instead of recipes exist.

adhamsalama 3y ago
Try 3blue1brown. You'll love it.
liammclennan 3y ago
It's not just the books it is the whole method of teaching. I remember learning the steps to calculate an eigenvector without a single comment on why one would ever want to do that. I think it is done so that the educator can claim "this course teaches all of calculus and linear algebra and quantum mechanics". To actually explain things properly would require more modest course goals.
- aerhardt 3y ago
  I was taught linear algebra and multivariate calculus as a business major. They could hardly justify why they were teaching it in that context - they were weeder courses - but I always wished they had at least tried to give us a hint of applications. Nothing, it was all algebra for the sake of algebra. Atrocious.

noduerme 3y ago

Hand up, I failed precalc twice and never made it past high school. But give me a blank SQL query where I can use window functions on a large dataset, I will tease out the differentiating variables that matter to a business. I think I missed the point of precalc. I never had to solve any real world problems, and the jargon was too impenetrable. Grouping and summarizing sets within sets is like, piece of cake. Just as is formal logic. The way it's taught is completely upside down.

woopsn 3y ago
The lead up to calculus is brutal. I did poorly in high school math, got my requirements finished by the grace of BYU online. Never took precalc. Then in college I crammed for the placement exam and got directly into the upper level calculus track - and from then on I loved math. Unfortunately in the traditional track, calculus is the first and only class that "makes sense", as in someone would want to take it. But it takes years of preparation (supposedly). In the meantime like you I took some interest in programming.
PennRobotics 3y ago
My brother had a strong dislike for math that wasn't immediately practical. He ended up understanding integrals (at least numerical approximation) after building a small boat and needing to calculate how much air it would hold: obviously less than the containing rectangle, more than the contained rectangle; and then he figured out you can split the boat into sections and get a good guess of each section's volume, move from rectangles to trapezoids, etc.
- noduerme 3y ago
  I'm a bit like that. I scored surprisingly high on the math portion of the SAT, but couldn't remember which thing was sin and which was cosine. In coding for 2D game platforms I think I've needed to reinvent parts of trigonometry and spatial geometry every time I encounter a new version of the problem. What your brother did makes perfect sense to me; not that I dislike math per se, but his way is probably how I would approach that problem too. I need to be able to visualize it in space. By breaking it up into discreet buckets if necessary. Once I do, I can figure out an iterative logic function that approximately describes it... although it may take me much longer if I'm trying to figure out an actual equation.

vixen99 3y ago

Just a reminder that this text was written in 1910 by https://en.wikipedia.org/wiki/Silvanus_P._Thompson https://en.wikipedia.org/wiki/Silvanus_P._Thompson

coolThingsFirst 3y ago

it's already easy -eastern european

coolThingsFirst 3y ago
mericans jealous
emeril 3y ago
some americans find it easy too!

cess11 3y ago

When I was in school calculus was the first time we encountered sleight-of-hand tricks as a technique in mathematical reasoning, along with math basically being a symbolic form of fiction it made it quite hard. While I enjoy Silvanus book, I don't think calculus can be made easy without preparing the students to accept the handwaving and trickery involved in the jump from an equation to its 'area' or 'velocity'. Compared to the solemn majesty of euclidic trigonometry or relatively straightforward step-by-step solving of quadratic equations the techniques foundational to calculus are rather devious (as are those that bring in complex numbers). In high school the combination with physics made it harder for some students, they had the impression that learning math amounted to learning about nature, rather than a language for expressing fictions about a view of nature. In turn the approximative nature of the problem solving and calculus didn't fit very well.

alberto_ol 3y ago

Here some of the previous submissions, with lots of comments. https://hn.algolia.com/?q=Calculus+Made+Easy https://hn.algolia.com/?q=Calculus+Made+Easy

melenaboija 3y ago

After 20 years dealing with calculus in school, professional career and spare time there is always joy and inevitably a smile when I see writings like this. The feeling is that the intuition that took me years to build it is a matter of minutes when properly developed. Things like: > Then (dx)^2 will mean a little bit of a little bit of x; is one of the pillars for one of my last struggles that has taken tens of my hours just to have a basic understanding of stochastic calculus and why this actually matters in this specific case. When I see things like this makes me think that humanity is progressing as new generations having access to this information will make them learn faster. Thanks :)

smatija 3y ago
The problem isn't that resources like this didn't exist in the past (Calculus Made Easy was written in 1910 after all), problem is that they aren't wellknown - and that is unlikely to change even today.
- melenaboija 3y ago
  My opinion is that the problem (difference) is that digital format and internet is not the same as paper and libraries.
  - smatija 3y ago
    I would argue that internet makes good resources even harder to find - there is so much of everything, with no curation, that any resource's chance of gaining long term recognition is practically zilch.
    - melenaboija 3y ago
      For myself (not a mathematician) I would find hard to argue that in 1920 would have been easier to have access to this specific information. I was sitting in my sofa when I found this. Also, it should be easy to ask people attending school now if they would prefer books to internet. And to me if the justification is that youth doesn’t know what they are talking about is like denying progress.

anthk 3y ago

Re-edit that with metrical units and I'd recommend that.

spinlock_ 3y ago

The last couple of months, I have been studying the fundamentals of algebra using Professor Leonard's YT Channel[0]. My goal is to fill in the gaps in my knowledge before I refresh my Calculus. It takes a while to go through all this stuff, if you do it right. But man, I have so much more confidence in my skills now than I had before, which to me is in itself rewarding and motivating. I had no idea how big my knowledge gaps in algebra were before I started going through his playlists. My end goal is to be able to follow Andrej Karpathy's "Neural Networks: Zero to Hero"[1] without any big problems So starting basically from "zero" in order to learn the prerequisites before learning what you actually want to learn on your own can feel daunting at times. But I think taking shortcuts will result in frustration. So, here I am taking algebra courses on YT with 38 years. [0] https://youtube.com/@ProfessorLeonard?si=0kiGvmbZv4b9Sgf9 https://youtube.com/@ProfessorLeonard?si=0kiGvmbZv4b9Sgf9 [1] https://youtube.com/playlist?list=PLAqhIrjkxbuWI23v9cThsA9GvCAUhRvKZ&si=jKv7dUgvpQmwFnJg https://youtube.com/playlist?list=PLAqhIrjkxbuWI23v9cThsA9Gv...

raverbashing 3y ago

> The fools who write the textbooks of advanced mathematics — and they are mostly clever fools — seldom take the trouble to show you how easy the easy calculations are. Yes this hits the nail on the head The explanations overall look simple, maybe too long sometimes, but that's a killer prelude

WhitneyLand 3y ago

How much UX design / usability tests do the authors of math textbooks do?

dagw 3y ago
When I was at university I had probably 2 or 3 courses where we were beta testing textbooks that the lecturer was in the process of writing. On the downside the book had some rough edges, on the plus side the books where free
- WhitneyLand 3y ago
  Interesting, did they ask you to look for mistakes, or also validate if things were easy to understand and learning efficient?

russfink 3y ago

WRITTEN IN 1910! Why oh why has every text written after this failed to include such a clear understanding of the concepts. There is no excuse for this, academia! Shame, shame! I should engage thee in a bout of fisticuffs were thou a being!

theonemind 3y ago
The 11th edition of the Encyclopedia Britannica is famous for being the last "opionated" edition, also from around 1910. It seems like there was a shift toward trying to be objective in academia around 1910 or so.

HarHarVeryFunny 3y ago

Easy derivatives are indeed easy - easier than that site makes them seem! We can approximate the slope (derivative) of any function y = f(x), at (x,y) as the slope of a straight line from (x,y) to another nearby point (x',y') where y' = f(x'). The slope of this line is (y'-y)/(x'-x). Since we need x' close to x to make this accurate, we use x' = x + dx where dx is small, and we can also represent y' as y' = y + dy. So, then our slope is (y'-y)/(x'-x) = dy/dx = (f(x+dx)-f(x))/dx And that's all there is to it. e.g. For f(x) = x^2 dy/dx = ((x+dx)^2-x^2)/dx = 2x + dx We get the true value of the slope at (x,y) in the limit of dx -> 0, giving us the derivative of f(x) = x^2 as f'(x) = 2x.

HDThoreaun 3y ago
Probably why derivatives are the first semester of calculus,. they are indeed the easiest part and a good way to introduce limits.

barfbagginus 3y ago

The title claims it makes calculus easy, and yet there is no category theory. "How can this be possible?!" I hear you ask. Well, the book was written in 1910, 50 years before category theory appeared. But do not worry! There is a book that uses categories to develop ordinary differential and integral calculus! What could be easier than that? I don't know! But if I find it, I'll let you know! Enjoy! https://books.google.com/books?id=gaE5EAAAQBAJ&newbks=1&newbks_redir=0&printsec=frontcover&pg=PP1&dq=info:e9n__jWuz-IJ:scholar.google.com/&hl=en&source=gb_mobile_entity&ovdme=1#v=onepage&q&f=false https://books.google.com/books?id=gaE5EAAAQBAJ&newbks=1&newb...

gen220 3y ago

I "knew" enough calculus to get good grades in math courses through high school and college as an engineer. But I didn't really "know" calculus until I read a book not too dissimilar from this one, "A Course of Pure Mathematics" from 1908 (!), which constructs calculus up from number theory (I think the fundamental theorem of calculus comes halfway through the book?). From that point it's impossible to forget. As for why it's not taught this way today, I'd blame our testing regime and large classrooms, which incentivizes temporary memorization of key formulae and knowing where to mindlessly apply them, over deep/lasting/semantic understanding. I'd also blame the fact that we have a different maths teacher every year, so students come in with heterogenous understandings of the pre-requisites for the next year's material, so the first part of a section is spent reviewing + consolidating. It takes maybe 10-20% more time to get a rich understanding of the subject that lasts a lifetime. But we value compression and instantly-measurable results more than actual learning. :/ Encountering material like this makes you really happy, but it's kind of bittersweet because it makes you realize that the modal state of modern pedagogy is pretty abysmal.

zackmorris 3y ago

This is great! For anyone reading this, I highly recommend attending the best college you can. The reason why is: the caliber of professor. I was lucky to receive (IMHO) a world-class engineering education at a university in Illinois. It would have been even better at an Ivy League school. But I have heard various accounts from students at other state schools and community colleges, and there's a distinct lack of nuance in their coursework. For example, my statistics and probability professor explained everything by counting apples and bananas on his fingers. It sounds hilarious, but that's all probability is - counting the number of certain outcome(s) out of all possible outcomes. He broke down even the most complex problems into a series of basic counting operations. A less dedicated professor might have just regurgitated the course material verbatim. Where this really matters is, I don't believe in complexity. Learning to learn is about breaking down problems into smaller pieces that are at most the maximum size that can fit in a human brain, and then working each problem until reaching the final answer. Once you can do that, you can tackle any challenge, like running hurdles, with full confidence that you will succeed in time and with enough resources. Which is why I'm so hard on fields which I don't feel are mature yet, due to their byzantine terminology and barriers to entry. Some low hanging fruit might be: AI's obsessions with tensors, CUDA and the entire concept of a GPU (instead of just using symmetric multiprocessing), quantum physics, organic chemistry, biology, medicine, genetics, law, accounting (the CPA test for example), imperative programming (vs functional programming and fundamentals like Lisp and spreadsheets) and design/repair of internal combustion engines. Now, paying for college is a whole different story. It stopped being affordable in the late 2000s, by design to enrich private interests and create a subjugated class that needs a job to pay off student loans, much like the scam of attaching health insurance to employment. The only feasible answer there is systems thinking. Which looks like working the problem backwards by enumerating the core problems and then deriving a general solution which allows everyone to attend for free, funded by taxes paid by automated robots and billionaires. I mean, that's how Star Trek works, why settle for less. Getting rid of billionaires is just a bonus.

bikenaga 3y ago

I got "Calculus Made Easy" back in high school, and it helped when I first took calc in 10th grade, and then BC Calc in 12th grade. It gave me some confidence to have seen the basic ideas presented in a gentle way. Actually, once you get past the old-fashioned language and layout many math books from the early 1900's are pretty good. They don't have a lot of the abstraction that came later, some of which makes you feel like you're trying to do surgery while wearing boxing gloves. Anyway, there are other calc books which have similar intents, though different styles. Here are two: Daniel Kleppner and Norman Ramsey, "Quick Calculus" - We used this for math background when I was taking AP Physics in high school. The physics textbook was Resnick and Halliday, which uses calc throughout. It is not particularly intuitive, but the "programmed learning" approach makes it very easy to pick up enough basic computational ability to do applications. (The phrase "programmed learning" is so old that maybe I should clarify that it has nothing to do with computers.) H. M. Schey, "Div, Grad, Curl, and All That" - This might be helpful for people taking vector calculus, or using it in an electromagnetism course in physics. In fact, a lot of his examples are from E-M. Very good at providing intuition without too many technicalities. I think the best uses for books like these are: (a) You want to get the "big ideas" without worrying too much about technical facility (maybe because you won't ever need it - e.g. you're just learning for your own interest); (b) To provide a fair amount of the math you'll need in (say) a physics course; (c) As prep before taking a standard calc course, or maybe as a refresher. Books like Thompson, Kleppner and Ramsey, or Schey will help with the big picture and to boost confidence but it's best to read them before taking the "real" calc course - since once the "real" course is running you'll be too busy with the work in the course to read something else on the side. But do whatever works for you. Calculus really is wonderful - have fun!

krick 3y ago

I appreciate the effort, but after a couple of pages I can already tell, that it wouldn't be something I wanted if it was the first time I'm learning calculus. And even for a refresher it isn't exactly perfect. IMO, the author is right about what's wrong with the most "proper" books, but overcompensates, and thus makes it unnecessarily complicated to understand, maybe even more so, than some "proper" textbooks. It is too wordy, too informal and pretty hard to read and follow. I don't need any Dean Swift poetry and references to "Queen Elizabeth's days", I just want to know what is calculus, why do we need it, and how do I actually do it, goddammit… That's if we follow the seemingly easier "engineering" approach. And we need it to be a bit more formal still if we follow even slightly more mathematical approach, which I think is actually necessary if you want even a bit of real understanding, because math is this thing where it's very easy to fool yourself into thinking you understand something, when you actually don't, and then get all flabbergasted and helpless against paradoxical "fake proofs" or being asked to prove anything yourself. And to tell valid derivation from invalid, yes, you do need some formal definitions. And in fact, the (formal) basics really aren't difficult to understand at all. Everyone can readily agree that x² grows faster than x, and being introduced into concept of limits to see why (dx)² can be considered negligible compared to dx. You don't need to be comparing weeks to minutes for that. If anything, the latter only throws you off. And I feel like it takes way less patience to read a handful of formal definitions than pages of these "old British"-stylized ramblings. Edit: oh, it isn't "stylized". Still, doesn't change the main point, that there are far better more recent resources to learn calculus.