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Calculus for Beginners and Artists (2003)

836 points by lewisjoe 7y ago

caymanjim 7y ago
Poking around this, it looks like there's a well-written introduction to calculus buried inside, but the navigation is terrible. They should come up with a better way to present this. A single long scrollable page, as clunky as that might be, would be far superior to this.
- myself248 7y ago
  It's the visual equivalent of a duet sung by Gilbert Gotfried and Fran Drescher.
- diggan 7y ago
  Great minds think alike! I linked to a single-page version here: https://news.ycombinator.com/item?id=20434118 https://news.ycombinator.com/item?id=20434118
adamnemecek 7y ago
I'll leave this here https://github.com/MikeInnes/diff-zoo https://github.com/MikeInnes/diff-zoo
sunraa 7y ago
Priceless: http://www-math.mit.edu/~djk/calculus_beginners/chapter00/section01.html http://www-math.mit.edu/~djk/calculus_beginners/chapter00/se... 0.1 What You Should Know To study calculus it is essential that you are able to breathe. Without that ability you will soon die, and be unable to continue. Beyond that, you will need some familiarity with two notions: the notion of a number, and that of a function.
steve76 7y ago
This is all you need: Replace the derivative with the little triangle. Replace the integral with the sum sign that looks like that weird capital E. Then just bash numbers into your calculator. Your professor will have to give you a C. If you are a good liar, you can hide your work, and then just con your way into a cheat sheet for all the theory garbage. PS: Newton was really really bad. Infinity sign means Snake. Eulers constant means throw snake into boiling oil.
lewisjoe 7y ago
Came across this just when I was looking to deep dive into calculus, typed lambda calculus, differentiation, etc. I somehow have a hunch learning these, will help me down my road on language design and implementing compilers. Only midway, but hands down this is the best material I came across on the subject. As light-hearted as much as detailed. Update: Thank you, all. I guess the dots would never connect down my road. But I'm midway and the topic's still interesting enough for me to keep going.
- mcncm 7y ago
  Just out of curiosity, why do you need to know differential calculus to write a compiler, and what does it have to do with lambda calculus? Learning it is a great thing to do, and it will certainly help you in other ways. But that kind of programming usually has little in common with the "mathematical analysis" kind of calculus (unless you're specifically interested in differentiable programming, which is actually kind of big these days). Anyway, happy studies!
  - lewisjoe 7y ago
    Oh, I have this bad habit of going down the rabbit hole of fundamentals if I feel I lack the basics of a concept. While I still haven't grasped the need for calculus in language design, I've come across pieces on how Lisp is based on lambda calculus vs Haskell based on typed lambda calculus. Also, that any problem that can be solved with turing machine, can be solved by lambda calculus as well. While not much about compilers, I guess it will solidify my understanding of language designs? I'm not sure. I'll rather learn and hope the dots will connect later than ignore the subject altogether.
    - p0llard 7y ago
      Just to clarify, you do realise that lambda calculus is an entirely different field to the calculus of mathematical analysis? They really have very little in common apart from the name. Sorry if you already knew this, but it seemed like you were making a connection that wasn't there.
    - 7y ago
diggan 7y ago
Seems like a handy resource but seems the navigation is a bit annoying to deal with. Quick hack with `wget --mirror` + `pandoc` for joining the documents, I got the following single-page layout: https://cloudflare-ipfs.com/ipfs/QmZetPzAP5bC3uXVmNUKU2DU22LPBmxfkSdGqYaGMqmgp7/ https://cloudflare-ipfs.com/ipfs/QmZetPzAP5bC3uXVmNUKU2DU22L... (I take no responsability for any formatting errors. Math seems to render correctly, but probably some links are broken)
nothis 7y ago
Artists?
- theoh 7y ago
  This comment is like a red rag to a bull for me, because why wouldn't some artists want a basic grasp of calculus? Bear in mind that a secondary education, for most people, involves learning the basics of calculus. You might be thinking that all artists are bad at math or too stupid to grasp calculus. You might be thinking that there are no applications of calculus in art. I don't know. But given the role artists are supposed to play in society, it seems pretty reasonable to expect that some of them will have an interest in learning calculus to a high school level like a significant fraction of the non-specialist public. A concrete example of this might be an artist who is interested in systems theory from an ecological perspective. Once you start talking about stocks and flows (of fish or minerals or greenhouse gases) then calculus and differential equations are really the next thing to tackle.
  - thethirdone 7y ago
    I interpreted the vague "Artists?" very differently from you. I thought it was saying "What about this makes it appropriate for artists? It is not showing off the beauty and simplicity of Calculus." > You might be thinking that all artists are bad at math or too stupid to grasp calculus I think labeling it as "for Beginners and Artists" is actually doing some of that. I certainly would not make a book labeled for artists unless I thought it would be particularly good for artists and from what I have seen this book does not touch on creativity or beauty. Therefore it must be labeled as for Artists because they can't learn calculus from existing books which might imply "all artists are bad at math or too stupid to grasp calculus".
    - nothis 7y ago
      Thank you!
- suyash 7y ago
  I am looking for the same but found nothing artistic in it.
kstenerud 7y ago
http://www-math.mit.edu/~djk/calculus_beginners/chapter03/section02.html http://www-math.mit.edu/~djk/calculus_beginners/chapter03/se... We can determine the linear function which takes value f(a) at a and f(b) at b by the following formula: x-b x-a f(x) = f(a) --- + f(b) --- a-b b-a The first term is 0 when x is b and is f(a) when x is a, while the second term is 0 when x is a and is f(b) when x is b. The sum of the two is therefore f(a) when x is a and f(b) when x is b. And it is a linear function. Linear functions have a term that is x multiplied by some constant, and may also have a constant term as well. Umm... whut? This has me completely lost. I don't see how this can be geared towards beginners and artists.
- baby 7y ago
  1. X-b befomes 0 if x=b right? 2. This means anything multiplying this x-b becomes 0 as well when x=b right? His definition of linear is also wrong in my book. Linear: f(x) = a x Affine: f(x) = a x + b (For some a, b)
  - kstenerud 7y ago
    Is that comment for me or about the referenced page? If it's for me, I'm not sure what you're saying.
    - baby 7y ago
      I'm trying to explain what he meant. Which step don't you understand specifically in my comment?
      - kstenerud 7y ago
        I'm trying to understand why there's f(a) and f(b), why is b being subtracted from x, why it's divided by a-b, why there's a similar thing for f(a), why the two are being added, why the zero values are important, and what the whole thing actually means.
enriquto 7y ago
I sort of dislike this kind of "philosophical" introduction to calculus. Maybe I don't have the spirit of an artist. The best way to start with calculus is the one by Gilbert Strang, who explains everything on the first page of his book (and the rest of the book are "just" examples). The first half of the first page shows a drawing of the speedometer and odometer of a car and it explains what they are, and that they are not independent but related in a special way. On the second half of the first page it says that differential calculus is the task of computing the speed from the distance, and integral calculus is the task of computing the distance from the speed. Then it says a lovely sentence "this is not an analogy, this is the real deal and we have already started with the subject, and this is actually all there is to it". Then in the rest of the page it explains in a couple of sentences how can you compute speeds from distances and vice-versa, and why you need a constant of integration, and so on. It also proves the fundamental theorem of calculus. The rest of the book consists in concrete examples and a few more constructions, up to Taylor series.
- lewisjoe 7y ago
  https://ocw.mit.edu/resources/res-18-001-calculus-online-textbook-spring-2005/textbook/ https://ocw.mit.edu/resources/res-18-001-calculus-online-tex... I guess this is the one?
  - enriquto 7y ago
    Yes, this one! Just read pages 1 and 2 of the pdf (I was writing from memory and my explanation is not exact).
- GuiA 7y ago
  Different people learn differently. Very few people learn all they know from a single source. Therefore, having many sources presenting a same subject in many different ways is a wonderful thing. Gilbert Strang’s book is wonderful! Some will find it to be the “best book” to learn calculus, others won’t. No need to poo-poo others’ work just to value signal. (I’m sure the authors would appreciate any constructive feedback though!)
thisguyuknow 7y ago
I noticed the use of jQWidgets and I wondered what people had to say about that.
- boikom 7y ago
  It's a UI Component library with a good quality at a good price
gibsonf1 7y ago
I know of no better way to both learn, truly understand, and enjoy calculus than the visual approach of 3Blue1Brown in the Calculus series of videos: https://www.youtube.com/watch?v=WUvTyaaNkzM&list=PLZHQObOWTQDMsr9K-rj53DwVRMYO3t5Yr https://www.youtube.com/watch?v=WUvTyaaNkzM&list=PLZHQObOWTQ...
- edjrage 7y ago
  I swear one day I'm going to write a WebExtension whose sole purpose is to auto pin comments about 3B1B's videos on calculus and linear algebra HN threads.
- hackermailman 7y ago
  I found the best way to truly understand and enjoy calculus was to learn it as it was historically developed https://youtu.be/HRD9X-2Bmdw https://youtu.be/HRD9X-2Bmdw and then learning about the problems they ran into and how we ended up with modern Calculus after Euler and Lagrange tried to correct these problems https://youtu.be/fCZ8jJCVinU https://youtu.be/fCZ8jJCVinU (these lectures cover Stillwell's book Mathematics and it's History) Interesting in the second lecture is how Australia does 'photo radar' on one stretch of highway, where it records you going through a gate at one point, then many kilometers away records you again from another gate, then establishes your average velocity between the two gates using Calculus and sends you a ticket if your calculated average shows you must have exceeded the speed limit during some point between the gates.
  - krick 7y ago
    > best way to truly understand and enjoy calculus was to learn it as it was historically developed In fact, I'd argue it's the best way to understand almost anything, especially in math. Many topics I found somewhat confusing at school or university or whatever got really simple once I learned about their history. Little by little I come to feel that most of great inventions or discoveries made by people we regard as geniuses are often brilliant at how clear, beautiful and somewhat unexpected the solution was, but it's actually very rarely complicated and usually seems like the most natural thing in the world, when told about how Fourier/Laplace/Leibniz/etc discovered it, and not hidden behind standard school math curriculum. That's a part of why I love 3Blue1Brown videos so much, and why I love Morris Kline books. And it always makes me kind of sad feeling how much time I wasted trying to come to terms with something that always was just unnatural explanation.
neom 7y ago
As someone with extreme dyslexia and dyscalculia, this is fascinating. Because numbers and words are images for me, my recall and then contextual association of them is very slow, building different frameworks to understand them is super important and I wish there was a greater emphasis on different memory styles in learning outside of rote.
undershirt 7y ago
> “If calculus is the language of the universe, then Steven Strogatz is its Homer.” I saw someone reading this at a coffeeshop, it’s a new one: http://www.stevenstrogatz.com/books/infinite-powers http://www.stevenstrogatz.com/books/infinite-powers
- gshubert17 7y ago
  This is an excellent book, which is full with historical insights, references to people I didn't know about (Sophie Germain, Mary Cartwright, Sofia Kovalevskaya), and emphasizes intuition and applications rather than proofs.
- vcavallo 7y ago
  i’m reading it right now. very highly-recommended!!
putzdown 7y ago
This site undermines its own intent of providing a friendly and accessible introduction to calculus through poor and erroneous grammar. An editor could make it very useful. Some examples: “OK, but how does calculus models change?” error: subject-verb number agreement. Try “model”. “The fundamental idea of calculus is to study change by studying ‘instantaneous’ change, by which we mean changes over tiny intervals of time.” So, a change is a changes? Stay with the singular: “by which we mean a change over a tiny interval of time.” “It turns out that such [tiny] changes tend to be lots simpler than changes over finite intervals of time.” Now we have a logical problem. Isn’t the set of tiny intervals of time a subset of finite intervals of time? In ordinary usage, “tiny” is finite, surely; a tiny thing is not infinite. So how can a tiny-time change be simpler than a finite-time change, when it is itself a finite-time change? Errors of this sort lost my faith early on. I hope they’re corrected, because the promise of the article is appealing.
- bardworx 7y ago
  I’m my opinion, your comment is very self serving and reminds me of a quote from Theodore Roosevelt: “It is not the critic who counts; not the man who points out how the strong man stumbles, or where the doer of deeds could have done them better. The credit belongs to the man who is actually in the arena, whose face is marred by dust and sweat and blood; who strives valiantly; who errs, who comes short again and again, because there is no effort without error and shortcoming; but who does actually strive to do the deeds; who knows great enthusiasms, the great devotions; who spends himself in a worthy cause; who at the best knows in the end the triumph of high achievement, and who at the worst, if he fails, at least fails while daring greatly, so that his place shall never be with those cold and timid souls who neither know victory nor defeat.” If you really cared about the subject you could have sent the creator/department a message with your constructive criticism. Instead, you posted on the form how a few errors invalidates the entire work that took significant amount of time.
jszymborski 7y ago
I've discovered Calculus Made Easy recently and it's wonderful. The edition edited by Martin Gardner is particularly good, with amazing preliminary chapters [0]. I've spent many years "pretending to understand" calculus, but things I remember gnawing at me, like limits & infinitesmals, are accompanied with context and history such that you can finally put yourself into the conversation and understand that my confusion is simply due to only getting a fraction of the story. You can read the full text for free here [1] [0] https://openlibrary.org/books/OL351037M/Calculus_made_easy https://openlibrary.org/books/OL351037M/Calculus_made_easy [1] http://calculusmadeeasy.org/ http://calculusmadeeasy.org/
- tranchms 7y ago
  Calculus Made Easy is the best!
- Deinos 7y ago
  Thank you for the links. I always did well with math classes in school, but my retention was always so so after passing the final. Will give this a try for a more lasting understanding of the subject!
- hyperpallium 7y ago
  [a critique, not a request for help] I found it relied too much on faith: you can ignore this small quantity, but not this one. What's the threshold? Why? Fractional and negative powers are assumed to work as a generalization of positive integer powers, without proof. But, TBF, all maths education requires a lot of faith. e.g. the unique prime factorization theorem is assumed in high school, not proven.
  - doliveira 7y ago
    Yes, but if you think faith alone is not enough you can always drop back to the sacred texts of the proofs. But it is weird not being shown the proofs for things like generalization for other situations. I studied Calculus while in engineering at college and we always got the proofs for the tricks we were using.
rayalez 7y ago
Great link! Now I only wish there were similar resources for Linear Algebra and Probability/Statistics. Any good recommendations?
- zrobotics 7y ago
  For linear algebra, the YouTube channel 3blue1brown has a truly excellent series on linear algebra; I found it was the best intro to the subject avaliable.
halbgnarf 7y ago
http://www-math.mit.edu/~djk/calculus_beginners/chapter03/section03.html http://www-math.mit.edu/~djk/calculus_beginners/chapter03/se... > if y = z + a * (x - z) then 1.) f(y) = f(x) + a * (f(x) - f(z)) This is incorrect, isn't it? It should be 2.) f(y) = f(z) + a * (f(x) - f(z)) if I'm not mistaken. For example, consider f(n) = n + 1 - z=1, x=2 - f(z)=2, f(x)=3 - f(z + a(x - z)) <=> f(1 + a); for a = -1, we should get f(0) = 1, Substituting into 1.) yields 3 + -1 * (3 - 2) = 2 (incorrect) Substituting into 2.) yields 2 + -1 * (3 - 2) = 1 (correct)
ClearAndPresent 7y ago
Speaking as an artist... The tone from the beginning (at least in Chapter 0) struck me as what I might class "fey wankery". The wry style gets into the way of communication. Is there some misapprehension amongst STEM explainers that people who are less familiar with maths need to be treated like skittish, possibly mentally-challenged deer? Or is it a nod to Socratic dialogue except one of two is a moron? Then Chapter One leaps into rational numbers, set theory and fractions and commits the usual errors: asymptotic curve into complexity, no history, no vivid metaphorical visualisations and (a personal peeve) no historical or causal explanations. Chapter One literally starts with a question ("What are numbers?") and does not ever answer it. You can count them, some of them are natural (what does that mean?), you can perform operations on them. But what are they, Professor Kleitman? An abstract concept that can be derived from our ability to discern a first order similarity between both similar and dissimilar objects? Is it too vast a question for a introduction to calculus? Then don't use the question in your section header. Operations? "There are addition, subtraction, multiplication and division." Why? Why isn't there redition? What's redition? I don't know, it's an operation I made up, but you seem to have plucked four relations between natural numbers from the air and asked me to assume that's acceptable. The text continues in similar, tedious terms, walking us through the basics on stepping stones of assumption and unearned trust. Educating beginners and "artists" doesn't require you to speak to us like five year olds. Take a leaf from Feynman or Fuller's books and talk to us like adults but do the work in creating vibrant metaphor. Just my opinion and apologies to Professor Kleitman. To end on a positive note, the opening pages of Gilbert Strang's book, linked by another poster here, were much more effective in conceptualising the need for and use of calculus by anchoring it in a strong metaphorical example. In barely a page or two, I understand a relationship between velocity and distance - and that time is involved - and how that relationship can be geometrically envisioned. Further, I'm already pondering how one might deal with a more realistic car journey with a variable velocity, something that calculus will address later on. The difference? No infantilisation; instead a clear and applicable metaphorical example.
- mathgenius 7y ago
  This question "What are numbers?" I find way more fruitful to contemplate than any answer I have so far found. And why should we be adults anyway? Adults already know all the answers. Ask this question like a five year old would.
prvc 7y ago
From reading a few excerpts only, I am reminded of a lot of popular science writing, which too often adopts a condescending casual tone full of flimsy, inaccurate metaphors and chatty asides, and which too often omits simple statements of the details of the subject at hand (often less complex than the verbiage used for the adulterated version). The writing becomes so far removed from the subject, that while those already familiar with it will be able to guess what the author was getting at, those who are not will remain in the dark, leading to a zero increase in the reader's knowledge in each case. Despite being in the former category with respect to basic calculus, I can't help but feel a sense of umbrage on behalf of the latter.
itronitron 7y ago
Khan Academy is a better resource for being introduced to calculus. The title of the course is unfortunately perpetuating the mistaken stereotype that artists are not mathematically inclined. The incoming class my freshman year of the art school at the university I attended had the highest median Math SAT score of any of the other schools (i.e engineering, science) in the university.
nemetroid 7y ago
Despite the title, I struggle to see who exactly this was written for. The Javascript widgets are quite neat. Fiddling around with parameters, and receiving instant graphical feedback, is great for developing an intuitive understanding. But I find that the writing has several issues that probably make it less clear to beginners than your average calculus textbook: The writing is very brief, with typical mathematical terseness. This is usually a good thing, since it lets you be very exact in your definitions. But to be accessible to a beginner, this writing needs to be backed up with concrete examples, and preferably a picture or two. In most cases I find that it isn't, so the reader needs to internalize a lot of things on their own before being able to move on. The text is also sparse on showing its work. A positive example is near the bottom of 5.1, showing how to derive the quotient rule, but most of the time it looks more like example 1 in 6.1, with a lot of information given inline before actually applying the mentioned operations to the expression: > Suppose we substitute the function g which has values given by g(x) = x² + 1 into the function f which takes values f(x) = x³ − 3. > The substituted function f(g) has values f(g(x)) = (x² + 1)³ − 3. > Let us compute the derivative of this function. The derivative of f(s) with respect to s is 3s², while the derivative of g(x) with respect to x is 2x. > If we set s = g(x) which is x² + 1, and take the product of these two we get: > [expression] This type of mental expression manipulation is fine for someone that's had practice with it, but probably not for a beginner, who would gain a lot from having these kinds of things written out in a more structured way. In these aspects, I find the text less clear than the calculus textbook i used at university (which was not directed toward beginners or artists). I agree a lot with prvc's comment[0] about being able to fill in the gaps, in that the terseness and handwaviness can make this look like a beginner-friendly version to someone that already is familiar with the subject, but I don't think changing tone is enough to make something beginner-friendly. 0: https://news.ycombinator.com/item?id=20435238 https://news.ycombinator.com/item?id=20435238
krick 7y ago
A point as minor as they come, but I really am not a fan of that embedded chapter structure. It's absolutely ok to have multiple sub-topics in a topic (that's why TOC exists in the first place!), but it also makes it so much more enjoyable to read, when you can simply go through text linearly, without worrying about what link on what level you clicked just before that.
jedahan 7y ago
I want the inverse so badly. Call it 'Art for Beginners and Mathists'.
- barry-cotter 7y ago
  If you want a book Drawing On the Right Hand Side of the Brain http://drawright.com/ http://drawright.com/ If you want a video course http://drawabox.com/ http://drawabox.com/
WheelsAtLarge 7y ago
I took 2 semesters of Calculus in high school. It was a "requirement" to get college AP. I was convinced that I would have failed at college if I didn't take it. I have yet to use it in my career. A waste of time. A 2 semester class of drawing or Art would have been way more useful. Why is calculus held in such high esteem in college prep? Most students can safely skipt calculus. A 1-day general introduction would be enough for 90%+ of college students.
- mlevental 7y ago
  90% of college students aren't stem majors and so what you're saying is almost tautology. for stem there is literally no branch that doesn't use calculus in at least some way.
RichardCA 7y ago
I never really felt like I understood Calculus until I watched the Mechanical Universe on PBS. Here's an example, demonstrating how Kepler's Second Law is an expression of the conservation of angular momentum. https://www.youtube.com/watch?v=0tSmY4fn5xY#t=7m50s https://www.youtube.com/watch?v=0tSmY4fn5xY#t=7m50s
theNJR 7y ago
Just last night I was complaining to someone about wanting this very thing. I suppose I should now accuse Hacker News or listening in on my phone.
deleted 7y ago
[deleted]
mrcactu5 7y ago
Robert Ghrist has written an colorful and innovative calculus text. https://www.math.upenn.edu/~ghrist/calculus.html https://www.math.upenn.edu/~ghrist/calculus.html https://www.math.upenn.edu/~ghrist/notes.html https://www.math.upenn.edu/~ghrist/notes.html https://sites.math.washington.edu/~morrow/334_13/FLCT.pdf https://sites.math.washington.edu/~morrow/334_13/FLCT.pdf

836 points by lewisjoe 7y ago

Poking around this, it looks like there's a well-written introduction to calculus buried inside, but the navigation is terrible. They should come up with a better way to present this. A single long scrollable page, as clunky as that might be, would be far superior to this.

myself248 7y ago
It's the visual equivalent of a duet sung by Gilbert Gotfried and Fran Drescher.
diggan 7y ago
Great minds think alike! I linked to a single-page version here: https://news.ycombinator.com/item?id=20434118 https://news.ycombinator.com/item?id=20434118

adamnemecek 7y ago

I'll leave this here https://github.com/MikeInnes/diff-zoo https://github.com/MikeInnes/diff-zoo

sunraa 7y ago

Priceless: http://www-math.mit.edu/~djk/calculus_beginners/chapter00/section01.html http://www-math.mit.edu/~djk/calculus_beginners/chapter00/se... 0.1 What You Should Know To study calculus it is essential that you are able to breathe. Without that ability you will soon die, and be unable to continue. Beyond that, you will need some familiarity with two notions: the notion of a number, and that of a function.

steve76 7y ago

This is all you need: Replace the derivative with the little triangle. Replace the integral with the sum sign that looks like that weird capital E. Then just bash numbers into your calculator. Your professor will have to give you a C. If you are a good liar, you can hide your work, and then just con your way into a cheat sheet for all the theory garbage. PS: Newton was really really bad. Infinity sign means Snake. Eulers constant means throw snake into boiling oil.

lewisjoe 7y ago

Came across this just when I was looking to deep dive into calculus, typed lambda calculus, differentiation, etc. I somehow have a hunch learning these, will help me down my road on language design and implementing compilers. Only midway, but hands down this is the best material I came across on the subject. As light-hearted as much as detailed. Update: Thank you, all. I guess the dots would never connect down my road. But I'm midway and the topic's still interesting enough for me to keep going.

mcncm 7y ago
Just out of curiosity, why do you need to know differential calculus to write a compiler, and what does it have to do with lambda calculus? Learning it is a great thing to do, and it will certainly help you in other ways. But that kind of programming usually has little in common with the "mathematical analysis" kind of calculus (unless you're specifically interested in differentiable programming, which is actually kind of big these days). Anyway, happy studies!
- lewisjoe 7y ago
  Oh, I have this bad habit of going down the rabbit hole of fundamentals if I feel I lack the basics of a concept. While I still haven't grasped the need for calculus in language design, I've come across pieces on how Lisp is based on lambda calculus vs Haskell based on typed lambda calculus. Also, that any problem that can be solved with turing machine, can be solved by lambda calculus as well. While not much about compilers, I guess it will solidify my understanding of language designs? I'm not sure. I'll rather learn and hope the dots will connect later than ignore the subject altogether.
  - p0llard 7y ago
    Just to clarify, you do realise that lambda calculus is an entirely different field to the calculus of mathematical analysis? They really have very little in common apart from the name. Sorry if you already knew this, but it seemed like you were making a connection that wasn't there.
  - 7y ago

diggan 7y ago

Seems like a handy resource but seems the navigation is a bit annoying to deal with. Quick hack with `wget --mirror` + `pandoc` for joining the documents, I got the following single-page layout: https://cloudflare-ipfs.com/ipfs/QmZetPzAP5bC3uXVmNUKU2DU22LPBmxfkSdGqYaGMqmgp7/ https://cloudflare-ipfs.com/ipfs/QmZetPzAP5bC3uXVmNUKU2DU22L... (I take no responsability for any formatting errors. Math seems to render correctly, but probably some links are broken)

nothis 7y ago

Artists?

theoh 7y ago
This comment is like a red rag to a bull for me, because why wouldn't some artists want a basic grasp of calculus? Bear in mind that a secondary education, for most people, involves learning the basics of calculus. You might be thinking that all artists are bad at math or too stupid to grasp calculus. You might be thinking that there are no applications of calculus in art. I don't know. But given the role artists are supposed to play in society, it seems pretty reasonable to expect that some of them will have an interest in learning calculus to a high school level like a significant fraction of the non-specialist public. A concrete example of this might be an artist who is interested in systems theory from an ecological perspective. Once you start talking about stocks and flows (of fish or minerals or greenhouse gases) then calculus and differential equations are really the next thing to tackle.
- thethirdone 7y ago
  I interpreted the vague "Artists?" very differently from you. I thought it was saying "What about this makes it appropriate for artists? It is not showing off the beauty and simplicity of Calculus." > You might be thinking that all artists are bad at math or too stupid to grasp calculus I think labeling it as "for Beginners and Artists" is actually doing some of that. I certainly would not make a book labeled for artists unless I thought it would be particularly good for artists and from what I have seen this book does not touch on creativity or beauty. Therefore it must be labeled as for Artists because they can't learn calculus from existing books which might imply "all artists are bad at math or too stupid to grasp calculus".
  - nothis 7y ago
    Thank you!
suyash 7y ago
I am looking for the same but found nothing artistic in it.

kstenerud 7y ago

http://www-math.mit.edu/~djk/calculus_beginners/chapter03/section02.html http://www-math.mit.edu/~djk/calculus_beginners/chapter03/se... We can determine the linear function which takes value f(a) at a and f(b) at b by the following formula: x-b x-a f(x) = f(a) --- + f(b) --- a-b b-a The first term is 0 when x is b and is f(a) when x is a, while the second term is 0 when x is a and is f(b) when x is b. The sum of the two is therefore f(a) when x is a and f(b) when x is b. And it is a linear function. Linear functions have a term that is x multiplied by some constant, and may also have a constant term as well. Umm... whut? This has me completely lost. I don't see how this can be geared towards beginners and artists.

baby 7y ago
1. X-b befomes 0 if x=b right? 2. This means anything multiplying this x-b becomes 0 as well when x=b right? His definition of linear is also wrong in my book. Linear: f(x) = a x Affine: f(x) = a x + b (For some a, b)
- kstenerud 7y ago
  Is that comment for me or about the referenced page? If it's for me, I'm not sure what you're saying.
  - baby 7y ago
    I'm trying to explain what he meant. Which step don't you understand specifically in my comment?
    - kstenerud 7y ago
      I'm trying to understand why there's f(a) and f(b), why is b being subtracted from x, why it's divided by a-b, why there's a similar thing for f(a), why the two are being added, why the zero values are important, and what the whole thing actually means.

enriquto 7y ago

I sort of dislike this kind of "philosophical" introduction to calculus. Maybe I don't have the spirit of an artist. The best way to start with calculus is the one by Gilbert Strang, who explains everything on the first page of his book (and the rest of the book are "just" examples). The first half of the first page shows a drawing of the speedometer and odometer of a car and it explains what they are, and that they are not independent but related in a special way. On the second half of the first page it says that differential calculus is the task of computing the speed from the distance, and integral calculus is the task of computing the distance from the speed. Then it says a lovely sentence "this is not an analogy, this is the real deal and we have already started with the subject, and this is actually all there is to it". Then in the rest of the page it explains in a couple of sentences how can you compute speeds from distances and vice-versa, and why you need a constant of integration, and so on. It also proves the fundamental theorem of calculus. The rest of the book consists in concrete examples and a few more constructions, up to Taylor series.

lewisjoe 7y ago
https://ocw.mit.edu/resources/res-18-001-calculus-online-textbook-spring-2005/textbook/ https://ocw.mit.edu/resources/res-18-001-calculus-online-tex... I guess this is the one?
- enriquto 7y ago
  Yes, this one! Just read pages 1 and 2 of the pdf (I was writing from memory and my explanation is not exact).
GuiA 7y ago
Different people learn differently. Very few people learn all they know from a single source. Therefore, having many sources presenting a same subject in many different ways is a wonderful thing. Gilbert Strang’s book is wonderful! Some will find it to be the “best book” to learn calculus, others won’t. No need to poo-poo others’ work just to value signal. (I’m sure the authors would appreciate any constructive feedback though!)

thisguyuknow 7y ago

I noticed the use of jQWidgets and I wondered what people had to say about that.

boikom 7y ago
It's a UI Component library with a good quality at a good price

gibsonf1 7y ago

I know of no better way to both learn, truly understand, and enjoy calculus than the visual approach of 3Blue1Brown in the Calculus series of videos: https://www.youtube.com/watch?v=WUvTyaaNkzM&list=PLZHQObOWTQDMsr9K-rj53DwVRMYO3t5Yr https://www.youtube.com/watch?v=WUvTyaaNkzM&list=PLZHQObOWTQ...

edjrage 7y ago
I swear one day I'm going to write a WebExtension whose sole purpose is to auto pin comments about 3B1B's videos on calculus and linear algebra HN threads.
hackermailman 7y ago
I found the best way to truly understand and enjoy calculus was to learn it as it was historically developed https://youtu.be/HRD9X-2Bmdw https://youtu.be/HRD9X-2Bmdw and then learning about the problems they ran into and how we ended up with modern Calculus after Euler and Lagrange tried to correct these problems https://youtu.be/fCZ8jJCVinU https://youtu.be/fCZ8jJCVinU (these lectures cover Stillwell's book Mathematics and it's History) Interesting in the second lecture is how Australia does 'photo radar' on one stretch of highway, where it records you going through a gate at one point, then many kilometers away records you again from another gate, then establishes your average velocity between the two gates using Calculus and sends you a ticket if your calculated average shows you must have exceeded the speed limit during some point between the gates.
- krick 7y ago
  > best way to truly understand and enjoy calculus was to learn it as it was historically developed In fact, I'd argue it's the best way to understand almost anything, especially in math. Many topics I found somewhat confusing at school or university or whatever got really simple once I learned about their history. Little by little I come to feel that most of great inventions or discoveries made by people we regard as geniuses are often brilliant at how clear, beautiful and somewhat unexpected the solution was, but it's actually very rarely complicated and usually seems like the most natural thing in the world, when told about how Fourier/Laplace/Leibniz/etc discovered it, and not hidden behind standard school math curriculum. That's a part of why I love 3Blue1Brown videos so much, and why I love Morris Kline books. And it always makes me kind of sad feeling how much time I wasted trying to come to terms with something that always was just unnatural explanation.

neom 7y ago

As someone with extreme dyslexia and dyscalculia, this is fascinating. Because numbers and words are images for me, my recall and then contextual association of them is very slow, building different frameworks to understand them is super important and I wish there was a greater emphasis on different memory styles in learning outside of rote.

undershirt 7y ago

> “If calculus is the language of the universe, then Steven Strogatz is its Homer.” I saw someone reading this at a coffeeshop, it’s a new one: http://www.stevenstrogatz.com/books/infinite-powers http://www.stevenstrogatz.com/books/infinite-powers

gshubert17 7y ago
This is an excellent book, which is full with historical insights, references to people I didn't know about (Sophie Germain, Mary Cartwright, Sofia Kovalevskaya), and emphasizes intuition and applications rather than proofs.
vcavallo 7y ago
i’m reading it right now. very highly-recommended!!

putzdown 7y ago

This site undermines its own intent of providing a friendly and accessible introduction to calculus through poor and erroneous grammar. An editor could make it very useful. Some examples: “OK, but how does calculus models change?” error: subject-verb number agreement. Try “model”. “The fundamental idea of calculus is to study change by studying ‘instantaneous’ change, by which we mean changes over tiny intervals of time.” So, a change is a changes? Stay with the singular: “by which we mean a change over a tiny interval of time.” “It turns out that such [tiny] changes tend to be lots simpler than changes over finite intervals of time.” Now we have a logical problem. Isn’t the set of tiny intervals of time a subset of finite intervals of time? In ordinary usage, “tiny” is finite, surely; a tiny thing is not infinite. So how can a tiny-time change be simpler than a finite-time change, when it is itself a finite-time change? Errors of this sort lost my faith early on. I hope they’re corrected, because the promise of the article is appealing.

bardworx 7y ago
I’m my opinion, your comment is very self serving and reminds me of a quote from Theodore Roosevelt: “It is not the critic who counts; not the man who points out how the strong man stumbles, or where the doer of deeds could have done them better. The credit belongs to the man who is actually in the arena, whose face is marred by dust and sweat and blood; who strives valiantly; who errs, who comes short again and again, because there is no effort without error and shortcoming; but who does actually strive to do the deeds; who knows great enthusiasms, the great devotions; who spends himself in a worthy cause; who at the best knows in the end the triumph of high achievement, and who at the worst, if he fails, at least fails while daring greatly, so that his place shall never be with those cold and timid souls who neither know victory nor defeat.” If you really cared about the subject you could have sent the creator/department a message with your constructive criticism. Instead, you posted on the form how a few errors invalidates the entire work that took significant amount of time.

jszymborski 7y ago

I've discovered Calculus Made Easy recently and it's wonderful. The edition edited by Martin Gardner is particularly good, with amazing preliminary chapters [0]. I've spent many years "pretending to understand" calculus, but things I remember gnawing at me, like limits & infinitesmals, are accompanied with context and history such that you can finally put yourself into the conversation and understand that my confusion is simply due to only getting a fraction of the story. You can read the full text for free here [1] [0] https://openlibrary.org/books/OL351037M/Calculus_made_easy https://openlibrary.org/books/OL351037M/Calculus_made_easy [1] http://calculusmadeeasy.org/ http://calculusmadeeasy.org/

tranchms 7y ago
Calculus Made Easy is the best!
Deinos 7y ago
Thank you for the links. I always did well with math classes in school, but my retention was always so so after passing the final. Will give this a try for a more lasting understanding of the subject!
hyperpallium 7y ago
[a critique, not a request for help] I found it relied too much on faith: you can ignore this small quantity, but not this one. What's the threshold? Why? Fractional and negative powers are assumed to work as a generalization of positive integer powers, without proof. But, TBF, all maths education requires a lot of faith. e.g. the unique prime factorization theorem is assumed in high school, not proven.
- doliveira 7y ago
  Yes, but if you think faith alone is not enough you can always drop back to the sacred texts of the proofs. But it is weird not being shown the proofs for things like generalization for other situations. I studied Calculus while in engineering at college and we always got the proofs for the tricks we were using.

rayalez 7y ago

Great link! Now I only wish there were similar resources for Linear Algebra and Probability/Statistics. Any good recommendations?

zrobotics 7y ago
For linear algebra, the YouTube channel 3blue1brown has a truly excellent series on linear algebra; I found it was the best intro to the subject avaliable.

halbgnarf 7y ago

http://www-math.mit.edu/~djk/calculus_beginners/chapter03/section03.html http://www-math.mit.edu/~djk/calculus_beginners/chapter03/se... > if y = z + a * (x - z) then 1.) f(y) = f(x) + a * (f(x) - f(z)) This is incorrect, isn't it? It should be 2.) f(y) = f(z) + a * (f(x) - f(z)) if I'm not mistaken. For example, consider f(n) = n + 1 - z=1, x=2 - f(z)=2, f(x)=3 - f(z + a(x - z)) <=> f(1 + a); for a = -1, we should get f(0) = 1, Substituting into 1.) yields 3 + -1 * (3 - 2) = 2 (incorrect) Substituting into 2.) yields 2 + -1 * (3 - 2) = 1 (correct)

ClearAndPresent 7y ago

Speaking as an artist... The tone from the beginning (at least in Chapter 0) struck me as what I might class "fey wankery". The wry style gets into the way of communication. Is there some misapprehension amongst STEM explainers that people who are less familiar with maths need to be treated like skittish, possibly mentally-challenged deer? Or is it a nod to Socratic dialogue except one of two is a moron? Then Chapter One leaps into rational numbers, set theory and fractions and commits the usual errors: asymptotic curve into complexity, no history, no vivid metaphorical visualisations and (a personal peeve) no historical or causal explanations. Chapter One literally starts with a question ("What are numbers?") and does not ever answer it. You can count them, some of them are natural (what does that mean?), you can perform operations on them. But what are they, Professor Kleitman? An abstract concept that can be derived from our ability to discern a first order similarity between both similar and dissimilar objects? Is it too vast a question for a introduction to calculus? Then don't use the question in your section header. Operations? "There are addition, subtraction, multiplication and division." Why? Why isn't there redition? What's redition? I don't know, it's an operation I made up, but you seem to have plucked four relations between natural numbers from the air and asked me to assume that's acceptable. The text continues in similar, tedious terms, walking us through the basics on stepping stones of assumption and unearned trust. Educating beginners and "artists" doesn't require you to speak to us like five year olds. Take a leaf from Feynman or Fuller's books and talk to us like adults but do the work in creating vibrant metaphor. Just my opinion and apologies to Professor Kleitman. To end on a positive note, the opening pages of Gilbert Strang's book, linked by another poster here, were much more effective in conceptualising the need for and use of calculus by anchoring it in a strong metaphorical example. In barely a page or two, I understand a relationship between velocity and distance - and that time is involved - and how that relationship can be geometrically envisioned. Further, I'm already pondering how one might deal with a more realistic car journey with a variable velocity, something that calculus will address later on. The difference? No infantilisation; instead a clear and applicable metaphorical example.

mathgenius 7y ago
This question "What are numbers?" I find way more fruitful to contemplate than any answer I have so far found. And why should we be adults anyway? Adults already know all the answers. Ask this question like a five year old would.

prvc 7y ago

From reading a few excerpts only, I am reminded of a lot of popular science writing, which too often adopts a condescending casual tone full of flimsy, inaccurate metaphors and chatty asides, and which too often omits simple statements of the details of the subject at hand (often less complex than the verbiage used for the adulterated version). The writing becomes so far removed from the subject, that while those already familiar with it will be able to guess what the author was getting at, those who are not will remain in the dark, leading to a zero increase in the reader's knowledge in each case. Despite being in the former category with respect to basic calculus, I can't help but feel a sense of umbrage on behalf of the latter.

itronitron 7y ago

Khan Academy is a better resource for being introduced to calculus. The title of the course is unfortunately perpetuating the mistaken stereotype that artists are not mathematically inclined. The incoming class my freshman year of the art school at the university I attended had the highest median Math SAT score of any of the other schools (i.e engineering, science) in the university.

nemetroid 7y ago

Despite the title, I struggle to see who exactly this was written for. The Javascript widgets are quite neat. Fiddling around with parameters, and receiving instant graphical feedback, is great for developing an intuitive understanding. But I find that the writing has several issues that probably make it less clear to beginners than your average calculus textbook: The writing is very brief, with typical mathematical terseness. This is usually a good thing, since it lets you be very exact in your definitions. But to be accessible to a beginner, this writing needs to be backed up with concrete examples, and preferably a picture or two. In most cases I find that it isn't, so the reader needs to internalize a lot of things on their own before being able to move on. The text is also sparse on showing its work. A positive example is near the bottom of 5.1, showing how to derive the quotient rule, but most of the time it looks more like example 1 in 6.1, with a lot of information given inline before actually applying the mentioned operations to the expression: > Suppose we substitute the function g which has values given by g(x) = x² + 1 into the function f which takes values f(x) = x³ − 3. > The substituted function f(g) has values f(g(x)) = (x² + 1)³ − 3. > Let us compute the derivative of this function. The derivative of f(s) with respect to s is 3s², while the derivative of g(x) with respect to x is 2x. > If we set s = g(x) which is x² + 1, and take the product of these two we get: > [expression] This type of mental expression manipulation is fine for someone that's had practice with it, but probably not for a beginner, who would gain a lot from having these kinds of things written out in a more structured way. In these aspects, I find the text less clear than the calculus textbook i used at university (which was not directed toward beginners or artists). I agree a lot with prvc's comment[0] about being able to fill in the gaps, in that the terseness and handwaviness can make this look like a beginner-friendly version to someone that already is familiar with the subject, but I don't think changing tone is enough to make something beginner-friendly. 0: https://news.ycombinator.com/item?id=20435238 https://news.ycombinator.com/item?id=20435238

krick 7y ago

A point as minor as they come, but I really am not a fan of that embedded chapter structure. It's absolutely ok to have multiple sub-topics in a topic (that's why TOC exists in the first place!), but it also makes it so much more enjoyable to read, when you can simply go through text linearly, without worrying about what link on what level you clicked just before that.

jedahan 7y ago

I want the inverse so badly. Call it 'Art for Beginners and Mathists'.

barry-cotter 7y ago
If you want a book Drawing On the Right Hand Side of the Brain http://drawright.com/ http://drawright.com/ If you want a video course http://drawabox.com/ http://drawabox.com/

WheelsAtLarge 7y ago

I took 2 semesters of Calculus in high school. It was a "requirement" to get college AP. I was convinced that I would have failed at college if I didn't take it. I have yet to use it in my career. A waste of time. A 2 semester class of drawing or Art would have been way more useful. Why is calculus held in such high esteem in college prep? Most students can safely skipt calculus. A 1-day general introduction would be enough for 90%+ of college students.

mlevental 7y ago
90% of college students aren't stem majors and so what you're saying is almost tautology. for stem there is literally no branch that doesn't use calculus in at least some way.

RichardCA 7y ago

I never really felt like I understood Calculus until I watched the Mechanical Universe on PBS. Here's an example, demonstrating how Kepler's Second Law is an expression of the conservation of angular momentum. https://www.youtube.com/watch?v=0tSmY4fn5xY#t=7m50s https://www.youtube.com/watch?v=0tSmY4fn5xY#t=7m50s

theNJR 7y ago

Just last night I was complaining to someone about wanting this very thing. I suppose I should now accuse Hacker News or listening in on my phone.

deleted 7y ago

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mrcactu5 7y ago

Robert Ghrist has written an colorful and innovative calculus text. https://www.math.upenn.edu/~ghrist/calculus.html https://www.math.upenn.edu/~ghrist/calculus.html https://www.math.upenn.edu/~ghrist/notes.html https://www.math.upenn.edu/~ghrist/notes.html https://sites.math.washington.edu/~morrow/334_13/FLCT.pdf https://sites.math.washington.edu/~morrow/334_13/FLCT.pdf