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Calculus Explained with GIFs and Pics (2014)
- wilkystyle 7y agoAs a visual learner, I have always struggled to intuitively grasp mathematical laws and concepts from words or equations in a textbook. This type of illustration makes so much sense to me.
- originalvichy 7y agoYeah for some reason learning mathematics in school always felt too abstract to me. The only courseS I really excelled in were probability and geometry because I could more easily visualize and conceptualize the topic as opposed to lines and positions in 2D space.
- GuB-42 7y agoYou should try to do 3D graphics and maybe some demoscene/shadertoy stuff. Helped me tremendously. 3D graphics is mostly vector maths and linear algebra but you can do some calculus too. Derivatives are tied to continuity and continuity is tied to smoothness, which make things easy to visualize.
- archibaldJ 7y agoauthor here; i wrote this quite some time back (2014). I'm now studying Chinese and would like to write a Chinese version for Sinophone audience (with more Chinese humor, etc). Anyone knows a good place to share articles like this if it's in Chinese? Thanks!
- tata12121212 7y agoThanks for writing and sharing this. I really wanted to read the linked essay "What do we talk about when we talk about limit", but the link is broken.
- hackermailman 7y agoat 5:30 Wildberger goes through the definition of Limits https://youtu.be/K4eAyn-oK4M?t=331 https://youtu.be/K4eAyn-oK4M?t=331 If you really want to understand these things, start here: https://www.youtube.com/playlist?list=PL5A714C94D40392AB https://www.youtube.com/playlist?list=PL5A714C94D40392AB and watch them as you work through Analysis I by Terence Tao, skipping to chapter 2 where he begins from scratch introducing the naturals, integers, etc. Wildberger even shows you how to do algebraic calculus which will help to learn the Tao book when he goes about rigorously introducing calculus defined using limits. For me that helped anyway, seeing Naturals, Ints, Rationals, and Reals constructed from scratch with all their laws/operations defined and proved by Tao then him using analysis to introduce how these tiny changes in calculus work.
- dhdc 7y agoChinese here. A good place to start would be 知乎，zhihu.com, or chinese quora. I'm by no means skilled in mathematics(still an undergraduate after all), so there could be some more specialized websites/forums I'm missing here.
- NieDzejkob 7y agoHi. Your math notation gets cut off on mobile. Consider allowing horizontal scrolling.
- jacobolus 7y agoLet me recommend scaling equations down to fit the viewport width instead of scrolling. If you shrink equations, then if someone wants a closer look they can zoom in/out. But if you have them scroll off the side of the view, it is impossible to see the whole equation at once.
- blululu 7y agoThis is very nice work. If your goal is disemminating this work for pedagogical purposes I would strongly encourage you to add some of these animations to the relevant Wikipedia pages. A lot of students turn to Wikipedia and these kind of animations can be very helpful.
- caro_douglos 7y agoAre there any self-taught type of classes that utilize Wikipedia for say history?
- deleted 7y ago[deleted]
- bobharris 7y agoI need this for the umbrella of statistical signal processing.
- throwawayhhakdl 7y agoCalculus is super intuitive, but the algorithms are both difficult to memorize and involve a lot of guesswork if your pattern matching isn’t well honed (integration, specifically). I would advocate teaching calculus with charts and stuff to demonstrate the logic of if, but not require the solving of problems. Most of it is a waste of time, but the concepts are useful.
- MrQuincle 7y agoIt is also nice for yourself to do. One time I created this visualization using vis.js of the Legendre transform: https://www.annevanrossum.com/blog/2015/08/08/legendre-transform/ https://www.annevanrossum.com/blog/2015/08/08/legendre-trans... It made it click for myself and it's fun to do!
- xerox13ster 7y agoI'm familiar with these concepts but the notation used in the article ironically looks like Greek to me. I'm used to classic mathematical notation.
- larnmar 7y agoI’ve often thought it very unfortunate that we leave calculus so late in K-12 education, and for some students never get there at all (at least in my country). I remember spending most of my childhood mystified about what this mysterious “calculus” thing was, and then finally getting around to learning it and saying “oh, it’s just that?” And then once students are initiated into the (disappointing) mysteries of calculus, what next? They spend the next few years mechanically carrying out analytic differentiations and integrations of a bunch of ever-more-artificial one-dimensional functions. Do you know what the integral of tanh (x + x^x) is? You shouldn’t, because it doesn’t fricking matter! Most of the actually useful integrals that anyone would ever calculate in real life turn out to be only solvable numerically. In conclusion, put an introduction to the intuitive ideas behind calculus earlier (Year 7), emphasise numerical over analytic solutions, and use the time saved to move on to things like multivariable calculus (and path integrals, which I still don’t understand properly).
- helpPeople 7y agoFunctionally I see many non math people struggle to understand rate of change. I find integrals useful. Also I see my computer science pals have a lack of faith in math. I don't understand why.
- kharak 7y agoWhat do you mean by lack of faith? Not trusting mathematical proofs?
- macintux 7y agoI can't speak for GP, but I'd speculate "lack of faith that understanding the math behind computer science will provide any real value as a programmer".
- kharak 7y agoAh, then I'm right behind. But it's not that more Math knowledge wouldn't help at all, just that the cost to benefit ratio seems not worth it. I never needed actual Math in my career so far.
- haolez 7y agoI loved the parts where the author shows who created the notations used and what was their context and way of thinking. I tend to get distracted when the notation in a field is awkward to me. Statistics confuse me with notations that resemble function applications, but are not. I like how physics customize the notation to their needs, like the use of brackets in quantum mechanics.
- Tainnor 7y agoUnfortunately, parts of the section about integration are wrong and contribute to some of the widespread confusion about the FTC, which often seems reduced to a mere tautology (which it's not; it's a deep result). First, the "indefinite integral" is really just the antiderivative (and as such, "the" indefinite integral is only unique up to a constant). This doesn't follow from the 1. FTC, but purely by definition. OTOH, the definite integral is not "defined" as being the difference of antiderivatives; it's defined in terms of Riemann sums (at least in elementary calculus), as explained further below. What the two parts of the FTC do is proving that those two notions, which have no a priori reason to be related, are in fact related in a particular way. The first part of the theorem says that, if f is "nice" (in particular: continuous), the antiderivative exists and can be expressed through the definite integral with a variable upper bound. The second part says that, if f has an antiderivative, the definite integral can be computed using that antiderivative. It's important to keep this distinction because there are e.g. functions that are integrable but don't have an antiderivative (e.g. a function with a "jump"). In such a case, the FTC tells you nothing and you have to go back to Riemann sums to compute the definite integral.
- swimfar 7y agoFTC=Fundamental Theorem of Calculus
- ajross 7y agoYou never got to the parts of the section about integration that are wrong. Broadly: your criticism is a mathematician's nitpicking. I don't know that that really has much value to someone learning "Calculus" in a practical sense about understanding and working with functions. And frankly... I mean, the fundamental theorem of calculus is a historically important result and a relatively profound truth, but it's not that big a deal. If you really want to play the "you really need to know this important truth more deeply" game with amateurs, start with the incompleteness theorem or something.
- yshklarov 7y agoThe article purports to teach calculus, so it should get the basics right. The Fundamental Theorem of Calculus is called fundamental for a reason: it's the most important concept in the entire subject. It's not just a technicality. The article defines things in a non-standard way so as to reduce FTC to a tautology, and nowhere does it correctly explain the connection between differentiation and integration (which can -- and should -- be understood visually and intuitively). In short, I'm not convinced that the author knows what he's talking about.
- Mugwort 7y agoA bird in the hand is worth three in the bush. If I could give someone from a non-mathematician background, e.g. programmer, who sincerely wants to get a peek behind the curtain and know what calculus really IS without diluting anything in an unacceptable way I'd show them I. Gelfand et al "Sequences, Combinations, Limits". Once you have the idea of a limit of a sequence down it isn't hard to learn the limit of a function (epsilon, delta) and move on from there. A good second step is Apostol's calculus vol. 1 just the first chapter where he calculates an integral from scratch with no "fluff" (details you don't need just yet). (Yes, learning integration first actually makes more sense than learning derivates.) Once you do that you can continue with Apostol (highly recommended.) or use Spivak's calculus (tougher but even more highly recommended). To wrap up Gelfand is one bird in the hand, Apostol chapter 1? with integration from stratch is another bird in the hand. That would be a good first two steps to get a sense of what calculus is really about.
- blazespin 7y agoI tried explaining this to someone but struggled a bit with Motivating. I think in math it’d be nice to start out by explaining why limits and integrals are useful.
- 40four 7y agoCool. Seems like a good, quick and dirty run down of all the core concepts. Funny, I don't think I realized it at the time, but is calculus the first time we are formally introduced to the concept of a function in math? If so, that gives me a new found respect for calc class :)
- billfruit 7y agoI think the concept of function is normally introduced during teaching set theory rather than calculus.
- p1esk 7y agoI think function is introduced in elementary algebra when dealing with expressions like y = 2x
- gizmo686 7y agoIn elementary algebra, functions aren't really first class objects you can manipulate. That is, if you have f(x) = 2x and g(x)=x^2 , you might ask questions like "what do they evaluate to at A?", or "what is their minimum?", but you wouldn't ask questions like: "what is f + g". Calculus is the first time in the standard sequence when you start manipulating functions as first class objects.
- p1esk 7y agoBut we are talking about introducing a concept of a function. Which is a mapping between values of x to values of y. That's it. No need to manipulate anything.
- hota_mazi 7y agoNice illustrations but that article makes the same mistake that a lot of posts of that type often falls to, namely, assuming the reader has a similar background as the author: > We all know that the slope of a linear equation/function has a constant value Er... no. We don't "all" know that. Actually, I'd say that 99.99% of the population of the planet has no idea what that means.
- katzgrau 7y agoEven the examples provided are kind of Math-heavy (for example, jumping right into the notation of the limit of x^5/25) - enough that a novice would probably feel overwhelmed. I find that people who are very familiar with a subject usually have a difficult time explaining it without slipping into jargon that will lose the listener.
- reubens 7y agoIs there a word for this effect in education? Where the teacher forgets the context of the learner?
- ci5er 7y agoAssuming a learner of calculus has already learned basic algebra?
- rohan_shah 7y ago"The curse of Knowledge"
- dredmorbius 7y agohttps://en.wikipedia.org/wiki/Curse_of_knowledge https://en.wikipedia.org/wiki/Curse_of_knowledge Also Shoshin, "beginner's mind": https://en.wikipedia.org/wiki/Shoshin https://en.wikipedia.org/wiki/Shoshin
- drsopp 7y agoExpert blindness.
- unethical_ban 7y ago>Gottfried Wilhelm Leibniz, a great German mathematician, came up with this notation in the 17th century when he was still alive. I would be impressed if he came up with it before or after that time.
- mraza007 7y agoTo be honest I love the resource man You have put great effort into this Love it keep it up!!!
- ducaale 7y agoThe essence of calculus series by 3Blue1Brown https://www.youtube.com/playlist?list=PLZHQObOWTQDMsr9K-rj53DwVRMYO3t5Yr https://www.youtube.com/playlist?list=PLZHQObOWTQDMsr9K-rj53...
- teichman 7y agoIf you like this site, check out (a) 3Blue1Brown's youtube channel for intuitive visualizations of (otherwise) hard math topics, and (b) the book Infinite Powers, for more calculus intuition + history. Both are outstanding.
- rsa4046 7y ago+1 3Blue1Brown's channel is indeed a treasure.
- happy-go-lucky 7y agoThe linked Calculus One by Jim Fowler is no longer available on Coursera, but at https://mooculus.osu.edu/ https://mooculus.osu.edu/ you still have access to the lectures and the textbook.
- mathgenius 7y agoThis presentation of calculus is bog standard. I can't quite fathom how a few little gifs can make people so excited. Is it the Will Ferrel meme? Apart from that, this is dull as dishwater.