Unfortunately, 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.
You 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.
I think I did, but I'll be more precise. The first problem is that the FTC is reduced to a tautology. But the more serious problem is that the definite integral is _defined_ via the antiderivative. This is wrong, as the existence of integrable functions without antiderivative shows.
I'm all for simplifying material, and I'm not saying this needs to be a mathematically rigorous treatise, but it should get the basic maths right, because there is already too much confusion out there.
FWIW, I think the FTC is deeply important and used all the time, but the first part of it - the one about writing antiderivatives as integral - could probably be omitted for non-mathematicians.
It would probably enough to establish:
- Here's what differentiation is. Here's some rules.
- The inverse of that is antidifferentiation. You compute it by going in reverse, but note that this is harder than differentiation (which is mechanical)
- Then we have the integral via Riemann sums. Maybe you can show some simple example.
- Now comes the deep result: the integral can be expressed via the antiderivative, if it exists.
You don't have to go into full mathematical detail, but this presentation wouldn't be getting it wrong anymore.
So... if it were made clear that the discussion was specific to continuous functions, your criticism would disappear? Is that really worth the level of your dismissal?
I mean, that's how it's actually taught to high school students, after all. Complexities like discontinuities are absolutely not part of the initial curriculum.
Alternatively, if you think a simple introduction to calculus requires a discussion of discontinuous functions and the fundamental theorem of calculus before it describes how to compute a definite integral, I retreat to my earlier point: that's ridiculous nitpicking.
Basically: you want a textbook. They make those. This isn't that, it's an intuitive guide for people who found the treatments in textbooks opaque. Those people exist, they want to learn this stuff too, and they aren't well served by people like you just yelling at them to read the textbooks they already have.
I don't feel you've actually read and understood my criticism.
The fact that the theory falls apart for discontinuous functions is just a visible symptom of the deeper problem that the author confuses a definition with a theorem. I don't think this is just theoretically wrong, it also makes no pedagogical sense. To say "we define the integral as being the difference of the antiderivative at the two endpoints, but we can also define it as the Riemann sum" doesn't tell the student anything except "mathematics is magic, don't even bother to understand it". Whereas to be genuine and say "we can prove that the integral can be expressed through the antiderivative (though we choose not to do this here)" doesn't treat the reader like an idiot.
I don't think it's impossible to provide an intuitive and non-proof-based account without sacrificing getting things right conceptually. In fact, I provided a template for doing so above.
> To say "we define the integral as being the difference of the antiderivative at the two endpoints, but we can also define it as the Riemann sum" is [like, really bad!]
How about if there was a footnote on that point explaining that these two definitions are actually equivalent (for the continuous functions we're talking about) and that the proof is really interesting and can be found in your textbook? Would that meet your requirements?
Again, this is just nitpicking. In fact those two definitions are equivalent, as you keep pointing out. You don't have to prove everything in an introductory treatment, and in fact even high school textbooks on this subject don't even try.
The 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.
Yeah well - next time I read a proof by a mathematician that skips several steps that to mathematicians are unimportant or "obvious" yet fundamentally blocked my understanding of the proof overall - I'll think on this reaction and smile...
Or maybe even just write some mathematical overviews myself that leave out stuff that drives mathematicians insane with frustration...
I sympathise with your point of view and I agree that sometimes mathematicians are not keeping their audience in mind enough. You don't always need completely rigorous proofs for non-mathematicians.
But I think that it's a false dichotomy to only consider "extremely rigorous" or "simplified to the point of being wrong" as the only teaching philosophies. There is a middle ground where you can be somewhat hand-wavy and inexact without getting the conceptual broad strokes wrong.
Would you write code with syntax errors and blame the compiler for not understand how smart you are, or would you just learn the language and write correct code? Your frustration shows lack of understanding and appreciation for mathematics, when in reality maths and computer science are two sides of the same coin. If you can be good at one, you can be good at the other :)
> Would you write code with syntax errors and blame the compiler for not understand how smart you are
No, but you might tolerate example code with mild syntax errors (like using a "..." to skip unrelated material, or skipping import declarations and boilerplate to better clarify the point being made). You don't demand that every example in a programming tutorial be a fully buildable piece of software, why are you demanding that a simpified calculus tutorial exhibit full rigor?
> why are you demanding that a simpified calculus tutorial exhibit full rigor?
Aren't they just suggesting a marginal amount of rigor to communicate that the "definition" is in fact based on justifications which render it non-arbitrary?
I think for concepts like this different people might prefer different explanations, but I did some googling and this page seems to describe the heart of the theorem pretty well IMHO:
my own strategy is to try to get it explained to me in as many different ways as possible; I think it's often easiest to learn by integrating a bunch of different viewpoints of the same thing.
on the other hand, I haven't had a lot of luck with calculus, so who knows.
IMHO many people struggle with maths because they're not used to the fact that you have to read maths differently. You can't just digest it quickly like many other subjects. You have to ponder over definitions for a while and play with them and try to come up with examples and counterexamples.
Nope, I'm not any good at any of the maths. My own model is that it's like reading, you need a critical mass of vocabulary and/or grammar (one can make up for the other, to some extent) before you can understand much of anything. with most subjects, if you understand 80% of the paragraph, you can fill in the blanks pretty easily, and thus your learning rate goes way up.
That, and I think like reading, it takes a lot of practice, and that practice is really hard before you are good enough at reading to see the story. I think I'm at the point in math where I can't read it well enough for it to tell me a story, so practice is a real slog, just like it was for reading at that point.
Don't despair too much; maths is a hard subject and even the best mathematicians struggle with new material. You're dealing with crazily abstract ideas and your brain just has to get used to them. That takes time.
If you're having trouble with specific material (e.g. calculus), sometimes the reason might be that there's something else that you don't quite understand well enough yet that is a prerequisite. In such a case, the first step would be to identify what that is (it could be unfamiliarity with trigonometry or a lack of skills in algebraic manipulations) and then go back and refresh that.
This 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.
Nice 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.
Even 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.
author 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!
Chinese 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.
Let 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.
Thanks 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.
If you really want to understand these things, start here: 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.
A 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.
If 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.
As 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.
You 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.
Yeah 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.
Cool. 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 :)
In 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.
But 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.
I’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).
I'd be inclined to agree, but parent poster was concerned about solving problems numerically, which is trivial for line integrals, as they are just sums of triangle hypotenuses.
I 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".
This 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.
I 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.
The linked Calculus One by Jim Fowler is no longer available on Coursera, but at https://mooculus.osu.edu/ you still have access to the lectures and the textbook.
I 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.
I used umbrella, because statistical signal processing seems to include DSP concepts + probability concepts + random process + Control Theory (Markov stuff)
Calculus 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.
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.