Essence of calculus, chapter 1

Hello Grant.. I agree with Erik Haugan- I would love to see you explain differential forms and Stokes' theorem. But I understand if that's not a priority right now. Brilliant videos- keep up the good work!

I've always thought of the "d" values not as actual values or "infinitely small" values, but as having its own sort of "dimension" and not changing its identity when interacting with actual numbers, somewhat like a variable. I would treat it like it has a value but one that's so small, in a level of smallness not comparable to any other number and thus behaving like it's a variable, that it has no significance.

Darwin Kim

Hi Grant, I'm a little late but amazing video! This video finally helped me understand why the derivative of the volume of a sphere is its surface area. Also, I don't know if you have thought about this in making your videos, but I recently came across a short article about mathematics educators using approximation as a unifying theme of calculus rather than other ideas such as rate of change: <a href="http://launchings.blogspot.com/2017/01/ijrume-approximation-in-calculus.html" rel="nofollow noopener" target="_blank">http://launchings.blogspot.com/2017/01/ijrume-approximation-in-calculus.html</a>

I thought I remembered seeing something similar some time ago. It was here: <a href="https://betterexplained.com/articles/a-gentle-introduction-to-learning-calculus/" rel="nofollow noopener" target="_blank">https://betterexplained.com/articles/a-gentle-introduction-to-learning-calculus/</a> It was a while ago but I remember that "unrolling the circle" idea being so illuminating for me I just couldn't believe I hadn't seen it before.

Jake Palmer

Not at all, please fire away!

3blue1brown

Hello Grant. New Patron here and probably late to the party. Is it too late to share an opinion on this video?

Hi Grant, I would just like to second the wording Dan Davison suggested about the error. I came away from the video feeling like calculus was more about approximations than I feel would benefit incoming students. I like the phrasing of the approximation "getting less wrong," but I think adding in the fact that 'the approximation becomes exact when the change is "as small as possible"' would help solidify the credibility of calculus in students' minds.

Hey again. Just wanted to say that when I watched the video when it was first posted, I had some of the same reactions as others here, in particular the way you treat errors and avoid limits. I watched it again just now, and realized that my reactions were really just caused by the surprise of not seeing the traditional approach. The second time, there was naturally no surprise, and therefore also no negative reaction. I really think your presentation is brilliant, despite the formal objections.

I like this, very promising. One thing I would love to see is vectors and differential forms, even if you choose to avoid the jargon and stick to one dimension. Or, still in one dimension, the difference between scalars and pseudoscalars/one dimensional vectors. An example: Imagine walking up a hill picking blueberries. In order to calculate the amount of blueberries you pick, you can integrate the blueberry density. Similar if you start at the top of the hill and pick your berries on the way down, it does not matter which way you go. Contrast this with calculating your ascent/descent by integrating the slope of the hill. Now direction matters! Why?

No particular reason. Maybe subconsciously I was cautious about saying 'inverse' because you then have to be a little more cautious about the specifics (+C's and all that).

3blue1brown

Very nice! I did think the explanation of dr as small but not infinitesimal was maybe a bit belabored. I think I like the idea of not distinguishing between upper case delta and d. Students are going to fall mostly into two camps I'd guess: (1) new to calculus, and (2) have been exposed to limits. You talk about an error term. Isn't there a risk that the audience in (1) will come away thinking that calculus is a rough approximation? Wouldn't both audiences be happy with saying something like "My take is that its best to think of this as a tiny but finite value like 0.00001, and not to worry about the error because you can choose as small a value as you like". Related, isn't 0.1 a bit big for an example number?! People can sort of visualize 0.1 (like mm on a ruler) whereas 0.00001 is "tiny"!

Dan Davison

As a high school calculus teacher, I cannot wait to share the complete series with my students. One question: Is there a reason you chose "opposites" instead of "inverses" to describe the relationship between derivatives and integrals?

This is somewhat of an odd stylistic nitpick, but I think the Pi creatures' arm movements could use a little more "snap" to them - quicker timing with a little anticipation and overshoot. Might make them feel a little bit more organic.

Kevin McCurdy

Hi, Grant, I guess it's fair to say that the video won't make much sense to somebody hearing about calculus for the first time. Viewers need to struggle a bit with the theory on their own and get at least some of the concepts in place before these videos (which are great, btw) will make sense. Each little example needs time. Maybe you should tell this to your viewers. Assuming that there is such a thing as the absolute best possible way to explain these things (and I reckon your videos sure do come close!), it will still require viewers to put in some time and effort the old fashioned way. Speaking of which, I have been struggling myself, for some time now and without much success, with the proof of Euler Lagrange equation (just out of curiosity, I am too old to ever need it for anything including exams). Do you think we will get to that in this calculus series?

Hey, Grant. This is a really fascinating video. But I think that a lot of your explanations work around differentiation from first principles - the old f'(x) = [f(x+dx) - f(x)]/dx. This makes concrete the dA you showed in the video, as (slightly larger area) minus (regular area). It also provides a clear link to the tangent-line interpretation of derivatives (since that formula is essentially identical to the slope formula of a line). I think including it in this video could be pretty helpful.

Karthik T

Great video, Grant. I was wondering about the function going through the mean or the ceiling. I'm not quite sure which is correct - if it's the RRAM (right Riemann sum) or MRAM.

Josh B.

Overall I really liked your video. One thing that made me think was the length of your presentation. Even though the video conveys the most important idea of basic calculus, 16 min. might be too demanding. Did you consider splitting this video into two (maybe right before introducing integrals)?

Hey Grant!

Hey Grant!

Wonderful explanation. But.. I know this is supposed to be an introduction to calculus but will you go over some functions that are not continuous and how the Riemann integral fails at those ones and that there is something much more powerful (Lebesgue) that can deal with it? Just to let people know, that there isn't just one way to visualise an integral.

Thanks for the feedback. I went back and forth on the delta front, favoring simplicity in the end. That said, I think you are correct that a quick nod to the alternate convention is worth throwing in.

3blue1brown

Hey Grant, my name is Oscar and I'm a math student living in Mexico. I would like to propose something, I saw that you wanted to make your videos in other languajes. So I would very much like to work with you on translatting your videos, you already have the animations and the script. Send me an email, an we could talk about it.

Oscar Ivan Miranda Alcocer

I agree about the speed. A bit fast for the calculus naive I suspect.

Wonderful video. I really liked the circle example. Perhaps it is because it is different than usual ones, but I also liked the way the graphics were able to tear the whole proposition away from coordinate systems and graphed functions. I was a bit surprised by how you treated limits, but especially the notation. You ask to think of dr as an error, but then you explain how it disappears. But this is just the limits approach, wrapping "delta" and "d" up into the same package. Maybe that makes sense. I can see the value of the simplification, but it also strikes me as potentially confusing. Maybe not, but if you have a limits and calculus naive test audience, probably not the Patreon crowd (or am I making unwarranted assumptions?), it might be good to take a test spin. Maybe a nod to "deltas" would be helpful in this or a later video, even if just "you might be used to seeing this called "delta", but we're going to simplify things and just think of it all as "d". Thanks for your work and the opportunity to weigh in.

I feel like the many equations at the start goes a bit fast/is a bit much. It feels like you're changing back and forth between the topics from about 3min to 4min. I'm really not sure how to change that though.

Scott Ramsay

Hey Grant, Loving your work, it has made my day seeing your new videos for a while! At 2:27 the bracket looks a little clipped at the top At 10:32-10:33 would you think it beneficial to highlight the extracted strip "dr" shown? At 13:12 &amp; 15:31, the function y=x lies along the minimum of the bars; shouldn't it go through the mean, rather than the ceiling function? All best, Jack

The circle example really works for me.

Doug Fort

Limits will come later, in their own dedicated video, but my philosophy is that knowing what derivatives/integrals are trying to be can be a good motivation for limits, which otherwise feel like kind of an arbitrary thing to be defining.

3blue1brown

I think you're right, the full unwrapping is too pretty not to add in somewhere. I actually wrote all the code with that in mind, then just never included it.

3blue1brown

Thanks Jake! Really good thoughts. The point about hairy algebra is a good one. On the one hand, I have to assume at least some fluency. At the same time, I think what most students struggle with when it comes to calculus is not the new concepts that come up in the course, but the need to have everything that came before it mastered. I should probably communicate this somewhere early on, and suggest that people who want to buff up on their pre-calc knowledge check out Khan Academy or PatrickJMT. I plan to talk about velocity/displacement as its own example, potentially as a video that parallels this one. Otherwise, they'd come up as common examples in future topics. They were actually in the original script for this video, but length got the best of me. Thanks Again, -Grant

3blue1brown

Hi, nice video. One thought, I always thought of the area of a circle as the circumference multiplied by the radius divided by 2. The reasoning that helped me remember this might help illustrate your point more cleanly. So my reasoning goes as follows: - The circle can be thought of as a bunch of slices/wedges/triangles that when rolled out make a spiky strip where the base of the strip is a bumpy circumference of the circle. - 2 of these strips of slices/wedges/triangles can be put together so the wedges fit together tightly and the resulting shape is a parallelogram. \/\/\/\/\/\/\ - the smaller these slices are the more accurate the area and circumference is, and the more straight the top and bottom of this parallelogram are. so the idea of making the slice smaller/larger would get you closer/farther from the exact value of pi, with some rearranging of-coarse. This may however be too simplistic to represent derivatives visually, and what works for me may not work for everyone. Anyway, loved the video, thanks.

Nice, Grant, I had a lot of thoughts. At one point early on I thought you already might have lost a chunk of your audience just when you expand a binomial. Of course, I'm sure you already agonize over details like this, but it raises an interesting question of what you ought to expect of your audience. Would you care to comment on what your philosophy will be toward the "hairy algebraic details"? Upon some thought, binomial expansion is unavoidable in calculus because, I could argue, the simplest "non-trivial" example of calculating a slope explicitly by difference quotients is calculating that of a quadratic curve. So what will you say to your audience to address that? Or not explicitly address it? Totally separate comment here: I was a bit surprised by your choice to make your first example of the FToC about the circle area problem. I mean, it makes sense I guess, but ... I dunno, I humbly believe the velocity-displacement relationship is more intuitive? Perhaps you're motivated by a desire to make this more of a pure-math and less of an applied-math channel? I'm curious if you considered using kinematics as a starting point for examples and decided against it consciously? Maybe there's a place for both examples in this series? I might watch again and send some time stamps of things I liked or didn't like, but these were my initial thoughts and questions. Thanks Grant! - Jake Mirra Department of Mathematics University of Pittsburgh

Jacob Mirra

Are you going to be introducing the concept of a limit? Or would that detract from the intuitive visual style that you're so darn good at?

Joseph Cutler

One issue you don't address is that stretching a ring into a line requires distorting the line. It's not just the little bits at the end that contribute to the error. Obviously it's a small amount of stretching for small lines, but might be worth explicitly calling out if you redo any of this. That said, it's fantastic and it's worth releasing as is.

Adam Berkan

I was sure that when discussing the integral, you were going to unwrap all the rings constituting the circle onto the coordinate plane. That would have made lots of sense, do you think it’s not yet time for that? Also when you explained the complex derivative elsewhere, as in, the more you zoom in, the more a function looks like multiplication, that actually made the most sense to me of all the explanations of the derivative I heard.

Roman Odaisky

Fantastic! I can't wait for the public version to go up so I can show my friends :D

Joseph Cutler

Overall, great video. Sticks to your style. I think it's beneficial to retain consistency among different series. As an engineer, I generally learn best through visual/concrete examples. Starting with dR as an actual number really helps to cement this concept. I also think symbolic representations for area/circumference rather than variables will help those learning for the first time that might otherwise get confused by too many variables. The graphics/animations were easy to follow and synced well with the audio. I do wonder if talking about the tangency relationship this early might confuse those seeing all this for the first time. I do understand your reason for doing this, and believe it's beneficial to point out since it is usually presented this way. I would just also be interested to see how it is received by a new calculus student vs. someone already familiar.

Trevor Bruns

Good point, I should probably start by writing dR, and call out when and why I write dr.

3blue1brown

I'm right there with you on the tau vs. pi front, but for a video like this it's best to meet students carrying what they're already familiar with.

3blue1brown

Beautiful as always. Being a militant supporter of the Tau Manifesto [1] I only lament that pi = tau / 2 gets in there to muddy the waters. ;) [1]: <a href="https://tauday.com/tau-manifesto" rel="nofollow noopener" target="_blank">https://tauday.com/tau-manifesto</a>#sec-quadratic_forms

Jared Tobin

Great video! I really felt a sense of accomplishment after I watched it, and I think it is very important to build self-confidence in students in that way. One little issue that I noticed is that at 9:58 Grant defines a ring's radius "r", but we have already used "dr" before (on screen at that time) to indicate the thickness of the circumference. I felt that something is wrong here since "r" is defined after "dr" (change in r) is defined. Shouldn't it be the other way around?