The essence of calculus
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- Опубликовано: 27 апр 2017
- What might it feel like to invent calculus?
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In this first video of the series, we see how unraveling the nuances of a simple geometry question can lead to integrals, derivatives, and the fundamental theorem of calculus.
Thanks to these viewers for their contributions to translations
Dutch: @LFWarsen, Lauri Warsen
Hebrew: Imri, Omer Tuchfeld
Hindi: Vinayak
Hungarian: homok43
Italian: @Deye27, @hi-anji, alberto alessi
Korean: Hana Seo
Portuguese: redkk123
Spanish: @agustin-j
Vietnamese: @ngvutuan2811, ngvutuan
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These animations are largely made using manim, a scrappy open source python library: github.com/3b1b/manim
If you want to check it out, I feel compelled to warn you that it's not the most well-documented tool, and it has many other quirks you might expect in a library someone wrote with only their own use in mind.
Music by Vincent Rubinetti.
Download the music on Bandcamp:
vincerubinetti.bandcamp.com/a...
Stream the music on Spotify:
open.spotify.com/album/1dVyjw...
If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then "add subtitles/cc". I really appreciate those who do this, as it helps make the lessons accessible to more people.
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My calculus professor usually just says “lol here are some formulas and rules to remember”. Higher education at its finest.
Everywhere to be honest not only the US. Its damn sad.
Im a physics student and I had to learn calculus like this. But when calculus was used in QM and EMT I shit my pants cause I couldnt picture anything. Nothing made sense. Now im going through calculus again by myself. -__-
@@OmarChida Not where I am studying. Where do you get these garbage professors? Even online courses in MIT do not do this.
@@RanEncounter why would mit even do this they have a face to save, greedy school and way worse in college hire teachers at low cost and are generally of these sort
@@kaus05 Have you even looked at the lectures I mentioned before you commented?
Can we all appreciate how much effort Grant puts into the animations in his videos so we can understand?
of course
It's Python! He animates everything using python! SOOO COOLLL!
A python tool he created btw
Actually!!!
@@adarshas8454 really? Which one is that?
“Math has tendency to reward you when you respect it’s symmetry” - has easily became my favorite math quote!
That quote is very relatable with my work.
Very true. It's thankfully easier considering we can work with geometry where all objects have equal height which thankfully allows us to calculate a change 'dr'
Yess.. Perfect lines said..
If it's your favorite quote then maybe you should use the correct "its".
Thanks! I'm a math tutor, and I must say that 3Blue1Brown is a remarkably interesting, insightful series. Thank you, Grant.
I'm a high school student from Brazil, and I can say that you've inspired a generation of people to pursue math and physics for their own sake. I started to learn calculus through these videos 4 years ago and now I'm studying for the AP Physics C exams (wish me luck lol). I know you'll probably never see this, but if you do, just know that we really appreciate your work.
Boa prova mano (se já não fez kkk)
Good luck buddy
Como assim tem AP no Brasil?
How did the exam go? I hope well :)
Excellent!
How I wish I'd had access when I was in high school and college to this information. We were 'taught' by Rote.. but it never made sense to me because the Purpose was hidden away..
Good luck with your studies!
Blessings!
The reason I like math is because I’m obsessed with using logic to find answers rather than memorizing. That’s why I loved these type of videos.
Winter Hippo same,
I also like physics
Agree. Logic to derive, in my opinion, is the understanding at it finest. A computer can calculate...it takes a consciousness to UNDERSTAND...to comprehend MEANING. This is overlooked these days.
Exactly! Logically understanding how math and physics work the way they do >>>>> cramming formulas
How do you feel about quantum mechanics then ? If you are a true beaver it will drive you nuts cause you can't find the missing stick....
Yeah thats great and all, but after the first couple of weeks in maths (University) you will have to memorize a lot too to formally prove things... unless you always want to start from scratch and that won't work out very well
5 years have passed but this video still brings inspiration to many to love calculus even now❤️
for sure.....
does anyone know the name of the song that played at the intro
thats gay
@@QmVuamFtaW4 Definitely gay, but not the Homosexual sodomy type,
@@martinladley lmao
I'm almost 30. Never in my whole life I had calculus explained to me in such an intuitive (and therefore effective) way. I'm watching the whole series and taking notes. Just priceless. Thank you!
How have you been taking notes? Please guide me.
@@ananyasaha12345 Like it was an online course, drawing most of the graphs and writing down most of the equations and conceptual analyses. I use a tablet but pen and paper would be roughly the same. I think it took a bit too much time but I like the result.
Same. None of the calculus books I've studied explain it so effectively. So grateful for this content.😊
this just proves that maths isn't boring, unenthusiastic teachers are boring. Literally 25 minutes ago I was in a lecture being taught this and almost falling asleep, watching this, it's literally like a completely different subject.
sometimes it's because your subconscious "recorded" the lecture for you even if you weren't consciously paying attention so when you watch something related to that lecture, you'd tend to understand it better.
that or it's because you actually wanted to watch this vid instead of the boring lecture. sometimes wanting to learn or do something makes us pay more attention to it because we have a genuine interest rather than being obligated to learn or do that thing.
@@unrested7294 Yes. Repetition is the mother of all mastering
@@LM-he7eb Repetition is based on rote memorization, not actual understanding.
@@KameraShy Not entirely.
Sometimes through repetition you fill blanks that were left on first attempt.
I hear you though
In many Western countries we spend a lot of money on fairly mediocre teachers. When I see material like this I wonder why we waste our money on a lot of the weaker teachers we employ. Kids would learn so much more from just watching videos like this. It is about time we had a complete rethink about education and moved forward from the 19th century approach which we are clinging on to.
Dude. AP Calculus BC Exam is 11 days away. You're saving lives out here.
Art Schell - Underrated comment
pretty much. I felt like I was going to die, even with my calc teacher loading a bunch of TI Nspires with programs to help us halfway skate through it. I feel like we have hope now
Tristan Saldanha unfortunately, the series won't be finished in time for my physics exam
Tristan Saldanha i dont think hes going to hit on bc topics much. looks like mostly ab, he did mention taylor series though so idk
proffessor leonard will save yo ass. just make sure to do the practice problems along with the video.
Coming to this video after my final class in my first calculus course is mind blowing. Spotting the connections earlier on would’ve saved me so much sleep. This is an amazing video thank you!
ho
I’m currently taking calculus in high school and I can confidently say, they do not teach you how to understand the concepts like this video does. In ten minutes, I was able to fully understand the countless hours of plugging in numbers my teachers had me do. I am deeply grateful for this video.
When I took AP Calculus I was able to pass the exam but understood nothing I was doing haha- I just was taught to get the answer.
What other Math topics are covered in your syllabus?
@@manahil558 He pretty much covers them all in this series
"Math has the tendency to reward you when you respect its symmetry" - Grant Sanderson
My new favorite Grant quote!
Bharath Bhat I was going to say the same thing... brilliant!!
That is an excellent quote.
"The character of a country depends upon the racial character of the men and women who dominate it."
-Madison Grant
My favorite Grant quote.
Bizarre notion that a abstract thing can reward someone: sounds like a human need projected onto an abstract thing - sounds like childhood trauma of longing for respect and love from parents... Math doesn't even know you, let along reward you, it is what it is.
It’s so weird seeing favourite without a u, but I guess that’s just the American spelling.
This channel is like listening to Mozart.
EagleSlightlyBetter and watching him too
"Screw Mozart" - Salieri
Or Bach
You should watch sciencephile AI
Yup
in my math class we have done nothing but algebra and geometry and when it came time for calculus, our teacher didnt spare any time for introduction. when i asked why the derivative of some function had that result, the teacher just responded: "there we go with those phylasophical questions". im not sure if the problem is in the teacher of the school system but either way im glad there are other sorces where i can learn and not memorise a chart
Do you learn derivatives at school?
@@ElioSch1423 everybody does. In my country, India, for example we are taught it in the final year of high school.
@@thesocialpi9451 I asked here, and o found out high school teach derivatives here too, but public schools here are pieces of shitt by the way.
the problem is not the school system (which also) but your teacher. a man who loves maths would NEVER give that answer to the question
little does he know that philosophy is the mother of all science and intelligence, without it there's no motivation to learn anything.
the sync between verbal and visual explanations you have here makes it feel so intuitive and easy to understand - this is awesome!!! Thank you so much for this great explanation!
As a school going kid, I always wanted to visualize math like this animation. But my teacher could never provide with any material like this. Now, several years after finishing my bachelors I can finally feel these ideas for real. Thanks a lot for the hard work and this fantastic motion animation. I wish, in future you will cover each and every aspect of math, science and technology. The world needs these kind of content more and more for the people to better understand the underlying rules of our world. Cheerio from Germany. :)
Calculus was the definitely the strangest thing I had to go through during college and I feel silly after watching this series. Such elegance and I missed it completely.
Same feelings for me and all people like us who need to visualize in order to fully understand the nature of things.
Jawad Tahmeed
I agree visualizing is the best tool
just did 900th like to your comment. :)
ich weiß nicht, aber unser Lehrer hat uns das nicht so sehr vor Augen geführt, vor allem nicht mit Kreide und Tafel. Aber trotzdem hat er es irgendwie genauso heruntergebrochen mit Worten. Man musste dann natürlich das ganze selsbt machen, und da kannste ja dir alles vorstellen. Aber an alle Kernaussagen hier in dem Video erinnere ich mich. Komisch.
This is probably my favourite moment of this week
Ilia Boitsov yup
Life lesson i drew from this VDO : whenever see a BIG problem, visualize it as SUM of smaller problems 🙏🏻
Great visual👌🏻 they r fresh even if i am watching them after 6 yrs in 2023.
Best math series ever. Could easily replace a semester of learning in the traditional classroom setting.
"Math has a tendency to reward you when you respect its symmetries."
"Mathematicians don't just care about finding the answer but developing general problem solving tools and techniques."
"Transitioning from something approximate to something precise is pretty subtle and it cuts deep to what calculus is all about."
As much as I love your clear cut explanations of the topic - these hidden gems of wisdom is what attracts me to the videos you create and helps me build an intuition around the subject which my university professors failed to. This is the sort of insight which is developed when you've spent enough time with the subject and examined it thoroughly from all perspective. I am very grateful of the effort you've put into making these series' and I will be coming back to it from time to time to jog my memory.
are u a JEE aspirant?
@@abhaysharmafitness I love 2 play with my little stink hole it's kind of itchy bc I bleed when my turds are too large in diameter but it just feels oh so wonderful boy
@@abhaysharmafitness i am a JEE aspirant
💪😭
Me: "What's a derivative?"
My calculus teacher: "Derivative is Derivative!"
Can't relate more
Its like im right back in the classroom again. Man.
I'm glad I saw this video before I got to calculus in school. Now I won't be drowning in confusion
@@TheMagicalTaco I do all my lessons outside of class and ignore the teacher and just do the homework. You learn it easier and instructors are terrible. Also when you try to take notes its difficult.
@@gregorybattis9588 thanks for the info, I'll remember that when I'm a senior
Wow! Wow! Wow! I'm an electronics engineer turned Software Development Manager over the course of 20+ years and I decided to go for a Masters in AI, mostly for intellectual interest in the area. So, I started revising my Maths when I stumbled upon this and was left literally speechless with the clarity, insights and clear explanations of the fundamentals of one of the toughest areas of mathematics. How so much fun and productive my uni years would have been had I had access to this type of free (!!!!) content. What difference it would have made in the training of engineers, many of which just developed strategies to squeeze through their exams, never having properly grasped the subject and its relation to their areas! Congratulations and thank you so much!
Would have never imagined myself crying because of the eloquence of one's teaching.
Amazing video!
This video blew my mind. I took calculus in college 45 years ago, and it nearly killed me. Dropped the class a couple of times before finally squeaking by with a sufficient grade to satisfy my geology degree. I haven't had much need to apply calculus since, but my lack of understanding has always bothered me. Our professors just blew through it so quickly, I never had a clear understanding about why some factors were important and others could be diminished and discarded. Grant's simple explanation of the area under the curve equating to the piRsquared formula actually caused me a physical reaction! Why hadn't that been shown in any of my classes? Now, at age 67, math comes a little harder for me, but I still plan to work through this entire series!
In a crowded field, Grant has the best maths channel on RUclips. I highly recommend going through all his videos. 'PBS Infinite Series' and 'Mathologer' are also great
Man I’m half a century behind you, I have to take this next semester! Would be nice to have a professor as good as 3b1b, but I’ll probably just have to settle for more videos
never too old to learn :D
Good for you, man!
Psalms 34
Brilliant: not just "what's the formula" but the deeper understanding of "how they figured out the formula"! This is the way to learn not only math but how to do analytical thinking!
correct
You can learn how to create a specific computer model, but if you don't know the computers parts work, you cannot improve the design.
@@H3XED_OwO the point here is to learn about the model and not to recreate it. These videos help in sparking the fire of curiosity and its the viewers choice how far they wanna take things. If you are really interested then after learning the model you will march forward and learn everything there is to recreate the model even if you have to learn whats under the hood
@@bavidlynx3409, I think he was agreeing
A much clearer way to phrase it is: "how to think analytically". How are you not embarrassed to have such clumsy and sloppy thinking?
as an engineering student working my way through an accelerated calc sequence having taken 0 calc in HS, this video is filling in the gaps for me with these visual aids. your answering all the side questions I wondered and small mistakes in my ideas as you go along. (and of course the prof is too busy to answer while they speed through problem explanations.)
amazing supplemental videos here! still holding up 5 years later. it's the high level concepts & gap-fillers tips/hints I knew I needed but couldn't find.
I watched your calculus videos a couple of years back and you genuinely inspired my interest in mathematics. You taught me to look past the formula and into the patterns and i'll be forever thankful for that!
The dislikes must have been from calculus instructors, who feel inadequate after watching this.
I am a calculus instructor and I love these videos.
That’s a pretty uniformed and moronic comment. True teachers crave new ways to present materials. You just sound like a bitter calculus student who struggled to barely pass.
@@DaveyJonesLocka it's a joke.
@@DaveyJonesLocka You have no idea how some teachers can be
@Davey Jones If this bitter student had trouble understanding what the teacher was trying to tell him, but had no problem understanding this video ... wasn't the teacher inadequate by definition? Probably not for all students, but for some at least ... maybe even for most of them.
What you said sounds like an insult to the student, but you are just reinforcing his point.
And if you think being bitter is not the adequate reaction ... maybe you should try being less bitter too.
Yes, I get it is a joke. But the best jokes always have a grain of truth.
I just broke down in tears. Why isn't math taught this intuitively at all levels, at all schools, by all professors? Math is so beautiful; It is upsetting it took me so long to realize it.
P.S. Grant, thank you for your earnest passion for teaching and mathematics, you've introducing me to world I did not know existed!
Because it is difficult to teach math. What 3b1b is doing is incredibly impressive, and there are simply not that many people that are able to break it down as well as this.
Can you please help...
13:57 but doesn't it depend on the angle of the graph (line) to the x axis , I mean it seems to be more correct in this case as it's almost parallel to x axis but not so much for a line which is raising rapidly, even though dx is very small but it is different although slightly but it is, isn't it?? ( I know I am wrong but how someone please elaborate)
bro calm
@@harsharya545 going back the the triabgle shaped graph at the beginning, as dx gets smaller the left and right side of the thin rectangle get closer in height to the point where slant of the graph is negligible
very true
The quote "The art of doing mathematics is finding that special case that contains all the germs of generality." really touched the deep of my heart. Excellent deep meaning in that. Thanks Grant for the wonderful work that you put in
it might seem like a lot to say this but this video is so spiritually uplifting. I do want to feel as if I could have come up with these concepts. I'm getting calculus next semester and I'm beyond excited! Thanks for your dedication!
I've now taken five Mathematics units at university on my way to becoming a Maths and Chemistry teacher. It blows my mind that I'm paying thousands of dollars for my 'education' when there is far superior content on RUclips that is essentially free. The worst part of it all is I've scored really well on all of my exams without truely understanding some basic calculus concepts. I've since forgotten most of the content from only two terms ago and I'm starting to realise it's because I've been remembering rules and not attaining a deep foundational understanding which cements it all together. I'm mad because it's not even my fault. I'm clearly striving for proficiency (I'm sitting here, in my own time, watching a Maths video while eating breakfast) but my university never explained calculus this well, or any Math concepts in fact. Grant, as soon as I have the means I will be supporting you and other cannels on Patreon so you can continue to produce quality content.
Can you help me ?
13:57 but doesn't it depend on the angle of the graph (line) to the x axis , I mean it seems to be more correct in this case as it's almost parallel to x axis but not so much for a line which is raising rapidly, even though dx is very small but it is different although slightly but it is, isn't it?? ( I know I am wrong but how someone please elaborate)
@@harsharya545 Even if the graph were to be rising rapidly, if you get an infinitesimally small dx you will get values that are very precise and can account for even the steepest of slopes. Yes, the example shown on 13:57 shows a big dx and maybe that would not work with that graph in the right extremities where the graph is increasing rapidly, but he only showed this big of a dx for visual purposes. He could have used a very small dx for that parallel section to be more accurate as well. Dont know if this clarifies much, but yeah
most of us have done the same thing ,just memorise it ,never understood it , the fault lies with the teacher too .
How convenient, I’m watching this video while eating breakfast too
In many Western countries we spend a lot of money on fairly mediocre teachers. When I see material like this I wonder why we waste our money on a lot of the weaker teachers we employ. Kids would learn so much more from just watching videos like this. It is about time we had a complete rethink about education and moved forward from the 19th century approach which we are clinging on to.
A few years ago, I got a bit bored, so tried my hand at deriving the formula for the area of a circle. I got there by slicing it into smaller and smaller pie sections where each pie section approached a right-angled triangle with the short side being some fraction of the circumference, and the long side being the radius (radius and hypotenuse would approach equality as the angle got smaller and smaller too). So the area of the triangles would be 1/2(r.((pi.2r)/x)).x with x being the number of slices. Simplified out it became pi.r^2 :D
Just this week I got bored again... and started wondering how pi got calculated - had some fun working that one out!!! Got to pi=(360/x).sin(x/2) with it getting more accurate as x approaches 0 :D
Sure, I could've just googled them, but it was so rewarding to have that AHA! moment when I figured it out! I'd encourage everyone to try to figure things out themselves every once in a while, because that is a very special feeling!
I aspire to work like this.
Similarly, but less interesting, is the insight that the golden ratio is (1 + sqrt(5))/2...
Technically using sin(x) to calculate pi is cheating
How long did it take?
@@idiotidiot2805 Not terribly long, maybe an hour or so
This video made me realize that the derivative of the area of a circle is it’s circonference
and your comment made me realize that , thank you stranger
@@reajuddinsarker1703 same
Real good observation...thanks for sharing the same
Found this video hanging out on my watch later and keeping it on mind to start watching it. Procrastination stopped me from playing the video earlier, but im so glad i watched it and getting satisfied with unanswered past questions i had when i learned math. English wasnt my first language so the visualization also adds a big help for understanding, thank you for the video! (and pushes my curiosity to love and learn more about calculus) :D
Interesting, I've never seen integration being taught before differentiation. I wasn't feeling super crazy about this video at first, but when you got to dA over dx I was genuinely impressed.
NetWalkthroughs I was astounded, not just impressed.
This is one of the things 3blue1brown does that I enjoy.
He has a unique way of doing things.
As History of Mathematics says, Newton found the derivative first mostly because your intense concerns about motion and derivatives is really related to that (and as well he did claims a special operator for it, the upper dotted notation). However, Leibniz found first the integral and later the differentiation!
Integration was generally discovered before differentiation - some integration processes (more aptly called "method of exhaustion") appeared even in ancient Greece (thanks to Eudoxus and Archimedes.) If I recall correctly, both were developed before limits were a well defined and established method of understanding derivatives/integrals as well!
Thanks for clarifying! I really wished my university offered a history of mathematics class since I know a huge number of mathematicians without actually knowing anything about them (Leibniz notation, Euler's constant, Simpson's Rule, etc).
Integration is presented first in the book by Apostol.
"Whenever you come across a genuinely hard question in math, a good policy is to not try too hard to get at the answer directly since usually just end up banging your head against a wall. Instead play around with the idea."
I wish someone had said this to me in college... I would have had a much easier time with calculus.
True
.
Can you help me ?
13:57 but doesn't it depend on the angle of the graph (line) to the x axis , I mean it seems to be more correct in this case as it's almost parallel to x axis but not so much for a line which is raising rapidly, even though dx is very small but it is different although slightly but it is, isn't it?? ( I know I am wrong but how someone please elaborate)
@@harsharya545 The dx being a tiny nudge is not a specific value. It's just a very very small amount. So small that no matter what the slope of that line is its still a very small change. Remember the answer gets better and better for smaller choices of dx. So we keep making dx smaller until it bares no significant difference on the answer at the precision we're looking at. While dx is not an infintessimal, it is also not an actual number. It is a concept invented to be the arbitrarily small number represented by a limit approaching zero.
This subject always intimidated me so I maneuvered around in school at every turn. This kind of instruction gives me hope I can learn just for the fun of it. Thank you. And thank you also for not running 10+ ads during the lesson like every other creator. The continuity of the lesson is so helpful.
I've almost finished an 8-week calc course, it's frustrated me because I couldn't understand why things work and haven't had time to play around with the ideas.
The first animations with the explanation of the development of the (pi)r^2 function and the derivative absolutely blew me away. Everything makes so much more sense and I can start to see where it connects. Thank you so much for this video series. I think you just made me love math even more. ❤
Make the "Essence of Probability" after!!! I love your animations!
I'm totally with you there. Probability is imo a very interesting, but also pretty hard field of mathematics. I would love to see a series on it.
... He would need to do quite a bit more analysis, to dive into probability. Like what means measure and to get a decent understanding of that topic, he would need to build a little foundation on topology, set theory and rings.
I would say, he should first go into topology and measure itself, before starting with something like probability.
PLEASSSE haha I would love to have a great understanding of probability. I was very disappointed by my university's highest level probability class and didn't get much out of it. They just didn't get to in depth.
And then essence of complex numbers
I think that the "Essence of Statistics" would be better, as it includes probability
I said it before and I'll say it again: best educational channel ever.
* there's always a prime between _n_ and _2n_
Jam where n is an integer
What if n is 1?
not an integer. n must be natural and it's inclusive until n=2.
Is 'There's always a prime between n and 2n' inclusive? Because it'd still work for 1 if it were inclusive.
Your channel is A GEM. I passed Engineering without truly grasping calculus and you make it seem so simple...This is beautiful work, THANK YOU!!
this just showed me how much highschool and uni just doesn't cover the explanations expecting students to understand without question - this changed my view on calculus and how it all works at 1:30 in the morning, thank you so much
You know, it's funny. I'm a math major and I'm already well beyond the calculus subset of math classes for the collegiate level - I felt I had a solid, *solid* grasp on this content. I mostly clicked on this video because 3Blue1Brown made it and his videos are just amazing - my linear algebra professor was terrible at giving us intuition, even if the calculations were rather easy, so his essence of linear algebra playlist actually helped me understand a *lot*. I honestly didn't expect to come here and actually get a key bit of intuition that I've been missing for a little while - well, not so much "missing" as much as I looked it over because my AP Calc class was a little rough on me initially. ^_^;
Keep up the good work, 3Blue1Brown. I look forward to your future videos in this playlist and hope they can be even more enlightening for me. :)
Thanks so much Daniel, I'm glad you got something out of it. I always find it challenging to target a math video to a wide variety of backgrounds, so your words mean a lot to me.
Im in college right now, and exactly the same thing goes with me. I've finished my algebra and calculus courses, but came here to become more intuitive about these subjects.
same here, i am a physics student(european student, i don't know if its the same) and i had a realy blunt way of doing mathematics, untill i started watching you're videos. thank you so much, you opend new dimensions for me.
This is a fantastic lesson. It is interesting that he starts with areas and integrals. All calculus course start with differentials followed by integration. This approach, with its simple clear explanations building up from basic math gives the student a clear explanation of the processes involved. It really accomplishes its goal of presenting the material so that the student says, "I could have invented Calculus."
I have never seen anyone start with the circle and turn the areas into small rectangles before. It is a great starting place, and accomplishes multiple goals. The students learn the thought processes used to develop mathematics in general and Calculus specifically. They gain deeper insights into how the integral is developed and it relation to real world scenarios. And they learn why the formula for the area of circle looks like it does as an added bonus.
Understanding why the calculations are done the way are and how they relate to real world problems give the student the deep understanding of Calculus so often lacking in traditional classrooms where the teacher just drones on about equations and applications.
I wish math teachers did better jobs of teaching calculus in this manner because it would have made my job of teaching Physics so much easier. As it was, I had to teach Calculus before and alongside Physics. I had a policy of teaching the students how the equations they used were derived and how they related to the real world situations they described. Just like this video, I started with simple concepts and built on them until we arrived at the result we wanted. It was more about the process and understanding of fundamental concepts than the final answer.
I have not seen any other episodes, but if they are all like this, I would recommend using this series in place of the distance learning now going on during the Covid-19 crisis.
Wayne Adams
B.S. Chemistry (ACS Certified)
M.S, Physics
R&D Chemist 9 yrs.
Physics Instructor 33 yrs.
Thanks Wayne
Thank you Wayne for your extremely insightful comments here as I start to enter this world for the first time at 65 to help my son see the deep beauty of math. When experts like yourself write these comments it just inspires us to find more meaningful ways to learn and to understand.
He also has a linear algebra series.
Thank you for this - I am a distance learner through Open University and luckily my tutor uses the same approach. Seeing the videos really helps me - and I can run the visuals through my head. Wow, 33 years of teaching experience - that is quite an endorsement! I dropped math in Grade 11 and regret - in my 40s now and using the pandemic time to catch up.
yessirrrr
I'm teaching calculus for the first time this year and I appreciate the input you have on how to relay what calculus is. I had a similar view but I do want to incorporate some of your descriptions
my whole life my math teachers have taught only the formulas and spent all class doing fentanyl and I never understood the math until I cam across this video. Your video enlightened the neurons in my brain and gave me a new will to live along with curing my fentanyl addiction. If it weren't for you I would have never escaped my junkie math professor's calc class. Much love,
- a concerned mother
Sounds like things were really hard. Jello can come in a lot of colors. Just like pills. Get them mixed and you’ll be drying your clothes with the moon light.
-anna (Luisiana swamp gater)
My good friend. You have been more influential in my life than my father my dog and all my teachers of the past combined. I won’t ever meet you but I embrace you with gratitude and respect. Thanks
Wow that's... More influence than dad and teachers, ok I guess. More influence than doge tho? I mean that's... ruff.
+Guillaume Perrault Archambault LMAO
+aboctok I hop you spilt too wrong on porpoise.
@@vwlz3603 lol
THIS MAN IS A GODDD
How does this guy make a 20 minute long math video that feels like it went by all too fast?
Because it's 17 minutes?
Haha!
+RonJohn63 - I believe with higher "dx" values, that error is acceptable.
he's a Professional...- A Master....and LOVES his Subject....
If you take 17:03 and multiply it by the height of the video, you won't learn anything.
The graphics are beautiful, the narrative is superb and the two are perfectly synchronized for conveying the point. Add to this, inflections of speech that seems to order a flow of thought that brings to the listener, understanding. What a pedagogical coup!
Every time I come back to these videos they become better and better, like fine wine
I study Civil Engineering. Today I was brainlessly doing homework. Determining the center of mass of a circle section.
So I had to get the mass first. Without really thinking about it I used polar coordinate double integration on it and ended at R^2*alpha...
Then noticed...Wait...that looks familiar...
So I plugged in pi for alpha and realized that thats what it would look like if it reached all the way around and....wait...thats the formula for a circle
Suddenly i had an understanding of what the fuck I was doing all the time and all clattered pieces that were just formulas in my mind came together in a beautiful moment where I suddenly had determined the size of a circle and actually understood why
Thats the beauty of Maths. Its so much fun when you understand what youre doing
Can you help me ?
13:57 but doesn't it depend on the angle of the graph (line) to the x axis , I mean it seems to be more correct in this case as it's almost parallel to x axis but not so much for a line which is raising rapidly, even though dx is very small but it is different although slightly but it is, isn't it?? ( I know I am wrong but how someone please elaborate)
@@harsharya545 You are not wrong in the sense that for relatively "flat" graphs (equations with slow change like (x+a)) the error for this approximation is smaller than for equations with rapid change (like exponential functions, b^2x). But as dx gets smaller, this becomes more or less irrelevant since when dx aproaches zero, soo does the actual change and thus the error also approaches zero, which means a better aproximation. The fact that the error is larger for more steep graphs can thus be viewed as irrelevant for the purposes of what is discussed in this video.
Not sure if this was the kind of answer you were looking for :/
@@johnnygustafsson525 Great answer!
@@harsharya545 As the dx gets smaller and smaller, the error gets smaller simultaneously, For very-very small value of dx, Error is soo small like it does not even existed.
I'm so proud to be the 314th like. :)
Genuinely hope this comment gets 31415 likes as well. I was going to flex my 9265358979323846 knowledge but let's not go overboard with the likes, I said. Still it's good to imagine.
I studied up to advanced calculus. But it was never this well explained nor intuitive. My grandchildren will have it easier. Thank you.
9:45 I can't describe how nice it feels when 3blue1brown says "...you, as the mathematician...". Like dang, it is my dream someday to be just that so how nice does it feel to be called that randomly in the middle of a calculus video xD
Absolutely beautiful.
Incredible animations, and an immaculate description for the essence of calculus.
These 17 minutes gave me more understanding then 3 years of studying calculus at university. Can't be thankful enough, imagine how smart I'll be after all episoeds :>
LOL I'm 12
mario mario cool?
Yeah this video gives us an other view of calculus like we invented it. Like if we invented it we should understand it!
Which university lol
Using derivatives you will be able to collapse the economy again.
0:58
STUDENTS HATE HIM!!!
See how this guy invented calculus with one simple trick:
Lmao good gone
Good one*
Meryl nst as gold or g/bx or underx commenx or not, anyx can b perfx
this comment deserves 5,000 likes lmao
was there not this one researcher in some field not entirely within maths that reinvented calculus (or at least integration by summing areas of rectangles), without knowing calculus was a thing ? (it apparently passed peer review too)
Wow!!! Getting the area of a circle from the area of a rectangle is mind blowing! Why didn't our teachers explain to us these simple concepts this way. Im doing my masters and I had never known that.
The first 7 minutes is exactly how integrals should have been taught. 😊 Great job! I thank you for this.
I'm a Computer Science student at University, and I've watched your videos on Linear Algebra and now this series. I cannot overstate how much of an impact these videos have had on my learning. I have genuinley learned more from you than my lecturers.
No amount of thank you will do justice to this series.
Yes 👍
I haven't studied traditional math since highschool about 12 years ago, but I've done plenty of programming since. So returning to this format is rather challenging, but I figure this series is a good place to jump back into it.
Your explanation of PiR^2 broken into the area of a triangle blew my mind. It was like i suddenly saw the whole thing in 3 dimensions, and finally that formula MAKES SENSE. I'm sure other parts will have their own "Ahah" moments, but I just want to say thanks for all your incredible work.
This is awesome. I'm a visual learner and didn't have examples like this when I was in college; I changed my major based solely on difficulties with Calculus. I'm looking forward to revisiting some of the things I missed then, and have forgotten since. Thank you!
2:26 "Math has a tendency to reward you when you respect its symmetries"
*30 seconds later*
2:56 *RECTANGLE-ISH*
hahaha
@Michael Gayle either works tbh. If he stuck with calling it a trapezoid then you'd have the height dx and the top and bottom lengths a and b. As dx becomes smaller a and b approach each-other. Putting this in the area for a trapezoid:
1/2(a+b)*h
1/2(2pi*r+2pi*(r-dx))*dx where dx -> 0
1/2(2pi*r+2pi*r)*dx
1/2(4pi*r)*dx
2pi*r*dx
So you'd game the same answer.
Yeah LOL
@Michael Gayle trapezoids would work the same exact way but be waaay less visual and ultimately no further help as dx approaches zero!
Mister 3Blue 1Brown you deseve a nobel, for real! I don't understand why universities don't teach math like you do here. You make it look so easy and instresting. We have somehow to promote those videos on every univestity in planet. I hope you'll never loose interest in making those amazing videos. You make the world better. (just a guy from Greece)
Dimitris Tzimikas too bad there are no nobel prizes for mathematics!
There's Abel prize for maths which is equivalent to Noble prize in maths....
Moreover your forefathers from Greece are responsible for the mess that is caused today in maths...how unnecessary and complicated have your ancestors from Greece have been...they should have been very easy and simple...they would have been really honoured by millions of students across the globe ... instead of abusing cus words🤣😃😅😄😉😋😆🤔🤗😝😜😛😝😝😥😥😓😐😶🙄😏😣😣
DUDE YOY ROCK! I love the way you said you want us to be able to feel like we could have created calculus ourselves. Sometimes the hardest part isnt trying to pick up where someone else has left off or their interpretation of math but the entire thought process itself from the ground up. Keep up the awesome work!
When you showed the (A(3.001) - A(3)) / 0.001 I instantly understood lim(x -> 0) (f(x + h) - f(x)) / h
Grant has such a fundamental understanding of math and we're lucky to have him guide us through it. Thank you
This should give 2x right instead of x^2
Me few weeks ago: has Math homework to do
My brain: *Let's procrastinate by watching video compilation of famous authors quotes*
Me now: has English essay to do
My brain: *This is it! Time to learn the Essence of Calculus*
Lol
So true
I have an english presentation to do but nope, calculus
and I only have calculus and vectors next semester lol
Oh, boy. I'm feeling this on another level.
is your name istvan?
I literally changed browsers from incognito to my acc just to comment on this video and I usually NEVER COMMENT, EVER. I want to THANK YOU infinitely much for your hard work making this series. It is literally the most well made, visual, audio math explanation video I have EVER seen and i'm sure a lot of people agree. Your approach of explaining it; us being the sort of mathematician and like trying to come up with it, explaining it to us from the inside out is just brilliant, and you nailed it. Intuitive, simple, beautiful visuals, beast voice ... just 5 stars. GREAT WORK, PLEASE KEEP DOING THIS STUFF. Love u.
Does that mean that I just read one of the very few comments you've ever posted? Wow, I feel special now
Incognito mode doesn't stop google from tracking what you view on their website...
Porn and math best combo ever
@@banemen27 maybe he felt this was easy to explain to his mom. He just had to switch the tabs.
@@Rojoyerf How can google track in incognito. You are not even signed in. There are no saved cookies or history.
This was actually really cool...Never thought of the area of a circle like that, but it really clicks.
I actually heard something interesting from my Calculus teacher earlier this month. She said "Mathematics is like world building, with the equations as the tools you use." It's honestly made math seem far more interesting to me now, and this is helping me to understand what she means. In a way, it is exactly like world building.
Dude I'm from Argentina and I honestly cannot believe how easy it was for you to give a proper introduction to calculus. If it was cheaper to buy a Plushie and bring it over here i would definitely do it.
I left school at 15 to start an apprenticeship in the Navy. Much later as an adult, I started Computer Science undergraduate studies at uni. I had zero exposure to calculus before. Wish I had seen this. It makes sense.
Please don't build the Deathstar after learning calculus Darth!
I feel you. After 4 years in Marine Corps intelligence, I got a C in pre-calc and didn't want to even attempt calculus. I changed major because of that, but this makes it look... tolerable.
@@billandpech I did 5 yrs Navy, started college, did precalculus and then did 2 semesters Calculus. Do not give up. I got a C in all 3 courses. That's all you need.
@@brendandrury2177 Thanks, good advice for a younger person.
I'm 15 and your essence of calculus series has really helped me understand calculus before school even started teaching it. Teachers in the lessons i've attended as extra activities in the past just made me remember the power rule and i thought it was the definition of a notation called dy/dx, which I was quite wrong with. Thanks to your incredible visualization and awesome method of teaching I'm now fully prepared to start doing questions from school and exploring further!
Yeah, same
Yeah, same
Yeah, same
14 year old here- same
Yeah, same
When I saw the title of this video I knew I would like it. When in high school I liked math a lot but I could not just "run problems" to learn it that made no sense to me. I had to be able to see the whole problem in my mind...I had to be able to see all the calculations in a running tally in my head otherwise I did not really know the answer. You are taking calculus and using the same logic derived from algebra...mathematics is simply measuring something in greater minutiae with a million little vectors..thanks so much for this video
I studied calculus at university 15 years ago. It's such a beautiful subject I'm glad to get the chance to relearn it with such stunning visuals and clear explanations.
teachers at school always explain the "How" but never the "Why"
Kids aren't interested.
@@raymondfrye5017 but if they explained, kids would better understand and it wouldn't be such a struggle for them to get good grades
@@raymondfrye5017 don't generalize. I personally love physics, and am currently studying engineering at university.
I didn't do physics in high school because of how infamous our teacher was for making the subject boring. I forced myself to learn physics myself instead, because I still wanted to keep up with physics, but I didn't want to do it at school.
Schooling =/= education
I was interested. There are kids like me. Rare, perhaps, but still existent.
@@PsychonautTV You are right. Some kids love it but inept teachers can't convey the problem-solving skills needed.
Regards
@@raymondfrye5017 Teachers suck at making it interesting. They do not combine our different interests with our long term wants acting as if we are all the same kinds of people that work in the same way, thus making the class fairly unengaging and boring.
One word for you is "Awesome!", If there were teachers like you when i was in school... I would have been really educated.
Sirus Das
Awesome factorial? Hell yes please
TheLenny27 Lmao
Lol you take your own responsability on your education. Studying is always active not passive
You are not thinking. He wouldn't teach you like this. He would have many students to teach and focus on. Also he would be restricted to the circulum to teach. Also he has to make sure people remember which in high school and would have to change the way he teaches. Also you are assuming your teachers are bad without thinking how hard there jobs are or the handicaps they have.
I mean, if you had put more effort in, you could have. If you changed this attitude, you could have. Hell you still can!
I am uni student from Czech, who have been strugling with physics so much, but this video made it super clear, and answered a lot of my questions. Thank you!
What a terrific explanation. I have watched this video before but each time I gain more insight. I think what scares many students off is the new expressions. I think this video does much to alleviate the fear. Thanks so much for taking the time to make these videos.
"Math has a tendency to reward you when you respect its symmetry" Beautifully put
It's poetic
BEST.MATH.TEACHING.EVER!
I always had trouble with math because teachers, even the good ones, couldn't find the time or (in case of the bad ones) didn't have the talent, to explain math in a simple, intuitive way. Breaking down the basic assumptions and connecting those ideas to the big picture was something I always missed and was a main obstacle for me. Remembering math as a collection of rules and tricks gave me a weird distrust for the application of math in new problems, it just never felt quite natural (and was also just hard to remember).
Your videos made me like math as I never really could before.
Keep up the good work. If you ever need help making more vids just ask your fans, I'm sure you have a lot ^^ - and now you have one more
NORMIE.
not lying, the best video that really touches on the essence of calculus
0:58 i felt like that cz i did actually think about finding solutions to stuff like that when i was little. ur videos are awesome. theyre a great help. over here at my place/school, teachers dont teach us anything about the ideation of the concept n hence everything is like my own struggle n i think n think n think to figure stuff out n then at the end of the day unless u can verify the kind of tricky bits of those concepts from someone it kind of takes a toil thinking how u'd have to keep going likt this n then theres the fear that u might be thinking too less or somehow missing something in thinking about it how the great men that helped humanity with their hard work did. ur videos are just what i need. it helps me verify my intuitions n concepts derived from it. just perfect. i cant express my gratitude good enough. thank you so much for helping mathematical intuition sense survive by letting newbies know about stuff properly. common sense n critical thinking is becoming less n less common especially when it comes ot learning concepts of maths and science. everyone jsut wants to memorise which is very sad. everyone should think about solving a problem, maths or otherwise, on their own before looking at established ideas for its but like memorising. i would suggest that u mention that in a video to the young audience that they think about a concepts say, calculus or trig or vector spaces all by themselves for a week before watching your videos. once again, thank you so damn much!
This guy alone has the power of increasing the effectiveness of education by 100%
More like its over 9,000 !!!!!!
Vegeta
No because a 100% increase in a value of 0 is still 0
@@awaken6094 it's still a 100% increase
@@lamichhane so it's a 0% increase?
@@awaken6094 so 100% = 0%
1 = 0 ?
I just started my Calculus 1 course at my university, and my professor sent us a link to this series (and to the essence of linear algebra series). I had already watched the series, and now I'm going to watch it again, hopefully, this time I will learn things I missed the first time!
It's great that your professor recommended it........ if I had enough authority, I would have made it compulsory to include these videos as part of all calculus courses
Please never let these videos disappear!
I watched your first lectures several years ago on Khan Academy and was blown away by it. Watched this series about a week after it was released.
Come here from time to time to brush things up.
billions and billions (-; are spent on "education" systems in countries around the world.
This was practically free. I feel conned.
Billions and billions are spent because:
#1 teachers need to get paid.
#2 buildings don't get built for free.
#3 buildings don't get cleaned for free.
#4 buildings don't get warmed and cooled for free.
#5 Nor repaired for free.
I'm sure there's a jillion tiny little other expenses I forgot...
yes, there is a massive cost behind school
but it must stand to reason that better facilities such as physical school should be better than a video floating on the internet that you literally pay 10 cents for(if you watch the ads)
why it isn't?
school has waaaaaaaaaaaaaaayyy more resources than him, and you can argue that he won by the economy of scale (he can reach millions when normal teacher can only reach thousand) but the argument still holds, school has more facilities to use and teacher to ask, why isn't it better
Read Bryan Caplan's "The Case Against Education". He will confirm your concern satisfyingly.
Unfortunately the us educational system is paid for by the students
Martin Watson yes they pay their school, that's why it suprise me that it is worse than europian school
Paid stuff is usually better but apperently most thing USA is an exception to this concept, expensive stuff that is also not good
Thank you for making this video. As an AP Calculus teacher I love it! Was looking for something interesting to suggest to my students as we work from home during Coronavirus. Thank you!
Wow, thank you all who took part in the making of the video, it broadens your horizons
16:18 0:08 This explanation doesn't seem normal to the people here. But I was taught calculus exactly like this way. I can't imagine how can one know calculus and still be oblivious to this visual beauty.
Me: what is a derivative? Where did we get it from?
My teacher: yes
lol
also they say it is what it is :) ......like WHAT?
Teacher what is integral?
IT IS WHAT IT IS NOW STOP ASKING QUESTIONS AND SOLVE THEM INSTEAD, IS YOUR FIRST EXCERCISE DONE????????
Every math techer ever:
I WAS BORN AT A VERY YOUNG AGE
Ok
So there is value to “reinventing the wheel.”
its the only way to learn, learning really is inventing things
@@fosspointer he most probably deleted it
@@fosspointer I dont know bro 😅😅 I don't know everyone who uses youtube hehe😁
Almost !
Just found out what I had been missing on. Didn't quite grasp the whole video at once but I understood a whole lot more than what I had been taught which were some formulas and a lousy explanation. Saying this was helpful and you're doing a great job would be a massive understatement. Love the video. ❤🔥
I am a high school student and this is sooooo helpful. Glad that I came across it. My concepts are crystal clear for this level, otherwise I my teachers tell us to read the whole book by ourselves if we need any explanation about the chapter.
When I learned calculus, it was as a mechanical process. I learned, and applied, the rules. I became quite good at it but never questioned the way it was constructed.
Thank you for this. It has plugged a hole I never knew existed until I stumbled upon your channel.
Can you help me ?
13:57 but doesn't it depend on the angle of the graph (line) to the x axis , I mean it seems to be more correct in this case as it's almost parallel to x axis but not so much for a line which is raising rapidly, even though dx is very small but it is different although slightly but it is, isn't it?? ( I know I am wrong but how someone please elaborate)
@@harsharya545 I suppose so, dA as a whole approximated as a rectangle so graph's tiny little curve has been overlooked. Like 1)Trapezoid shape was assumed a rectangle 2)The Rectangles don't fill the graph but about to fill it for dr ->0. There are always assumptions but I'm still on my way to explore it. Cheers
My first ever youtube comment: Thank you so much for these videos! Very rarely nowadays math is taught in such an intuitive way. If only this was how math was taught in school! Keep up your amazing work.
This is by far the best video that i have seen on youtube about the basics of calculus. Thanks a lot for uploading this.
I am seeing this video on 19th December,2022. And I must say teachers, professors must teach more through visualisations than through direct formulas