Differential equations, a tourist's guide | DE1

[@3blue1brown]

Some notes on the intended use of this series. I was deliberate in using the phrase "tour of differential equations", as opposed to "introduction to" or "essence of". I think of the relationship between watching this series and taking a course as being analogous to the relationship between touring a city vs. living in it. You'll certainly see a lot less with the tour since you're spending less time overall, but the goal will be to walk around some of the most noteworthy monuments and town centers with helpful context given to you by a guide. And just as someone who lives in a city may very well have never gone to visit some of the historical sites of their town, despite living there for years, many differential equations students may not always get the chance to zoom out and appreciate the central cornerstones of the subject amidst all the computations they are learning. I hope you enjoy the tour, but at the same time know that it is, by design, very different from taking courses on the subject.

[@lukewaite9144]

I have just started a course in PDEs, I always find your videos fantastic for intuitive understanding so I look forward to supplementing my course with this series.

[@aboutmedicine]

I have very little experience with calculus, but am finding this an excellent introduction nonetheless. It's the effort of simplifying the message that is so appreciated for the relative newcomer.

[@mahmoudomar2922]

Where is the rest of the series ??, I hope you continue teaching math in such a spectacular way , thanks a lot

[@DudeWhoSaysDeez]

I am curious though, are you considering doing a more in-depth review of differential equations?

[@naught_]

I'm an undergrad physics student currently taking a class on DEs, and though I'm only a halfway through the class this video still give me quite a few aha! moments based on the material I've learned so far. Great vid as always, you're channel does quite a service to the math community.

[@Alexander_Sannikov]

I basically solve all kinds of differential equations for living: modelling soft body dynamics, rigid body dynamics, optics, whatever. I just wanted to give an advice to anyone aspiring to learn all this stuff. What they often teach you at school/uni is how to solve these equations analytically : sure, this is sometimes useful. However, in most real applications, it's more productive to be able to write a program to solve them rather than to do that "manually". Even the simplest explicit solver that's literally 2 lines of code, instantly covers 95% of all differential equations, even though its performance and accuracy are not the best. It's kind of a brute force solution. One might even start thinking that "hey, if my computer can just solve all these equations, why even bother studying them?". And the reason is very simple: by applying knowledge of certain types of DE's and by using more advanced solvers, one can increase performance and accuracy of these solvers by orders of magnitude. Which might make a huge difference: physics simulation in your game becomes more stable, your rendering algorithm can run your effects 60fps instead of 5fps, you can render your image in higher resolution on the same hardware, etc. My idea is, don't think of your computer as a replacement for your knowledge. Think of it as of an incredibly powerful tool, that still fully depends on your skill.

[@DarkLight-hl2do]

It's funny that I see this because one of my professors was literally telling me the same thing 2 weeks ago. He was basically telling my friends and I how understanding the concepts in our circuits 2 class is much more important than just solving the differential equation on paper. He was relating this to the same fact that technology has evolved to the point where computers can solve almost any differential equation much faster than any human and also went onto say that with advancements in AI, we could see even more changes in the engineering field. Then he talked about the importance of creativity and how it's one of the most important traits to be successful in any field.

[@piman7319]

That's a nice comment; very interesting insights. For me it's sort of an evolutionary addition to the value of the video itself.

[@ArdeshirBanerjee]

What resources would you recommend for learning to solve these DEs in a computer program? I am currently enrolled in a atmospheric science program and need all the technical skills I can get by myself, as I can't depend on my professors.

[@Alexander_Sannikov]

@@ArdeshirBanerjee For me the most important part of finding a good solver(and just learning too) is to have a good framework to test it on. Something visual where properties of your solver have apparent influence on what you're simulating. Crunching abstract numbers is boring, in my opinion and does not give you intuitive understanding of how solver interacts with the system. In my opinion, learn theoretical background from the university course. While doing so, pick any application that's relevant to your field and that you like playing with and just start writing simulation programs. For example, if you're studying atmosphere, you can start with fluid dynamics -- implement any simple 1st order projective Navier-Stokes solver, then go conjugated gradients, then multigrid, etc. Or simulate atmospheric scattering by simple raymarching and then try to improve performance/precision with more advanced methods. Simulating subsurface scattering in cloud rendering works great too. For every narrow enough field (e.g, cloud subsurface scattering) most up-to-date resources are small research papers that you can find find for example on google scholar.

[@onemillionhits9297]

Yes. You are absolutely right. I am a student and I love studying fluid flows, vibrations, etc. Most real-world problems require large amounts of calculation that a human being cannot do. it is more efficient to code these equations in a computer and compute the solutions. Sometimes even when you do it on a computer, the simulations take hours or even days. In my case, this is called computational fluid dynamics. It is very essential to have knowledge of programming languages such as Python, C++. The same goes for Matlab.

[@nickm1902]

3B1B making differential equations videos? Two years of optimistic dreams have reached fruition! Thank you so much. The amount your videos help visualise and truly understand concepts is incredible.

[@青雲浮遊]

I almost cried. New a season is coming.

[@roygalaasen]

I love the way 3b1b presents the material it goes way and beyond topics other channels are not talking about. If you are interested in diff eq course in a more traditional setting, Professor Leonard’s channel just had a round of differential equations recently. (But I am sure people who subscribe to these kind of channels already are aware.)

[@oo88oo]

"I am reaching my final form." - every serial killer in a movie towards the end (True Detective, e.g.), but this is the other side to that dark side

[@assootoshmotah2350]

I am studying differential equations this year😂

[@DavidLindes]

roygalaasen: there will always be new folks who were not aware! Thank you for sharing that!

[@zsoltlendvai5864]

,,To be realistic, let’s add air resistance”. All of my phisics books in my life: we don’t do that here.

[@mickolesmana5899]

opening my engineering textbooks. me: ah sh*t, here we go again

[@marksullivan1173]

Yeah, surprising. Although I think air resistance is actually proportional to the square of theta-dot.

[@archismandas7760]

😂

[@spoicydeemer985]

@@marksullivan1173 Depends on the mass and velocity of the particle,

[@antoanphu272]

all textbooks be like: "ASSUMING THAT THIS SYSTEM IS IN A PERFECT VACCUM WITH NO FRICTION"

[@zelllers]

THIS IS THE BEST SUNDAY IVE HAD IN WEEKS. Every single teacher I've ever had in my life has failed to make me excited about math (except for one of my physics teachers, but I think that was more because I was seeing math applied for the first time than anything). YOU have made me more excited about learning these abstract ways of thinking than anyone else in the entire world. Thank you

[@englemanart]

Math is beautiful! Too many people who teach it lack the imagination to show its aesthetic appeal. Dig deep and enjoy.

[@e2win389]

I am a math major, and have taken classes in both ODEs and partial differential equations, and I still find my mind being blown at how this field is so important

[@mayankndasannacharya6292]

Literally same lol

[@vineetjoshi1740]

How to put a search symbol in a comment?

[@dontseemyprofilepic3157]

@@vineetjoshi1740 it's a new update from within 2 weeks ago that is completely random. Some external bot or program identifies a word that people might not know, and automatically makes it a search term that you can click on. I took some time to get used to it too. It's not something that you can manually add yourself.

[@jorgegutierrez3782]

Grant, I doubt you will read this (I know from your Numberphile interview that you don’t read these so much anymore), but you are just an amazing presence in my life. You are a teacher to me of the highest caliber but most importantly, you awakened a love of math and physics in me. You are part of the reason why I am getting through college and why I decided to do Applied Math and Computer Science. God bless you and in everything you will do in the future. You are a master of your craft.

[@roygalaasen]

At least you got a heart! 😄 We all agree.

[@roygalaasen]

MATH Genius don’t let that bring you down. Math is a subject that needs to mature over time. And while you may always struggle with the subject, you will have moments of enlightenments that brings you forward. Small things like realising that no physical representation of things is ever exact. Perfect circles only exists as ideas, or mathematical objects, that no perfect circle has ever been actually observed by any living person anywhere in this universe. Infinity does not exist either, for the same reason. Infinite precision or size or anything at all are only possible as ideas. Take PI as an example. It is a transcendental number. This basically means that no one has actually been able to see the whole number PI. It goes on an on forever, with no (known) way of predicting the next number in the sequence of numbers. As soon as you say to yourself that you found enough digits of PI, you just literally cut off PI at som finite precision. And now we are back at the fact that there are no perfect real world objects of anything. Just ideas and math.

[@roygalaasen]

... and you probably heard that 0.999999... forever essentially is the same as 1? That 0.99999... = 1. 0.99999... exists only as an idea. As soon as you have a physical representation of the number 0.9999... means you cut it off at some point, making it finite contrary to infinite. If you actually think through all the number 9’s all the way to the end, well you can’t, because it never ends. So the only thing you can say about the number is that the more nines you add to the end, the closer and closer it goes towards 1, that in the border or transition to “infinity” you realise that you can’t put any numbers between 1 and 0.9999... neither can you distinguish the numbers from each other. They are really the same number.

[@carbon4183]

@@randomdude9135 MYD is focused on using a cool trick or perception to make the problem easier, so you do "less math" if that's what you wanna call it. Numberphile (through my experience) introduces a concept or question, and presents it iin a cool way. Or just general math news, information or topics. Their problem videos are sort of like MYD... or just cool proofs (Idk anymore I don't follow Numberphile that much). 3B1B is meant to create a more deeper and truer understanding of math. I know that sinking feeling of frustration where you just can't get it, but the other dude answered my points. My biggest advice for 3B1B is simple. PAUSE. At 3:08 for example, simply pause, think it through and continue. This WILL take time, but you have to pause to understand Grant's intuition, rather than seeing it as a math virtuoso sharing his stuff with other math virtuosos. At 3:08, I'm assuming you don't understand how the derivative graphs check out. It's actually simple (see what I mean? PAUSE). So the first function (the blue one) is the function describing the ball trajectory. Like, the function (call it f) that when you apply it against time, gives the trajectory of the ball. Now, you might be confused why its derivative is a straight line (I'm assuming you know derivative roughly relates to slopes/gradients). If you think about it, that downward line is showing the VERTICAL VELOCITY of the ball. When the ball is thrown, its upward velocity decreases. Why? Because gravity is constantly reducing how fast it goes up (because gravity) pulls down. Now, per unit time, gravitational acceleration (or deceleration) on earth actually stays constant. It's not like at this time you are pulled hard to the ground but at other times you can float. That doesn't happen. So, for the red line, it actually makes sense that it is linear. How fast you are going up will slow down by some force, g (the gravitational attraction). G doesn't change, so it makes sense that the amount the graph is going down (upward velocity) ALSO doesn't change Now, if the slope of the red line is the same, the YELLOW line (that measures the slope of the red line) will be flat. Like, the slope of the red line doesn't change, and what happens when you have a line graph of a value that stays the same? That line will be flat (draw one in WolframAlpha or something to check if you want) So, the yellow line is flat! As for the length, that is a tradeoff for getting a good understanding

[@carbon4183]

@@randomdude9135 No Problem. Try googling more youtube channelsbbthat approach math visually. Easiest way is to go to the 3B1B channel page and on the bar that says Videos, Playlists, Community, etc. Go to the Channels tab and just subscrible to all of them. Solved.

[@ColoredScreens]

I had no idea we could model "playing hard to get" with differential equations, but you did it quite elegantly! Bravo!

[@HafidzNoorFikri]

I couldn't agree more 👍❤

[@adityakhanna113]

Yes! Actually, it was originally done by Strogatz. His book has a whole series of exercise where you get to model different relationship dynamics and their results

[@UsatMVS]

I still didn't get it. When you add "resistance" in love the spirals tend towards zero, that means both people are super in love with each other or meaning they are totally indifferent to each other?

[@bismajoyosumarto1237]

@@UsatMVS The origin would imply the exact middle between those two possibilities: they are equally neutral with each other; the amount of love is exactly in the middle between "totally indifferent" and "totally in love".

[@ivanenev323]

Yes definitely elegant! I wonder how one would model the fact that "the one who can easily exit the relationship, basically have control over it". :)

[@garnetedinger1126]

I'm taking an engineering course on differential equations and this is super helpful. My teacher gets so caught up in the specifics of methods that it's hard to understand the applications for what each term actually represents. This video is an excellent complement to the course in actually understanding what I'm learning. Keep up the great work!

[@vk2ig]

I did my engineering course (given by the Mathematics department) on DEs a long time ago, and all I remember is the phrase "taking the D operator". Where you take it, I can't remember ... out the front of the equation, I believe. Anyway, it was so unrelated to anything we were doing in the rest of the course that sadly, we never used any of it. In fact, our electrical engineering lecturers effectively taught us what DE solving we needed to know using other methods. This led to strange anachronisms such as in 3rd year we were taking a mathematics class for solving the wave equation in two dimensions - on the first day I asked the lecturer if we would be looking at three dimensions, because we were already solving DEs in those situations for electromagnetic wave propagation in waveguides. Looking back on it all, I think we were being taught certain things just so the university administrators could tick an accreditation box somewhere saying "Yes, we've taught them how to solve DEs", but there was little or no thought towards how to teach the subjects in an integrated way (pardon the pun).

[@gilernt]

@@vk2ig I'm only a 2nd year MechE, but I can say that I already see a bit of this kind of thing happening with our degree pathway. Our minds are crammed with the idea of the importance of second and third order derivatives, but our various physics classes are genuinely too afraid to use them. On the flip side, within the first month of my statics course, we've already been expected to know some linear algebra, which for a lot of people is completely unknown at this point, and way later in our pathway.

[@zachh6868]

Aha! I seem to have found someone else inspired to learn physics due to watching too much Doctor Who

[@erikziak1249]

There should be a Nobel prize for KZbin videos and this channel should be the first one to win it!

[@SadmanAhmedShanto]

AGREED!

[@ArpanD]

This channel, along with DrPhysicsA and Physics videos by Eugene Khutoryansky - these 3 channels should win the Nobel Prize for Creative KZbin Videos

[@Umenemo]

I see what your intention is, but you should remember that the Nobel prize is awarded for new (and revolutionary) discoveries in a given science (physics, math, medicine, etc.). An educational video can't exactly be called a breakthrough in science, can it? (Considering there is no Nobel prize in teaching/pedagogy). This would require a whole other 'reward' for the field. Again I know you mean well, I just don't want people slapping a Nobel prize on things that are only distantly related to what the prize actually entails.

[@najmussaqib7542]

@@Umenemo Bro he was taking figuratively...

[@technoultimategaming2999]

or Vsauce but I would vote for 3B1B because his presentations are amazing

[@mesharial6676]

*just finished an ODE class and I don't wanna look at another DE again in my life* me: oh 3blue1brown video about DE *click*

[@lukedavis6711]

My ode was way to easy :(

[@eliasrodriues6614]

Read Vladimir Arnold book on ODE will see all geometry into action. So I suggest you to take a compact smooth manifold and let things flow.

[@yoink6830]

It's rewind time!

[@nadie-qm8rq]

relatable

[@zoltankurti]

@@eliasrodriues6614 is the manifold simply connected.... to the domain of the functions in question?

[@aleksanderboci9059]

I understood about 15% of what i saw, but i loved every single second of it. Great work!

[@sh1w5z]

same

[@alphaleader3450]

@@sh1w5z ombu

[@martinskypala8929]

I stopped at 4:35 and went to an easier lecture, derivatives may help

[@riazo4910]

lmao so dumb

[@chrismarklowitz1001]

The derivative is the vector. The path follows the vectors. The vector describes the "state". Which is the velocity and position specifcally for the pendulum problem. If you understand calculus I. Those are the same thing except the vectors are more simple.

[@3blue1brown]

Chapter 2, on PDEs: kzbin.info/www/bejne/oqqXhGOlnpiMsJo

[@ThePaulahoo1]

Awesome!! I can't wait!!

[@grekiki]

Nice!

[@alin_ilies]

All your series, videos have a base in your previous videos. Good long therm thinking!

[@racimeexe9868]

Dammn that's also some dope cheat code

[@amritkshetri5528]

please make a vieo on finite element method and on weighted residual also

[@EntropicalNature]

Being a math teacher myself, I truly feel humble. By far the best math visualizations in combination with a concise way of delivering abstract concepts available on KZbin! Thanks so much 3Blue!

[@simonrichter3950]

Dude thanks, those graphics are freaking awesome

[@ashmood7551]

Any idea which software is used for those animations? Or is it all Python ;)

[@simonrichter3950]

@@ashmood7551 my idea/guess would be blender+python or something like that Python is not directly optimised for fancy rendering stuff :) (thought it my be done in python but would mean quite a lot of additional development time)

[@steveburton5825]

@@ashmood7551 I know Grant has developed some software to do these using Python but most of this looks to be done using the app Grapher which comes with every Mac. It's a brilliant app even if you just like to play with the examples.

[@isaac10231]

Diff eq was a nightmare, I regurgitated info, got a bad grade, and forgot everything. I was so apathetic to the entire class, I just didn't give a damn about any of it. Now I have the ability to actually get a deeper understanding of the class I spent God knows how much money for. Thank you.

[@jeffspaulding9834]

It was kind of the same for me, although I believe I got an A in the class. I had no idea what I was doing, though - I'm just good at regurgitation. Calculus I could connect with the real world and gain somewhat of an intuition for. Diff Eq was the moment that I lost touch with math altogether. It didn't come back until I took Numeric Computation in my last semester.

[@amrrelsheikh]

I'm speechless about the beauty of this video. I want to become a mathematician now.

[@riazo4910]

dont do it lil bro

[@evadeecho]

bro i just see you in the comments acting like u all that like kid shut up 💀 @@riazo4910

[@Maverick56912]

@@riazo4910 why not

[@tk27a]

This is a remarkable video. It illustrates many of the tools we can use in this era. While watching, I was thinking at how much work and time you spent to come out with this quality of teaching such a fascinating and complex topic. Thank you so much for doing this kind of video. And BTW, your Python code is small, clean and correct!

[@haakonvt]

Just like to add that calling numpy (the sine function in the video) with scalar input (inside of the time-loop), is waaay slower than just using math.sin (due to overhead). As you move to vector input, numpy of course quickly outshines python's inbuilt math functions by orders of magnitude :D

[@spunit262]

@@haakonvt If you care even an iota about efficiency you don't use Python at all.

[@Aj-zr8dz]

@@spunit262 what do you prefer/recommend?

[@haakonvt]

@@spunit262 In a general sense, I agree. However, using JIT tools like numba I've come close enough to "compiled speed" that I'd rather stay in python where I can run the rest of my pseudocode :P

[@tk27a]

​@@spunit262 I was once worried about Python's performance when I started to use it in early 2000's. At that time, we worked on Pentium 120 Mhz and I had to sort a list of 50,000 3-tuples many times and I wondered how to do it efficiently. I had the chance to have a colleague who was a friend of Guido van Rossum, the author of Python, and knew about the design of Python so I asked my him for is opinion. His answer was: "Don't worry, just use the .sort() method." I was skeptic so I did benchmarked it and was flabbergasted by the speed which was almost instantaneous. I went back to my colleague to ask him for explanation. He told me that when Guido did designed and developed Python, he took precious care to well structure and design Python as for excellent performance. He told me that as a rule of thumb, if you use the proper methods and/or statements, you should expect performance to be 90% of the speed as if you wrote it in C. I have witnessed these results for the last 18 years and every time I encountered slow performances, I was able to rewrite a few lines of code that would get me back on track, with tremendous speed.

[@1224chrisng]

22:22 "and another equation from... uhh... chemistry" 🤣 i died lol

[@oldcowbb]

typical physics/math guys reaction to chemistry lol

[@dekrain]

@@oldcowbb *reaction*

[@エディー-h1k]

But, I mean, he is ...uhh... kind of right

[@Hecatonicosachoron]

It is the system for a particular chemical reaction. Somewhat idealised of course. It is the simplest class of oscillating reactions (although this particular one doen't have a counterpart in reality, but it illustrates the general principle). See e.g. the Belushov-Zhabotinsky reaction.

[@randomdude9135]

Didn't k a dead person can comment ! Just saw Ted talk's Mary Roach vid. I guess it's similar to the Lazarus reflex 😂

[@HalcyonSerenade]

Most 3B1B videos: - Average viewer: "math is hard" - 3B1B: "Well, look at it like this and it's not so bad." This 3B1B video: - Average viewer: "math is hard" - 3B1B: "yea u rite, but look at it like this anyway"

[@joshuasusanto4258]

ikr haha

[@tonaxysam]

@@randylejeune Sort of... I am a computer scientist student and solving problems is not math; it is an important part, not gonna lie, but building a good and deep understanding to be able to see the patterns that no one else can see because they are not understanding it (just solving problems) is; at least in my opinion; what really maths means. Without exercises you cannot test what you think that you know, neither see those patterns so clearly, so your knowledge is useless; but without deep understanding of the topics; if you cannot see any patterns or connexions, in the modern days, you are as useful as a computer program... or less I would say, because at least a computer can repeat a task 1000 times a second.

[@ammyvl1]

@@randylejeune that's like saying it's not music when you listen to it.

[@Gameboygenius]

Actually... - Average viewer: "math is hard" - 3B1B: 9:47

[@ivoryas1696]

​@@ammyvl1 Bro made a... fascinatingly on the nose point 💀

[@MyTBrain]

Your positive influence on this world is so damn high. Major karma points to you!

[@handsome_man69]

he doesn't give a damn about my bad case of hemorrhoids though...

[@BenjaminFife]

I so so agree

[@maicee7603]

[Everyone likes that]

[@stgo_g]

You could say he quickly converges to a stable point of illumination

[@facitenonvictimarum]

His influence on this world is almost NON-EXISTENT.

[@miguelangelromerorodriguez490]

I've been scared of differential equations in college. After watching your video and the way you've made the connections between models and ways to represent and understand them, I felt my eyes water a little bit with joy. Now I am super interested in this particular field of Math. Thanks for the inspiration and beautiful way to describe such complex and intimidating concepts.

[@naej3721]

Thank you so much for guiding us into complex things so well! Watching this even only the first 5 minutes makes me know DE so much more. I am new to DE and is still confused by "in what circumstance do we use DE to describe?" and "what is solving DE about?" " DEs arise whenever it is easier to describe change than absolute amount." "....figuring out what a function is based on information about it's rate of change." That's why you are such a great teacher. Not only are you explaining the content superbly, but also explaining what we are doing now in course.

[@jayamitra4656]

Now this is gonna be epic... Love all your essence series, differential equations will be even better! I always love your geometric interpretations and intuitive explanations for all concepts. Keep up the excellent work!👍

[@Jeffrey_Wong]

My 8th grade math teacher insisted that math was beautiful. I didn't understand what she meant until I started watching your videos.

[@shadowcoder19]

Is this math or art?

[@christianartmann1395]

@@shadowcoder19 both

[@Aurora-oe2qp]

@@shadowcoder19 Those two things are equivalent.

[@theboiyoulove5124]

@@osamamohsen4554 i dont think so, you are the crazy one here

[@johannesfortmann8697]

so true

[@douglasheld]

Wow, thank you so much. My brain (and feelings) were still hurt from my experience in 1992 of failing and dropping ODE, only to attempt it, better rested a second time and only barely chug through with a C - and still feeling like I had missed something fundamental. The pendulum, and the similarity with the predator/prey models finally showed me not only what was the fundamental concept that my education somehow sidestepped, but also showed me, in a sense, why I think I would never succeed in groking the formal approach any further than I did. In the end, I found success with other fields of Math but the failure in ODE has always gnawed away at me.

[@ehsan_ul_haq]

I switched from Medical to Math and I am proud of it. A lot of creativeness otherwise hidden is out now. You are the reason. Thank you 3Blue1Brown. God bless you.

[@dominionprodeus5883]

Can we learn together I've got discord and KZbin of course

[@Inndifference]

Same for me I switched from health care to mathematics. Really challenging though 😥but I love it.

[@genessab]

Omg yesss you knew I was failing diff eq and came to help

[@someonesomewhere3542]

I KNOW RIGHT??? I HAD NO CLUE WHATS GOING ON IN CALC 2

[@TOMENTIS]

I'm failing Partial Diff Eq...

[@georgebareiro]

Me too, I was failing in understanding CFD equations, but now I have a better starting point. I'm glad to see people in similar situation ;)

[@brianevans4]

I'm not the only person in Year 2 maths that doesn't know what a differential equation is? That is a satisfying thought. I had mocks last week, and they came up in two questions, both of which I didn't get haha (2018 past papers)

[@idontlikesand1346]

you're not alone

[@tielessin]

*A couple goes to a consultation Couple: "So it's always going up and down in our relationship" Grant: "How familiar are you with differential equations?" 21:01

[@memesarehealthy7818]

Omg this has to be the best explanation of DE I have ever seen, this is actually art My hat is off to you sir, I am speechless

[@DavidPace]

Beautiful Equations Explained Beautifully

[@ashwith]

During my engineering undergrad our Control Systems teacher skipped state space entirely. I struggled to study it on my own several times but beyond going through the mechanics of solving the equations, never got a sense of it. This helps immensely. Thank you so very much! I can give this another shot with real hope of understanding it this time.

[@rabbitpiet7182]

Ashwith Rego oh my gosh, I get to see state space!!

[@keithwins]

I can hardly watch your videos because I want to stop every minute or two and write a long screed thanking you, or contact everyone I've ever met and demand they watch them. Just amazing, over and over. I do wish these had been available 35 years ago when I was a physics undergrad, it would have been a very different experience.

[@romangrachev8716]

Physics undergrad here living the dream.

[@VictorChavesVVBC]

The quality of your animations is something I can't get used to. It's so absurdly impressive. I wish I could give you a thousand thumbs up without infringing KZbin ToS.

[@yoink6830]

*T H E Y `R E R E A L L Y F R E A K I N G H A R D T O S O L V E* had me laughing more than it should have...

[@paradoxicallyexcellent5138]

I thought the best joke, which made me literally lol (llol?) was when he said, "one problem is from Physics and the other is from... Chemistry?"

[@zairaner1489]

Best part of the video, especially that the music cut off.

[@officialnoria]

*cries in math graduate*

[@retovath]

@@officialnoria _cries in engineering_

[@cicik57]

yeah a serious channel would never say that

[@ophirwesley4424]

this is without a doubt one of the best videos on this channel. did everything I hoped you'd do. the perfect balance of computation, theory, motivation, real-life intuition and rigor

[@miguelcamargo8312]

Hopefully this will explain Laplace transformations better than how I understand it now.

[@s3cr3tpassword]

Miguel Camargo I think of Laplace transformation as a more general case of Fourier transformation.

[@nazishahmad1337]

@@s3cr3tpassword both are transformation laws,only difference is one uses imaginary kernel whereas other uses real kernel

[@ErkaaJ]

@@s3cr3tpassword Fourier transforms usually exist, Laplace transforms don't but are easier to work with. In a larger context, they are special cases of Stone-Weierstrass theorem.

[@karolakkolo123]

@@ErkaaJ no sir, the opposite... the laplace transforms usually exist, the fourier transforms usually don't. Just look at the transform tables. The fourier transform isn't even defined on polynomials... (of course I am talking about the full transform, with bounds -inf to inf). Then look at periodic functions. Fourier gives dirac delta distributions or dirac trains which is much more ugly than the rational functions that laplace transform gives

[@nikhilsen9007]

You never cease to amaze all of us STEM enthusiasts. Thank You.

[@laurabarbara2001]

I wish I'd seen this video before I did my my computational physics project on applying the Runge-Kutta method to investigate a forced damped pendulum's motion - your depiction of the phase space (yes I'm a physicist and I'm calling it phase space) using a vector field was so beautifully done and explained. This channel has been getting me through so many of the tough parts of my degree since 1st year. Truly one of the best channels around!

[@aok76_]

I never thought that after 4 years of mechanical engineering I would learn something new about a pendulum. You just opened up realms of understanding in my mind. Thank you!

[@Zifox20]

The animations are soooo beautiful, you are the best maths channel on KZbin without a doubt !

[@lerthadensock3385]

Boy, are you one hell of a great teacher. I can’t imagine how happy my school years would have turned out with you as my math teacher throughout

[@marirsg]

".... converge towards mutual ambivalence. I hear wedding bells already" Hilarious !

[@xyzct]

Hahaha ... you beat me to it.

[@DavidLindes]

I'm just trying to decide whether he's properly using the term ambivalence in its ambi-valent (both extremes) sense, or in the all-too-frequent misunderstanding of it to mean indifference. I can't tell!

[@ChaosGodII]

I loved the "uhm... chemistry" right after that

[@rayniac211]

"repulsion" :D

[@xiaoboyu7517]

wait a minute, should the origin (0love, 0attraction) be the point of breakup?

[@ze714]

3b1b videos are so well crafted, with so much mathematical rigor yet SO intuitive to follow! Thank you so much man, keep up with the INCREDIBLE work

[@bambangkamubisaapa]

I'm a fresh grad architecture student, which as you all may know, didn't actually care a lot about these kinds of things. I studied building simulations for calculating the heat flow within the building envelopes. But I always use these computer tools just like a "black box" in which I could just assign the input and see the output. Later on, I was forced to know what this black box was all about. I read a few journals and all these papers have all the mathematical terms that I don't even know, such as finite element method, partial differential equation, etc. I really thank god I stumbled across this channel. It covers up so much, I studied all the way from the start. The calculus series, the linear algebra series, and finally I could watch this and finally (kind of) understand how all the knowledge is really important in developing these computer tools that help the architecture field. I really hoped that a lot of people watch your videos because it kind of helps the way you see these scary mathematical theories.

[@ricardobetancourt8933]

This is one of the BEST videos I've ever had the pleasure of watching on KZbin, and an AMAZING explanation of DE's. I love the presentation, the artwork, the "simplicity" (not of the subject, but of the conveyance), and it should pretty much be shown at the beginning of any course in DE's or Numerical Methods.

[@arminveres850]

I am a computer science student and I am taking a minor in computational science. I have neither learned nor ever looked at DEs, until the the lectures of said minor. I didn't get most of it, but it was easy to program. Thanks to this video, I now finally understand what I have been solving! Thanks a lot

[@karihotakainen5210]

How do you program something you do not understand?

[@DetectiveMar]

@@karihotakainen5210 Equations.

[@beyond4047]

How can this guy be soooo clear with alllllll topics? One who can truly understand one topic is already kind of outstanding...

[@TheTimeDilater]

I always wanted to learn differential equations with this kinda animations and Grant Sanderson is like a legend who makes my wishes come true. Eagerly waiting for next chapter...

[@DarkLight-hl2do]

I have to say... this video is amazingly well done. I can only imagine the amount of work that goes into making these visuals.

[@Ohmau33]

If anyone asks me why I want to study physics, this is why. This exact video brings out every part of me that loves math and physics.

[@NessHX]

As Richard Feynman has said "Why do I need an infinite number of logical operations to figure out what a tiny stinky part of space is doing"

[@NessHX]

@A person with much less free time kzbin.info/www/bejne/pZOmm4J6mtR_jNk I would recommend watching the whole vid though

[@NolanZewariligon]

@A person with much less free time I'm just going to say that even if that legend is true, the fact that Feynman did it doesn't make it OK in any way. Going to strip clubs isn't something that should be supported. It is helping maintain the current societal norm where women are constantly used and abused, objectified and degraded.

[@Gverri]

@@NolanZewariligon And also, no one is asking you to speak for them. Also, there are many male strip clubs also. Everything is objectified in out capitalist society.

[@Raneriu]

@@NolanZewariligon Sex work is real work

[@NolanZewariligon]

Either you guys are living on some different planet or you are clueless. It's 2019. We have Internet. Please inform yourself about the world before forming opinions about things.

[@amzfishman1]

The visuals and editing on these is absolutely incredible. Truly amazing work, thank you so much!!

[@tribo327]

This explained the concept of ODE's better than any text book or lecturer I have come across. Thank you for this, i'm going back over my old university topics that I simply learned the method to solve and trying to understand whats going on and your videos are a massive help.

[@MrLP10o]

"ODE are really freaking hard to solve" Pierre-Simon Laplace : Hold my glass of wine

[@PetaZire]

I would like to resolve that example (pendulum) with Laplace. However, there is no nice function to solve Laplace[sin(θ(𝑡))] ... Any solution? TT

[@julianbell9161]

My boy Laplace comin through clutch helping me solve DC circuits.

[@sphesihledlamini1944]

But the Laplace transform only exists for functions of "exponential order" (in addition to f(x) have an integral at some interval starting from 0 to some positive number)...not all functions involved in ODEs meet the criteria

[@jibran8410]

Whenever 3B1B uploads a vid, my heart skips a beat, similar to whenever I have a huge realization....Seems like my body has related 3B1B to a time of mathematical realization waiting to happen

[@somethingness]

Me too!

[@leonplank1343]

With your videos it's always the same: I can't decide whether I'm more fascinated by the topic itself, your seemingly simple and natural explanations or the visualisations. Thank your for making every single one of them.

[@jorgeromeu]

I love the tau on the graphs, show the world that Tau and pi can coexist in harmony!

[@altrag]

Or at least mutual ambivalence.

[@ClokworkGremlin]

Protip: set your time steps when programming to increments of 1/(some power of 2) seconds. Because of the way floating-point numbers are stored in a computer, this generally gets you slightly better resolution than decimal values. (the decimal 0.1 is irrational when represented in binary)

[@pineapplewhatever5906]

Not irrational, but infinitely repeating. Still annoying though.

[@KnakuanaRka]

Ah, floating point errors. (>_

[@timdo190]

1/1024?

[@matthewe3813]

another option is to use higher precision numbers that will get a lot more of the inaccuracies there out (even just standard doubles will be a lot better)

[@JuliettKilo]

@@matthewe3813I don’t think these simulations in the video requires double precision. 3B1B probably set the time step too large or implemented some bad algorithms that accumulate error too quickly.

[@RodHartzell]

I studied Differential Equations back in the mid 80s. I remember it being pretty mind expanding. But your videos take it to a whole new level with the visualizations. Absolutely amazing. Keep it up.

[@ferax_aqua]

Watching a Numberphile video. Receive a notif from 3blue1brown. Numberphile closed, 3b1b opened.

[@arnavgoyal7952]

@@no-one-in-particular Wtf

[@user-wf2fm1yj4k]

sjg1984 lol what

[@emperorpingusmathchannel5365]

Numberphile is pseudomaths. Don't watch them.

[@emperorpingusmathchannel5365]

@@no-one-in-particular It is true tho.

[@fabulator2779]

@@no-one-in-particular lmao wtf

[@davidtolle9533]

"This equation from (pause), chemistry..." Lmao

[@romawang9212]

hahaha I love this

[@romawang9212]

Ardalan G. Haha by chemistry he meant love

[@snipy3685]

so funny!!

[@enderredacted112]

*now i have more chem joke*

[@yanry7152]

@Ardalan Ghazizadeh 22:22

[@achalakankanamge9109]

Every 3B1B video is a work of art- but this is truly one of the masterpieces. The sheer dedication to conveying the beauty of math, with nothing hidden and as simply as possible is impressive. Conveying physical concepts in our universe as well as abstract concepts is no easy feat, so your service to humankind is truly appreciated, as you enable a large audience to grasp these concepts! Much love from Sri Lanka!

[@rafaelaassuncao9729]

PLEASE PLEASE PLEASE DO AN ENTIRE SERIES ON THIS THE SAME WAY YOU DID WITH CALCULUS AND LINEAR ALGEBRA I BEG YOU

[@MrMineHeads.]

He is, my boner can't get any harder.

[@SkaterProSeven]

isnt this...calculus?

[@MrMineHeads.]

@@SkaterProSeven this falls under "analysis" which calculus is also a part of.

[@ananyapathak8312]

Kurzgesagt and 3b1b uploaded on the same day! Edit: And Mark Rober too!

[@bodhisattva9762]

Kurzgesagt is over-rated. 3B1B is a gem of a creator.

[@shreeganesh441]

Again!

[@swarajsonavane2789]

@@bodhisattva9762 why do you think kurgzgesagt is overrated....just asking

[@swarajsonavane2789]

@@berserker8884 the one by coffee break?

[@berserker8884]

@@swarajsonavane2789 indeed

[@ethan6708]

My hair has never been this blown back in all my life. What an amazing video. The more you know, the less you know. Thank you for putting this together.

[@hal6yon]

Oh dear God I'm gonna cry

[@gaboqv]

you are not alone

[@GusTheWolfgang]

Though we're suffering from constant, unpredictable power outages in Venezuela right now, and I have a ton of work to do, I'm immensely happy to spend half an hour watching your amazing content. It truly made my day

[@PlatnmBeatz]

You can thank the CIA for that. Good luck with your work though!

[@Dissandou]

If they're constant, then they're predictable. f(x) = k, so f = k forall x

[@biblicallyaccurateangel2476]

i honestly understand none of this as a freshman in highschool, but i put 3blue1brown videos on while working on essays because it helps me get into the writing mood, and its so relaxing. i love this dude’s vibe, it’s so chill and you can tell he really loves making these videos.

[@wavez4224]

This stuff is very complex, honestly understanding any amount of this as a freshman would be impressive. I recently graduated high school and still struggle to understand parts of it

[@chamidumadumal7130]

Me: Gotta sleep early. should go to work tomorrow. *3b1b notification* also me: Naahh... work can wait...

[@HemantSharma-bk8ek]

But learning can't

[@miloradowicz]

Me: the video is 30 minutes long, but I'll need at least an hour to make sense of it, if I don't already know the topic.

[@rrequelme7196]

3Blue1Brown = the Richard Feynman of our generation :) Greetings from Ecuador bro! You are one of the best teachers on KZbin, hands down :)

[@yourlordandsaviouryeesusbe2998]

Umm...I love Grant Sanderson and all his efforts. This channel is by far the best mathematics educational channel in KZbin. But comparing him to Feynman is a stretch.

[@vestieee5098]

I'm a senior in high school, and I knew what differential equations are, how they often come up in physics everywhere. For around a year or more, every once in a while I've come back to this video out of interest, even though this depth of DEs is completely unrequired for me at this stage. Sadly each time I've found myself struggling to follow with the video, as in I understand what you're saying but I've been unable to grasp and visualize the animations. Today however (maybe my 5th or 6th) attempt, I could understand this entire video with a bit of pausing and thinking, and WOW! The feeling of understanding the idea of phase space and the vector field as well as phase flow blew my mind!

[@prateekbansal8302]

I know Grant you must be online now and scrolling through the comments. (I hope, atleast 🙈). I just really wanna take this opportunity to thank you for what you do. Seriously, you have done what no math teacher can, for me, personally. The amount of effort and research you take to make each one of your videos, is immense. And for that, I wanna thank you from the bottom of my heart. I have been a subscriber since you had only 10k. Been following for a long time. It would really make my day if you see this. Once again, thank you. You are the best.

[@3blue1brown]

Thanks so much for the kind words, and for being here since the start :)

[@prateekbansal8302]

@@3blue1brown Holy Shit you saw this. Thank you Grant! Keep up the awesome work!

[@ArpanD]

3blue1brown has really played a great role in my education, and especially motivation. Really nice channel. Short, concise explanations thanks a lot.. go ahead...

[@gabe-d8u]

Thanks so much man! Your course on linear algebra was already a banger, as well as your lecture on divergence and curl, so I expected no less of this video. And I was correct to assume so! You've taught me what uni never did, intuition, and I thank you for that.

[@josepha.cardenas9168]

Hi Grant (sorry for the friendly treatment). I am an engineering student. Though I have a lot to learn, I am truly inspired and motivated by your videos. I would like to be a Patreon and contribute, you're the man.

[@zakariyaal2848]

Sir, allow me to say that your videos are gold.

[@cescu2]

The visuals are amazing, I'm blown away by the quality. I came here as a musician and animator. I just love natural motion and chaos and apparent randomness. I love it when humans find patterns in apparent noise and share that with the rest of us, it's just amazing.

[@vUvesten]

You're amazing! I study Mechanical Engineering and don't have time study more advanced math and this realy suits my needs, thank you!

[@brogcooper25]

I am a mechanical engineer. My university had some math classes available under what they called tech electives. It's a good chance to get credit for learning math above your degree requirements.

[@Gabriel-nd5rp]

You are such a freaking asset to this world. Thanks you so much for the playfulness you throw at math. Truly inspiring!

[@אדםפליישר]

I study mathematics in the university and been trying to make my little brother be interested more in what I do. So I gave him to watch some of your videos (including this) and 2 things happened: 1. He enjoyed it a lot and asked me to help him with some of the concepts in this videos! 2. I watched the video till the end and enjoyed every second of it!!!!!!! Your explenations are intuitive yet full of knoweledge. Thank you very much for your work :) p.s sorry for my english

[@shrimatkapoor2200]

I love this video, probably one of my favourites on this channel... Really digs down into the roots of the links between the physical and the abstract.

[@DavidPace]

This video was extremely insightful for what I'm working on, and spookily served up to me without a search. I especially love that you gave a sample/example Python script. Screw King of Thrones, I'm about to binge watch this series to gain a more solid understanding of how to use the formulas, and hopefully with more possible script modeling setups.

[@rahulpatil5397]

The way you teach each math topics in a visual way is just awesome. By watching your videos I have realized how important are the visual interpretation of maths problems for deeper understanding. By the way, I have checked your series on linear algebra, calculus, etc. But I wonder why I haven't you created such more series on other remaining topics of maths. I hope in future you make these series and I would love to watch them.

[@OrangeC7]

I think this is actually my favourite video from you so far

[@fpereira77]

Everyone who's interested in Mathematics should follow your channel. EVERYONE! It doesn't get any better than this.

[@3xistism1.0]

5 minutes in and I have already learned so much. I genuinely get the idea of how to solve an ODE, albeit simple ones, but still. This video is amazing.

[@ohno5559]

I literally start my differential equations class tomorrow thank you for a thousand years

[@projektb4rd0n53]

oh no

[@SuperMiketron]

Holy shit I am so excited for this series. Keep doing what you're doing please, 3B1B!

[@personism]

@10:05 - the little pi's look so hopelessly defeated. Really captures the feeling of being stumped by a challenging problem lol. Love this video - another great math tour!

[@OCD.Reader]

Oh my god. I honestly never thought my flirting skills could be modelled by a phase flow. Unfortunately my phase space would be at least 5 dimensional. Ah stupid love. But hats of bro. Honestly. They way you describe. Really. I wish all my teachers were like you.

[@claycleveland4143]

As soon as he said stability my eyes lit up. Yes! Let’s talk about eigenvalues and damping!

[@lktyagi88]

I have never come across anyone who is this smart and finds the easiest way to expalin things. May be your expalinaiton becomes so easy because you know it so well. Amazing job man! Keep up the good work.

[@sujals7108]

I love it when he talks like a philosopher at the end of his videos

[@MaxxTosh]

I can’t wait for the next video already! And I’ve been waiting for a visual understanding of the Laplace transform!

[@AG-hl7bg]

In the meantime I suggest you to check Eugene Khutoryansky videos on the same topic

[@pnachtwey]

This is pure gold. I know it took a lot of time and effort to make. I am retired now but I wish I saw something like this 60 years ago.

[@unstoppabletachyon6779]

Damn, this channel is a drug.

[@SkateGeneva]

are you equal to E psi ?

[@unstoppabletachyon6779]

@@SkateGeneva Only if I am an eigenstate of the Hamiltonian. Generally speaking, No :(

[@leguminosa9]

finally! I really like stuff about differential equations! Thanks 3Blue1Brown, as always.

[@ncjanardhan]

My life path would have been different if these videos were available during graduation!! You're a boon to this generation ❤

[@KendoSonic]

Does anyone else feel depressed that there often isn’t an analytical solution? It makes me feel gross lol

[@whatitmeans]

its much worst: due to the Identity Theorem, no power series can represent a solution that stop moving, at least if is not defined piecewise.

[@user-gx7gs4xi9x]

Wonderful, wonderful. Charts and examples reinforce intuition. It turns out that a lot of time and effort has been made. Thanks very much and thanks. It was great, very much