Divergence and curl: The language of Maxwell's equations, fluid flow, and more
ฝัง
- เผยแพร่เมื่อ 5 พ.ค. 2024
- Visualizing two core operations in calculus. (Small error correction below)
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Special thanks to these supporters: 3b1b.co/divcurl-thanks
My work on this topic at Khan Academy: www.khanacademy.org/math/mult...
Timestamps
0:00 - Vector fields
2:15 - What is divergence
4:31 - What is curl
5:47 - Maxwell's equations
7:36 - Dynamic systems
10:37 - Explaining the notation
13:28 - No more sponsor messages
Correction at 4:55, the narration should say "counterclockwise rotation gives positive curl, clockwise rotation gives negative curl". The diagram is correct, though.
For more fun fluid-flow illustrations, which heavily influenced how I animated this video, I think you'll really enjoy this site:
anvaka.github.io/fieldplay/
Music by Vincent Rubinetti:
vincerubinetti.bandcamp.com/a...
Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
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![Div, Grad, and Curl: Vector Calculus Building Blocks for PDEs [Divergence, Gradient, and Curl]](http://i.ytimg.com/vi/lKXW7DRyyro/mqdefault.jpg)
I'm a 64 year old ex-Physics teacher. Where were you when I was studying in 1971? I needed someone like you - it was so hard then - you make it so clear and beautiful. I hope this generation know how lucky they are.
Stephen Hicks
Some of us definitely do!
Ditto
I gave up polytechnic for this exact reason.
What I would have needed is brilliant teaching like this.
I couldn't get my head solving equations for concept I couldn't clearly understand and visualize.
This video is beyond brilliant...
I'm 41 and in my third year back at school again pursuing different degrees. The pure amount of resources available now compared to when I first went to college in the mid 90s is crazy. And pushing it back to the 70s I can imagine is only more apparent. These kinds of animations are so nice for visual learners and makes it more intuitive when it comes to applying the concepts.
Stephen Hicks they dont
if there were a nobel prize for educators, you'd definitely get one.
There should totally be one though!
@@adityaarora2162 true
Can't agree with you more 👍
You can win nobel prize as educator. Mr. Kailash Satyarthi was person who won nobel prize as educator 😊
Yes sr!!!
I teared up when you stated your goal to "teach/remind people that they love math." I'm in my third year of studying electrical engineering, and that's just what desperately I needed; this video did that for me today. It helped rekindle the enthusiasm I had when I started down this path. You are helping so many people, and making the world a better place by encouraging them about what they can understand and do. Thank you! I appreciate your thoughtfulness about how to sustain your content while sticking true to your desired level of value to people, and about what incentive is driving the content. I'm happy to help support your channel so that the world can experience more of your work!
My exact sentiments.
Getting close to studying electric engineering, do you recommend anything to learn more about the field?
The fact that the same concept can explain both electromagnetic waves and foxes eating rabbits is one of the main reasons I have chosen to pursue mathematic studies.
It can also be used to explain why shops sell multiple items and the structure of the menu.
@@lordturnip4731 WHAT IS THE EXPLANATION OF SPACE AND TIME ON BALANCE:
Invisible AND VISIBLE SPACE in fundamental equilibrium AND BALANCE. This necessarily and clearly involves interaction, on balance. Consider what is the eye (on balance). Logically consider what is a two dimensional surface OR SPACE ON BALANCE !!!! Notice the associated DOME AND black “space” of what is the eye as well. Really think about it all. Consider WHAT IS THE SUN ON BALANCE. (NOW, think about time.) Outer “space” involves full inertia, AND it is fully invisible AND black. Consider one and three dimensions ON BALANCE !!! Now, consider what is the fourth dimension and the term c4 from Einstein's field equations. Think about ELECTROMAGNETISM/energy AND think about gravity (ON/IN BALANCE). Consider what is the man (AND THE EYE ON BALANCE) who IS actually standing ON what is THE EARTH/ground (ON BALANCE) !! Think about TIME !! Think about why there is something instead of nothing ON BALANCE. Consider that time is NECESSARILY possible/potential AND actual ON/IN BALANCE. Think about the man (THE EYE) that actually IS IN what is outer “space”. Think about time. The stars AND PLANETS are POINTS in the night sky ON BALANCE, and consider what is the speed of light (c) ON BALANCE. NOW, consider what is THE SUN. Think closely about everything in this writing. Balance and completeness go hand in hand. Magnificent. ❤️
By Frank DiMeglio
@@frankdimeglio8216 u good bro?
math is wild
@@frankdimeglio8216 Who is ur dealer and where can I find him
Small mistake: for curl it’s counter clockwise = positive, and clockwise = negative
Was about to mention that the graphics show counter as positive but the voice over says the opposite and I didn't know why. Thanks for the clarification!
Bro I thought I was tripping, glad someone noticed too
so many dickriders in the comment section, glad you noticed that.
@@hogo21Aahahahahahahha me too
We can take sign conventin c.w + or a.c.w +.
I'm currently applying to phd programs for pure mathematics now. This is in no small part to your videos over the last few years. They inspire me to see math in a new light. I watch this video having used divergence and curl for many semesters and I forget, or never realized how beautiful they are.
Adam Nelson I’m having the exact same experience.
This is literally what Grant wanted to do... To spark or fan the interest of a student in Mathematics
How did your application process go? What are you to now?
@@jorgerodriguez-anton7990 Ha, this is a throwback. I'm at Northeastern University now, thanks for asking. I work mainly in neural networks now, which of course I was introduced to with Grant's series on Machine Learning
Yes, the videos are visually quite beautiful as well as instructive.
Sir,
I had a teacher (in a small town in India) who explained things as you do. He taught physics. Very passionate. Lived it. Had youtube mobiles been at his time, we would have been fortunate to listen to him even today.
Mr Sanderson,
You are an absolute joy and delight to this world. I am so glad to have the fortune of discovering your channel and learning from you. You brighten my day when I watch your video and truly begin to understand what math is about. For that, I am grateful. Please don’t stop!!!
Since you talked about your viewers, I'd like to share my experience with 3blue1brown!
I am a german math student in 2nd semester and I am SO greatful for every single one of the videos I watch. I always keep recommending this channel to my peers and believe it to be the most... beautiful and informative one on maths on youtube. I love the colorful animations, your voice (it's very soft and motivating!) and - well, in fact, all of it.
This channel is a super good addition to my studies, and I want to be honest? I don't think I would have passed my first exams without constantly viewing and re-viewing these videos throughout the semester. Every single one gives the visualization and background I need to understand the topics (and not forget them 3 days later). Thank you very very much for supporting me in, well, the part of my life I need most help with. This might sound dramatic, but I really need you to understand that I mean the words I say; I wouldn't have developed to be the person I am today without all of this ^^ These videos are just fundamental to my studies - everything would be different without them.
Keep going!!!
3b1b Essence of linear algebra is my favourites
@Farran Khawaja if you start that young and keep going, you're gonna be brilliant in your 20s. I started studying electrical engineering with 30 and wished I had started as early as you. I hope you never stop being motivated!
Dein Englisch ist toll.
@Farran Khawaja Hey, I hope all is still going well for you :) motivated and intelligent kids is what keeps this world going in a positive direction!
3b1b is one of the most important sources of mine for understanding a topic. Being only 14 means that my teachers generally don't answer any questions other than those related to the topic we are taught at school. Idk where this will take me ahead but rn I am studying math to satisfy my curiosity and not for future prospects
Animations are just mesmerizing. This channel is so good!
And his voice!!!
And Python scripts!
They're amazing! I would love to learn how they're made. Personally, more than intuitions for divergence and curl.
Giorgi said it. He coded all of his own visual animations in python. Everything you see on the screen is his coding. Quite remarkable.
It is remarkable. I'm just learning to program, and a lot of these videos are over my head, or I don't know the practical application. I feel like if this were taught through making the visualizations, I would understand more. Of course, this one video would probably turn into a long series, though. Maybe I could find some resources that teach math through code.
Edit: Alright, I just searched it and there's many results. Maybe I'll understand these videos better in the future haha.
Hey Grant, I've had anxiety towards math for years.
I had a friend who introduced me to your videos years ago, and recently I revisited them to go through some of my modules.
I can't believe how helpful your material is. I've gone from being completely afraid of math to feeling like I can teach math as well someday!
Heck, I even cried watching this video at the end. I'm very inspired by the work you do and I hope you have loads of success in your journey. Thank you for this beautiful work!
Haha you cried because of math
@@billymays495 Billy, crying because of math is a sacred thing. Don't be mean billy
As I mature as a scientist throughout my years in schooling, I come more and more to the conclusion that your methodology of imparting sound concepts through visualization is superior to not using visualization abilities/techniques to try and understand a concept.
I always find myself coming back to your videos for refreshers or solidifying a new concept. Thanks for your service to humanity. You're a hero to many.
You, sir, are a gift to humanity. Only if we acknowledged that.
I have 2 degres at university in sciemce but I feed incortable, I should play with basic math for quantum mechanic and +, but I do not; dam it.
Hola, estoy tratando de entender bien las ideas expuestas para modelar un fenómeno físico con los conceptos de campo.
@@jorgegarcia6055 amigo, el comentario y sus respuestas están en Inglés. Lo más probable es que no te entiendan si escribes en español, así que escribe en Inglés
We are all gifts. Don't kiss ass. You'll ruin a good thing by watering pride
@@mathscience757 You should learn English before math.
Godzilla tried to read your comment and fucking died.
I love the confidence with which he says " The understanding for what it represents is more important." It feels like a SLAP to all those teachers who jumped into calculations, in order to hide their ignorance.
Or, as I think, to hide their incapability of teaching understanding, because of their own lack of deeply understanding the topic.
@@dombowombo3076 Yes that is what I meant by ignorance. The fact that they don't even know the subject in depth.
Its not ignorance ,they just dont know 😂
@@arshi4210 I thought that's what it meant. It doesn't mean "to ignore", it means "to not know".
Oo
As an electrical engineer with 28 years in the nuclear power industry, I find beauty and clarity of your exclamations to be absolutely breathtaking way for me to remember what I was talking to the 90s and God bless you and all of your endeavors! I am planning to make a contribution to your work once my next bonus clears. Charlie
I LOVED the fact that you're not selling out to sponsor content and that you literally just teach math in this artistic vibrant way! Keep up the great work!
Dude, you might just be the greatest explainer since Feynman!!
@Isaac Dweck Self-describing comment, I see.
Why not , these lectures are perfect.
@Isaac Dweck you seem very flattering, don't you?
I genuinely think so... Only difference is that he has the necessary tools to help people visualize, whatever he explains..... That makes it better!
Yea I agree but think what Feynman would have been able to teach with today technology
Last year I had a phone interview with a great company after I got my degree, and they were asking technical questions, one of which was about Fourier transforms. I used the concepts from your video to explain it over the phone and the interviewer was really happy. I got hired. all I can say is thank you, you’re making science tangible and graspable again. You’re doing a service.
at what kind of job you're being asked about fourier transform?
PinkySuavo a rotation program where one of the rotations was materials metrology with a bunch of different instruments, one of which is an FTIR, so they were just curious if I knew the concepts.
Grant, you have absolutely no idea how valuable you and your way of understanding and explaining things are. I mean it when i say i wish all teachers would teach like you. I'm certain if they did the world would be a better place. Thank you for your insights.
It's insane how born for this hugely important job you are. Your videos remind millions of mathcomputation-tired students why they chose this as their education in the first place and why math is so beautiful. Thank you.
Holy shit man, your outro makes me wanna cry. Your are such a genuinely good content creator. I really understand the need to somehow monetize your content if making videos takes up a significant amount of your time and becomes a large portion of your income, but some of my favourite channels are slowly getting rotten by the sponsorship, to the point where videos basically become unapologetic ads (looking at you, Wendover/Half as interesting, Smarter Everyday, You suck at cooking, Nile Red, Theneedledrop). The click bait is killing me too (seriously Cody's Lab? Wild Wilderness? AvE?). Fuck it, I'll support you on Patreon, something I never do. You are, by and large, the best educational channel on youtube, period. My only regret is not being a professor so I could recommend your videos to my students.
More than just a "significant part . . ." He doe's this exclusively
To be fair, "Divergence and curl: The language of Maxwell's equations, fluid flow, and more" qualifies as clickbait to math nerds. I mean, who *wouldn't* click that?
Seriously though, I dunno if I'd count AvE's titles as clickbait. If it was someone else, sure... but given the half-drunken excited bravado AvE fills his videos with, I think the titles are usually pretty descriptive.
My vector Calc professor recommends 3blue1brown videos to us. Made me happy to see the channel get a shoutout for excellence in a university setting.
NileRed?
Humanity was good to me. I've spent the last 5 years of my life trying to give humanity a massive gift, with absolutely no chance of me benefiting personally. I know the feeling. Thank you for you beautiful wonderful works of art, 3b1b.
12:04 “What are you doing, step-vector?” Sorry, the Internet has broke my moral compass
I was literally going to comment the same thing lol
@@fusiontricycle6605 Hey bro nice propic
hAHAAAAA
I wanted to comment this
Beat me to that lol
I am doing a postgraduate maths course and when I tried to understand these concepts using only equations, I was completely lost. Then my tutor simply shared this link and said he would never be able to illustrate it as well as this video.
I smiled when I realised it's 3Blue1Brown. I'm proud that the channel I watch for my own curiosity and enjoyment is often recommended by maths professors, and for good reason -- the beauty and quality of these visual animations is unparalleled. Thanks for demystifying advanced maths, Grant. I know you know this, but you are doing a great service to humanity.
Man I am a mechanical engineering student, but I must admit I learnt it for the first time in my life. Even though I took multivariable calculus in my college first year, I couldn't understand even a single thing. But you have encapsulated everything so beautifully and intuitively that I must say it's better to learn science and mathematics from you rather than going to college.
"Part 1"?
As in... a new series :D??
Oh hell yeah !!
I think it's more of a two- or three-part thing, not a proper series :/
No, as in a "this is a difficult as hell topic and takes more than 20 minutes to understand, an entire quarter is spent on it in calculus
Where can I find the next videos?
I am getting envious - as a "dinosaur" - how brilliant ways and tools You have TODAY to explain these things so _fluently_ that even I can grasp them finally. When I had some of that stuff over 30 years ago in school, it remained horribly abstract and left me clueless, almost. Especially the question how poeple like Maxwell, having not even an idea of digital computers or visualizations, could "make" those fundamental equations ... so I always stood kinda stuck in between: neither be a total ignorant for math, but also not even close to mathematicians, who always remained as one of the strangest species on earth to me (no offence intended =), as they would thrive in their abstract world called Math, and leave everyone else behind, not even understanding or being interested to understand, why non-mathematicians do not understand them and the matter which is "so obvious" to them. That mind gap is just astronomically great.
For me, it is these visualizations being the KEY to grasping. Unfortunately, too many people, even if educated, still mix up grasping with memorizing ...
Thanks for this video, most excellent!
Sorry to disagree with everyone, but it is still totally abstract to me and still leaves me clueless.
bless you man, to see that something so good exists for free , especially for us students, and that you're not even earning from sponsors to do these videos is unbelievable and so kind of you, thank you!
Thank you! This is where I come to to get intuitions about most mathematical concepts. If I ever become a teacher, I want to be like you :)
😇
15:15 "What matters more than wanting something is to actually align incentives". Dropping math knowledge and generalised wisdom in the same video, very nice.
Always high quality videos for free, the best channel on you tube.
Watching this right after your essence of linear algebra series and your neural network series literally opened my eyes permanently. Sometimes I have been swept up in the undertow of your other videos, but whatever happened this time that primed me to learn, you have taught me the most in the shortest amount of time.
I am ecstatic watching this interpretation... you have answered a lot of my questions in a single video ...just made my day...God bless you
i am a mechanical engg now training students for physics olympiads and JEE ADVANCED , YOUR VIDEOS ARE AMAZING , for the first time in my life ....there is a clear idea of curl and divergence . awesome content .
orzzz 🤩big fan
Legend is here
You are a legend sir! I absolutely love your videos!!
Come here after recommendation by you guru ji❤🙏🏻
holy how did you manage to animate this??!? You are truely out of this world grant!
yes he is :-)
@@kristoferson_0423 And I actually am, thanks man!
@@kristoferson_0423 wow! Ive been searching thru many vids to find this answer, ty!
@byzef a you just gotta remember that manim allows you to only draw simple shapes and lines. All the hard stuff has to believe be handed coded
@@ammyvl1 yeah its freaking hard. Manim allows for the animations to be made, but you need to know a fair bit of python to tell it to do anything more than simple drawings.
You are one of the best teacher i have ever seen. I have seen many videos but all of them are focused on solving problems rather then concepts. Thank you sir.
I cant appreciate you enough for your videos, and the inspiration you give to people for mathematics. thank you Sir.
I always come away from 3b1b videos feeling so **good**! Enlightened by the new intuitions, interested in the math behind it, calmed by Grant's voice, amazed by the beauty of the animations... They're just so wonderful. Thank you, Grant!
Mianne Kahkol I feel exactly the same, thanks a lot 3b1b !!
Same
Wait, one of the listed sponsors is Markus Persson.
Better known as Notch, the creator of Minecraft.
The creator of Minecraft is a Patreon supporter of 3Blue1Brown.
Huh.
He probably just has the same name lol
@@mathematoligiser-iserist2007 Nah I'm pretty sure it's him.
Desmos also supports him on Patreon. You know, desmos.com
You should be new around here. Look back few video's comments section and you see the same realization like yours. From those I heard, yes, he's the same Notch. He was even tweet about 3blue1brown
I wouldn't be surprised. He is a genius developer after all, and those brilliant ideas and concepts had to have roots from somewhere. Somewhere like these highly accessible, informative videos...
Re-learning this topic to start my second degree. 3Blue1Brown explains it better in 15 mins that any of my lecturers did in 4 years! Finally understand the point of div and curl for the first time! Thank you :D
Wonderfully clear exposition of what could easily seem a complex subject. Many thanks. There seems to be a great resurgence in the communication, or teaching , of maths, physics and biology on the web today and your offerings are right up there with the best. Keep it up!
This fantastic video not only helped me pass my minor subject studies in Physics about a year ago, it also helped me visualize and intuitively understand elementary electrodynamics, which in turn has helped me a ton with developing my intuition of chemistry (my major). And that is only the most concrete example in which being a 3Blue1Brown watcher has benefited me. Grant's videos continuously help me think about math in new ways, and that is already in the process of changing the way I think about science problems in general.
Your videos are a phenomenal treasure for us applied math regulars, perhaps even more so than for people who work with pure math. Either way they are consistently interesting and eye-pleasing. Keep up the good work!
I am in love with this channel. I teach everyone whenever I learn surprisingly beautiful ideas about the same old concepts of fundamental physics. This channel make me so happy and full of energy to share such amazing visualization. Thanks for existing on internet.
I went through Calculus 1-3 with absolutely no struggle to understand anything thrown at me. However, when we got to vector calculus, and especially circulation and flux/divergence and curl, I was kinda lost. I could do it, but I really had no clue why I was doing what I was doing and just plugging and chugging. I could not really grasp these concepts in any way beyond a vague sense that circulation/curl is a measure of how much the vector field flows with a curve and flux/divergence being how much it flows across the curve. Now what this meant or what the resulting answer actually represents was lost on me and I had no intuitive sense of what was happening. I’m glad to say that this cleared things up for me substantially, especially the portion about the dot and cross products and how they fundamentally relate to divergence and curl as the algebraic machinery being used just didn’t make sense to me before.
I was an ordinary person, Grant and Khan made me love Mathematics...Thank you grant from the bottom of my heart..😊
I know this might be totally inapporpiate, but watching your videos has made it so clear for me that I can in fact learn these type of things that I'm being taught in my university classes and that I struggle so much with there. That struggle has led me through a rollercoaster of anxiety, depression and suicidal thoughts, but watching your videos makes it all much better. Thank you for keeping at it and please never stop releasing videos :')
We believe in you!
Hi, I'm an engineering student from Italy. I really have the impressione that to understand everything you said in the video I shall watch it many and many times but in any case I want to thank you for doing such high quality videos reguarding math. Keep doing it and never stop! :)
4:49 is it just me or does the video show the opposite to what you're saying. Does curl F > 0 mean the twig would spin clockwise or anti-clockwise? Nevertheless still epic video
I think F>0 means anti-clockwise and F
Yeeeh was checking if anybody else noticed ...
Counter-clock wise is the default direction (positive) so it would make sense curl F >0 is counter clockwise and that what it shows in the video. AS for what he says, he makes a mistake.
He also mentions the duality does not merit a MUST law..
Check the description , he mentioned
I put this in the description, but let me just link this field flow interactive here because I think you all will like it so much:
anvaka.github.io/fieldplay/
3Blue1Brown it's so beautiful 😭
can you do quaternions? +what software do you use for the animations?
Quaternions are on the list. I've got a nice visual for them that I'm excited to share.
@3Blue1Brown
Your videos are the hands down, the best and of the highest quality, I have ever seen. I really love your animations, concepts, and thorough but really intuitive explanations.
I was travelling when the notification of a new video popped up. The first thing I did after reaching home was to start my laptop, grab my earphones and watch the video. Every time you explain something new a weird sense of joy flows in. I almost wanna cry with joy when I see it. Your intuition of fluid flow in the context of electromagnetism, it makes visualising those concepts so much easier! I am really grateful you make these videos.
I love you , thank you!!!
Man i'm crying out loud I wish they would have told me about this before I started doing electromagnetic field courses and stuff.. your video is enlightening
Thank you so much for making these videos. Previously in my life I had a fear of math and over the pandemic I found a love for it. When it comes to certain fields such as linear algebra, I have found that educators seldom take the time to explain what some of the content means. The content that you provide is super helpful from a conceptual standpoint as a student and I am very grateful that you continue to upload content. I can imagine that your work has done a lot to help others overcome the fear that many feel towards this field.
You are doing great work, keep it up!
Your videos are amazing: graphics, voice, audio, the whole package. I know all this stuff and I still find myself fascinated by all your work! I recommend it to any student having trouble with, say, vector calculus. Kudos!
I wish I saw these videos while I was in engineering. Watching the videos makes me jump from my seat and I feel like a child who just saw something very very beautiful and is excited to share it with others, talk with others about how cool is that! Thanks and lots of love.
Just when I started studying Gauss law in physics, here comes my favorite youtuber explaining the maths behind it!!!
Aditya lucky you. I am sort of jealous cant lie.
aboctok yea i didnt have blue brown holding my hand and literally showing me how its done.
Thank you for this incredible video. I learn a lot better with visual intuitive stuff, I am in my mid thirties and always struggled with mathematics until the last year or so. You're good at breaking it and breaking it down for us artists who felt like we could never grasp the secret language of math. Thanks for guiding us through these advanced topics with such care and not leaving me behind like my high school teachers did.
Absolutely breathtaking. Such a wonderful way to polish up my electrical engineering fundamentals as a 28-year veteran in the nuclear power industry. God bless you, Grant!
There is a kind of mind-blowing simplicity to everything that you do.
I feel like whole weeks of confusion in my life get suddenly unraveled with these visualizations that showcase _exactly_ the insight needed to understand something intuitively, without even needing any math at all. That is one hell of a talent that you possess.
Thank you.
Love from India! I'm an aerospace engineering student and your videos are very insightful
Came for a lesson on magnetism, left with a very prolific quote at the very end. Thank you for all the work you've put into this channel and spreading the joys of math, and aligning those incentives.
My professor just introduced the rot operator in fluid mechanics without explaining what exactly it means, just how to calculate it. Maybe my professor doesn’t know what the rotation of a vector field means. Thank god a genius like 3b1b can simplify the explanation so well.
Hi Sunshine, I want to thank you for the no commercials, an allowing people on a very tight budget the opportunity to gain knowledge from your videos. I've been watching videos on vortices in water flow, an also design for propulsion, I was noticing your wing design at the end of this video. I have also received a great deal of knowledge from Viktor Schabeger videos, the Water Wizard, so I'm looking forward seeing your next video. May peace an prosperity be with you always.
I always had a sneaking suspicion you where a math man Steve-O
Seriously...........
I always get a kick from your videos!!!
I ,being a 15 years old student, would not be able to understand these thing if I had learned it from somewhere else, but your way of introducing concepts and teaching maths intuitively is the best I've ever seen......
You're literally the most consistently amazing math teacher I've ever seen. Keep it up, the content is fun for those who know it and useful for those who don't.
One year ago, when I started watching your chanel, I knew only some basic arithmetic, geometry and maybe pre-algebra. Now, I'm learning Multiveriable Calculus.
Thanks to you for teaching me algebra, trigonometry, logaritms and inspiring me to fall in love with math.
Thanks to Professor Dave for teaching me Calculus.
Thanks to Brilliant for teaching me Linear Algebra and now Multiveriable Calculus.
You all have made a 12 year old know college level mathematics
i remember seeing the very first video you posted and immediately wishing i wasn't broke so i could help support this channel, but in a non direct sense I've still tried to support this channel. i have used it as a supplementary tool in tutoring jobs and have even shared it with college professors to give them more tools to teach classes with. i wish i could do more, but this channel is genuinely amazing and consistently keeps me engaged in my love for mathematics despite my resentment for the current college setup. i wish you the best of luck and would love to see you grow this channel and if you found people you click with and share your views add other topics in the same method, such as engineering principals, or possibly others that I have less knowledge of how to translate to videos like these. wherever you do take it, i wish you the best and hope the internet community can start to make a genuine growth effort for teaching each other productively and more efficiently that previous efforts. good luck and thank you for all your hard work. {end ramble}
Thanks for that fox rabbit equation. I thought it was funny to replace variables with pics too! 😄
Simeon Trieu, simple lotka-volterra system
What a beautiful way to TEACH those complex concepts that has been always in dark.
Congratulations, this is fantastic!!
The way u speak conveys much of the thought pattern and approach to maths/physics. Wonderful. Thx.
This is a pure embodiment of beauty, something unarguably good. Thank you for pouring your heart and soul into these lessons, sir. They hold immense value for our kind.💎 Just beautiful!
You are a godsend to anybody interested in science. This is how education is meant to be. From the application to a pursuit of intuitive understanding. Thank you.
I’ve struggled to understand some of these concepts and watching this video was truly eye opening. Thank you!
Que hermoso es este canal. Se ve a leguas el empeño que le pone, no solo en la animación, sino que lo explica de una manera tan sencilla que hace las ideas más complejas sean obvias
Electrical Engineer here, where have you been when I learned electrodynamics. This is gold, pure gold. Beautiful. Thank you!
Can your next video please explain how to send this one back in time to when I was in vector calc? I could handle the equations, but had no clue what I was actually doing.
Exactly what I was thinking
Same. First math class where I got lost conceptually.
Same here. This videos should be obligatory to watch in every math class. That would inspire so many people to study math because of its beauty not only to pass exams and forget.
Same with me. I remember crying because I could not understand what exactly I was doing. Even the lecturers were not able to tell.
Me too. Div and curl, vector calculus and that nabla thing were where my grasp on mathematics snapped. I passed the course but I never had a clue what I was doing.
A lot of math students didn't receive this depth of explanation nor with stunning visual aids during all their primary (and maybe even secondary) schooling years, especially in public as opposed to private
Thank you for working towards an easily accessible, professional format for mathematics
This honestly helped add another element to my physics paper, thank you
Oh this is mind blowing! Haven't seen a more concise and insightful explanation of Maxwell's equations. You are truly inspirational, thanks sensei!
Where was this video some months ago when I needed it? I've failed my Electromagnetism course because I couldn't visualise grad, rot (curl), and div and didn't know what they represent.
I passed it the 2nd time though, but still, it's great watching this video!
As a second year student in mathematics and physics, I find find this representation of those operators very enlightning! It has always been important for me to be able to visualise the objects I'm learning to wield, and I had trouble with this, especially the curl. I like that you only focus on helping us buid our intuition, we will eventually get to learn the laws in the university.
I also appreciate your efforts to be independant from sponsors, it adds value to your work, in my opinion at least.
Keep going on, the quality of your videos are through the roof!!!
i just obtained newfound appreciation for these div and curl functions after seeing this video. i've been using it a lot on my electromagnetics classes ad really needed this visualization to appreciate what exactly are these equations ive been solving
I love how even in his explanation for how much he cares about his audience, he uses math. He is clearly very passionate about his craft and that comes across in his content.
Writing my master thesis right now, your videos are like fresh air in a very demanding time! Thank you!
THIS is simply a piece of art!!! amazing explanation!!! i'm graduating in aerospace engineering and this was the best illustration i had in years!
I don't think you understand how much these visuals help us learn!! Praise to you and thank you!!
Very clear explained. Gain a "feeling" on the Div and Curl after seeing the video
so, i didn't think i would understand any of this video with my lack of knowledge of vectors and anything related to them, but remarkably; this was such a well made video that i enjoyed it still and even understood some of it. thank you!
You have done so much for my learning throughout the years, Grant. Thank you.
What a legend..!! One of the best Channels on TH-cam made to actually spread the love of math and science..!! Respect.! Thank you for sharing..!!
I have been looking for a video to offer this kind of illumination of Maxwell's Equations all my life! Thank you so much!
You have an amazing gift for intuitive explanations. I keep saying this on all of your videos but I've never come across a deeper explanation that was, at the same time, so understandable.
Agreed completely. He needs to be a teacher if he isn't
Thank you so much for everything you do. I will never stop recommending you to everyone who will indulge me in my love of math.
I have found your channel at the right time. I m currently in my 1st year engineering course. Thnx for these mind blowing math videos :)
There is so much to learn. U make all of us realise that we all can learn so much, Thanks for inspiring in every single video of yours!!
I don't care. I am going to keep repeating my praise for you in every single video you make. This is just too good man. Thank you so much for the hours you put into this. I'll be sure to revisit these videos and write some relevant comments too but for now let me try to convey how much these videos excite me.
No math text I've read, ever explained these principles so concisely and vividly. This content is exceptional.
You either don't read enough or don't understand enough. I learned this by myself from the textbook (Calculus by James Stewart) because my professor never got around to this topic. I aced Physics 2 because I learned this while my classmates didn't. In fact, if you think you understand these topics after watching this short video, you really don't know much about learning.
Thank you very much. I am an electrochemist and recently also previously, historically a fan of physics and math. Throughout my educational course, I very often there is something not yet there as most teachers they themselves doesn't understand that deep. I always think science is elegant, and they should be expressed beautifully. Thank you for your beautiful video.
These are videos that must be part of a curriculum on mathematics. The illustration and intuition of these series of video gives another layer of comprehension.
This is absolutely beautiful. I just learned how to apply Maxwell’s equations in practice in my physics class and finally chose to watch this video, this is such an amazing new perspective
This is one of the best illustrations of the concepts of divergence and curl! Absolutely loved it!
Ayyy it's about time I *converged* to the comment section on a 3b1b video. As usual, great lecture!
Faculty of Khan wow:-)
That was totally DIVERGENT dude! Also great intuition!
Man I always like to *curl* up with a nice math video
I conform to you
Ok that sounded like nothing to do with conformal mappings and there is no way I can make a joke out of holomorphism...
It is actually so beautiful to observe the mathematical transformation of water at 4:41, when the wind velocity changes in a semi-controlled environment.