But what is a partial differential equation? | DE2

The heat equation, as an introductory PDE.
Strogatz's new book: amzn.to/3bcnyw0
Special thanks to these supporters: 3b1b.co/de2thanks
An equally valuable form of support is to simply share some of the videos.
Timestamps:
0:00 - Introduction
3:29 - Partial derivatives
6:52 - Building the heat equation
13:18 - ODEs vs PDEs
14:29 - The laplacian
16:04 - Book recommendation
Typo corrections:
- 1:33 - it should be “Black-Scholes”
- 16:21 - it should read "scratch an itch".
If anyone asks, I purposefully leave at least one typo in each video, like a Navajo rug with a deliberate imperfection as an artistic statement about the nature of life ;)
And to continue my unabashed Strogatz fanboyism, I should also mention that his textbook on nonlinear dynamics and chaos was also a meaningful motivator to do this series, as you'll hopefully see with the topics we build to.
Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
------------------
Animations 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 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.
------------------
3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with KZbin, if you want to stay posted on new videos, subscribe: 3b1b.co/subscribe
Various social media stuffs:
Website: www.3blue1brown.com
Twitter: / 3blue1brown
Reddit: / 3blue1brown
Instagram: / 3blue1brown_animations
Patreon: / 3blue1brown
Facebook: / 3blue1brown

[@matthyslaubscher8151]

I can not give enough praise on the quality of animation coupled with the explanation. Amazing - Thank you

[@shubhmishra66]

Same here.

[@arsd9021]

yes, its amazing!

[@mikhailchannel8122]

I want my student loans back, amazing!!! I spent hours and loans trying to figure it out. Thank you thank you

[@AG-db6ef]

Totally agreed, it's phenomenal.

[@bclamore]

Ditto.

[@smartereveryday]

A modern mental masterpiece.

[@vishank7]

Yepp it is truly beautiful! Also, it is a delight to see you here sir! Please keep up the good work😄💎

[@VinOnline]

why only 3 replies?

[@athul_c1375]

@@VinOnline now 3

[@rahulchandra8938]

4

[@josephtraverso2700]

5

[@MrMaltheWG]

I have studied mechanical engineering for almost 5 years now, and this is the best explanation of PDE´s and fourie series I have ever heard. Thank you very much for fantastic illustrations and intuitive explanations.

[@freedomchinaify]

Malthe Wellendorf Gissel i studied electrical and we also need to be comprehensive in PDE

[@Guztav1337]

I'm in research physics, and this is a helpful starter

[@pablofernandezmartin2221]

P

[@johnmelon45]

1 year after my msme lol

[@binodtharu8348]

@Kaynbock Mehr being a high high school student who has only learnt 1st order "simple" differential equations, these series of 3-4 video of differential equation literally took me 4-5 hours to grasp.

[@kiwipaddy10]

3B1B deserves a Nobel Prize for explaining math! Simply outstanding, yet again.

[@MrRipper9]

Pero vaya!!!

[@divyansharora6788]

It is the Fields Medal i think...

[@duruguner6239]

@@divyansharora6788 In field of math education, it is Leelavati Award

[@madhavsirohi2225]

AGREED 100 PERCENT, but I think it is is called the field's medal 😂

[@ignacioaguirrenoguez6218]

A Nobel Peace Price

[@Alex_1652]

This channel made me realized that I don't actually hate math.

[@zyansheep]

Same...

[@segmentsAndCurves]

@Leopold Maximilian von Hagen Believe me, you can't hate math when you're self-learning. Except if you hate yourself, of course.

[@segmentsAndCurves]

@wickedcat We are far from being that rational, my fellow human.

[@fabio19h]

@@segmentsAndCurves we are rational but not in the philosophic way. In the mathematic way we are pretty much rational

[@segmentsAndCurves]

@@fabio19h You got me good. NOW GET OUT OF HERE.

[@debblez]

3:45 i love how x goes from pi to e^2 to phi. Amazing little easter egg. Edit: what the heck likes

[@dcs_0]

and then to tau

[@joelsagflaatholmberg3922]

I saw it too.

[@anirudhsathiya3516]

You guys definitely got some sharp observation skills.Thanks for mentioning!

[@theavo]

Yeah, nice one, guys!

[@moritzw.8730]

The Easter egg at Easter Sunday

[@saidelcielo4916]

I've watched countless 3B1B videos and they're all illuminating and high-quality. This one, among those, is an absolute masterpiece.

[@dainmeister]

2:23 Mathematicians: function Linguists: letter Musicians: forte Gamers: respect

[@subhramitbasu3886]

Hotel: Trivago

[@Some.username.idk.0]

@@subhramitbasu3886 hotel: Frivago

[@arvidjohansson3120]

Big pp

[@anteconfig5391]

@@arvidjohansson3120 I'm so confused... is this from somewhere?

[@arvidjohansson3120]

AnteConfig Yes it’s from “somewhere”, the land of the 9 year olds. If one these formerly mentioned 9 year olds posts a good meme. The highest level of respect a 9 year old can show another 9 year old is to typ Big pp. One thing that could be off use is to clarify who a 9 year old is. The concept of a 9 year old is bigger then age. You have to be apart of something much more meaningful. You have to be apart of Pewdiepie’s loyal following.

[@3blue1brown]

Next up, we'll look at how to solve this with Fourier series, including some discussion of boundary conditions. Stay tuned! Edit: I see a number of questions about changes at the boundary, so maybe I’ll add a quick note. For example, some of you ask if a function with constant non-zero slope is stable. Indeed, this is something that needs to be specified for a well-defined solution, so good question! The heat equation as described here only describes the interior. The easiest boundary condition to work with is when each endpoint is held fixed, e.g. if the ends of the rod were somehow constrained not to change temperature. In that case, straight lines are a stable solution. But other conditions can be specified too, as you’ll see!

[@schokoladenjunge1]

Cant wait to see if you use the only good convention for the prefactor

[@aksel9290]

Thanks for this, writing some coursework on this very topic right now and this really helped clear some stuff up, certainly one of your best series. Also nice choice of numbers @ 3:45

[@joshuagross6022]

What if the temperature distribution is linear? Then the second partial with respect to x is zero but my gut says that temperature should still ballance out

[@eunhyoukshin7777]

@@joshuagross6022 There was no assumption about the boundary in this vid, so it would correspond to a constant heat flow

[@randomdude9135]

@@aksel9290 1) pi 3) Golden ratio 4) tau What's 7.380? Edit: Sorry, 7.389 Edit: Nvm, I read it in comments. It means e^2.

[@flirkami]

Those graphics are really getting exceptionally good! The 2d representation animation at 4:13, for example, is just gorgeous.

[@paradoxicallyexcellent5138]

Yeah that moment struck me too. He's getting quite good.

[@xyzct]

I love the Fourier series section.

[@sds123faf]

I came to say the same thing. I hope it does not take a lot of time to generate them as it looks so

[@pedros8681]

does anyone know what tools he uses?

[@MariaNicolae]

He makes his own tool in Python, you can download it here github.com/3b1b/manim

[@Brynmawrhill]

5:30-6:15 Got it! Partial differential equations aren't actually partial. They are complete - but only as a description of part of the system. Thanks!

[@mtthwberops]

I wasn't understanding a thing until I read this comment... Thanks 👍

[@codersgarage2279]

And, I just actually think that this is the best channel to gain rock solid intuition in math. Thank you ☺️

[@Noah-rp7rv]

I’ve done the multivariable calculus lessons you recorded on Khan Academy and I gotta say, I’m getting flashbacks

[@evrik78]

I've just had an orgasm...

[@michaelwang1730]

Me too!

[@Confuseddave]

Good ones, I hope?!

[@Noah-rp7rv]

Confused Dave Of course mate

[@shakencocktail7137]

Can you put the link of these lessons to all of us?

[@MrBrightlight66]

I used to hate maths in a very profound manner. The reason was that it was presented in a manner like a recipe...do this that and the other in such and such a way and good food comes out. There was no understanding of what was going on and this was most frustrating. The internet, through people like 3 blue 1 brown, helped to propel me to the opposite side of the fence. Although I am no mathematician, I can now appreciate and admire the beauty and power of mathematics. There was clearly a lot of work behind this video to illustrate a complicated subject relatively easily. I wished I had access to such a tool when I was much younger. It would greatly help the younger generation to give them access to material such as this and to encourage them not to shy away from maths. As with most things beautiful, their creation is not easy but nothing to be terrified of either.

[@sukhendramajumder7656]

same here

[@giovoncolon9104]

I feel you on that!

[@elsamhariyanto1972]

preach it bro..

[@MikesKorner]

I definitely agree

[@Leyrann]

This is why a good teacher is so extremely important. Bad teachers present the subject matter, and the students are obligated to practice is, and start to dislike it. Good teachers explain with passion what their subject matter is used for, why it is done this way, and how amazing it is to capture something complex in such a simple way, and the students become intrigued and start to share in the teacher's passion.

[@olehborys1462]

Just want to thank you for what you do for me and milions of other students, trying to understand such a difficult (but very intuitive after your videos) area of math. Your explanation if very clear, animation is outstanding. I really hope you will continue doing videos this way.

[@idobenamram3743]

seconded

[@Gabriele_Oliva]

I'm Italian, and I haven't yet formally studied these topics, but your explanatory skills are incredible, you are able to discretize a complex problem in all its basic steps and then put it all back together. I understood practically everything. I don't think I can solve a similar problem by myself, but it's a step forward, really thanks

[@KraylusGames]

I'm a computer science grad and I've taken a handful of classes which required the use of ODEs and PDEs. After watching this video, I feel like I have finally gained a full conceptual understanding of how they fit into the larger scheme of operations in math and cs. The visualizations you created were absolutely incredible, but the explanation was perfect. Thank you so much!

[@antigy7962]

can you reference some pdfs or channels to learn odes and pdes related to computer science?

[@tjoohsanehwai]

Yeah @anti gy I have the same question, any recommendations @Kraylus?

[@GamerTheTurtle]

tf2 engineer tf2 engineer

[@connorseunninga2324]

comment exeptional n a created have you

[@alyriatutoring5697]

Likewise...The only classes I used DE’s in really were for Engineering Dynamics and Vibration Analysis (Civil Engineer) but I never really understood what was going on, this is a great video

[@rrrprogram8667]

Oscars award for animation to be given to this channel

[@BlueyMcPhluey]

move over Spiderverse

[@education4949]

Omg !, he should be given the highest award in today's world for explaining partial differential equations. Respect and a salute

[@wyboo2019]

this series makes me feel like i know more about DEs than i actually do. so i watch this, get confident i can solve some problem on my mind, and then realize that understanding the general mathematical ideas behind the concept does not automatically translate into being able to solve specific problems

[@myrobotfish]

Nothing more wholesome and satisfying than a 3Blue1Brown video

[@peterfazakas7499]

If i could like your comment 100000000 times......

[@raphaelfrey9061]

A video from eugene khutoryansky is also a dream

[@ChrisAthanas]

This is so great, I finally understand this concept after 32 years

[@nourghazal296]

wait what

[@user-hh7ge2zn1v]

Wait what 2 !

[@miffyn1737]

Congrats for not giving up learning this. I just got introduced to this topic last week by my lecturer and almost losing any interest to look it out. Thanks for the inspiration

[@howmathematicianscreatemat9226]

Dont need to be ashamed. Most textbooks are not really made to understood, they are just products to remember.

[@ChrisAthanas]

@@howmathematicianscreatemat9226 that seems to be the case Thank god for KZbin

[@hotsaucebeliever]

I'm a postgrad and your videos have saved me more than once. Thank you and please continue to remind people of how beautiful math is

[@abc3631]

The animations are beyond phenomenal, so original and so effective in putting the ideas across. Can't appreciate him enough for putting together all these lessons.

[@vg5028]

pleasantly surprised to see 3k views in 10 minutes on a math video

[@AK-km5tj]

@O r a n g Now 25k

[@AK-km5tj]

@O r a n g You're right. People love 3B1B more than I thought. The only person at my high school I know who knows who 3B1B is. is my math teacher. It's amazing how he can inspire people to love Mathematics. Especially math of Multivariable Calculus.

[@MindMathMoney]

I love this channel! ❤️

[@paradoxicallyexcellent5138]

@@AK-km5tj Yeah, I've done a math phd, a job in math education, and now I'm at a tech company, and regardless of where I am, I find that fairly few people know 3Blue1Brown. So I evangelize a bit. I'm giving a presentation on Friday in which I'll be referencing him.

[@natehoxie1914]

@@AK-km5tj When I was young I found math difficult and frustrating, but then once I got decent at it, I started to see how beautiful and extraordinarily useful it is. If you can understand this stuff in high school, you have a great start on life. Good luck, and keep learning friend!

[@googlesellsmydata]

3Blue1Brown, this kind of stuff is what I use on a daily basis as a software/mechanical engineer. One of the only disappointing aspects of my work is my inability to describe what I do to the layman. I have to omit huge elegant ideas. I really like that you've captured these ideas in a beautiful and easy to digest way. I cannot wait to see and share the rest of this series.

[@flaviusclaudius7510]

I'm a quantum physicist faced with the same difficulty; this channel is a great resource.

[@connorskudlarek8598]

What do you mean software/mechanical engineer? I'm mechanical and really don't know how I could be both, so I'm very curious as I enjoy some software work and more complicated math.

[@maniys]

Mecatronics?

[@googlesellsmydata]

@@connorskudlarek8598 sorry for the slow reply! I develop internal software tools for an AM startup. We have a welding system with a high dimension parameter space. My team and I do everything from simple GUI design to thermal simulation to g-code. Right now we're testing the waters with ML.

[@nandakumarcheiro]

Kindly explain the pi shifted heat transfer as isothermal expansion.

[@pkmpkm03]

Man....I feel so lucky to have found this amazing channel for learning. I want to thank and give my gratitude to the person/s behind all this hard work.

[@sekaranchandrakumar5774]

The visualizations, together with the explanations, are absolutely amazing!

[@CStrik3r]

Me: "I'll just watch the first 5 minutes to see what it's about and get back to work" *watches the whole 17 minutes*

[@DyslexicMitochondria]

A constant and e^x are walking down the street. Suddenly the constant screams a horrible sound and yanks e^x in an alley. "There's a differential operator coming against us - we must run or he will reduce me to nothing!" e^x answers calmly "Ok, you run - I'll be fine, the diffop can't hurt me at all". While the constant makes for his escape, e^x keeps walking. When approached by the differential operator - e^x says in a smug voice Hey, how's it hangin'? I'm e^x. The differential operator replies Oh, hi. I'm d/dy.

[@tomhappening]

haha

[@pawelrozkrut157]

All these series are beyond phenomenal! To say they are ultra-meticulously designed and edited- would not do the justice. This is a masterpiece! Hats off!

[@joelklein3501]

I am a student, currently at my second year of my bachelor degree at Physics and Maths. The past semester was more hectic than ever. Especially now while doing the finals of this semester's courses, watching a video like this is so refreshing, and reignites my drive towards what is bound to come along the way. I love talking and listening about Math and Physics, in a figurative and passionate way. I like seeing the beauty of it, that's why I'm doing my degree. Thank you for lightening up my morning!

[@MindMathMoney]

*The essence of mathematics is not to make simple things complicated, but to make complicated things simple* 🕊️

[@schokoladenjunge1]

3rd semester mathematics students would like to know your location

[@TrinityPR]

Me too

[@timh.6872]

It's more that the essence of mathematics is not to solve complicated problems, but instead build a landscape of definitions around the problems so that they become simple.

[@MindMathMoney]

@@timh.6872 Yeah! The question is.. Is math a language created by us humans to understand the world OR is math a universal language? 📚

[@schokoladenjunge1]

Having a framework which just exists outside of life implies math to be embedded in the foundations of the universe, which isn't the case If that were so, every single mathematical expression would have physical implications Or more explicitly: there are unphysical solutions of General relativity. GR is math. So that math has to come from our minds since it doesn't describe anything in our world.

[@rogerdotlee]

At last -- a channel that actually shows the guts of the math that everyone else glosses over. Yah. I'm hooked.

[@jiffylou98]

Well, there are a lot more guts he's not showing, but he does better than pretty much every youtuber that's not just a professor giving a lecture.

[@mathpgm601]

I bet most university teachers don't have a clue of what they are teaching their students. Imagine if math was taught like this in colleges. (chuckles)

[@ParmitaMishra]

Even when I "know" (am aware of) a concept in one of your videos and have been practicing it for months or even years, watching your videos still implants a new way of thinking about it. Takes the "intuitive factor" up a notch. This is brilliant.

[@michal9775]

At 3:35 I have a feeling, that the script goes: "...what the f. is going on in here" and you somehow saved the situation at the very right moment

[@yoavmatia]

AMAZING!!! I am so happy to be a PATREON member of this exquisite channel. I have never in my life seen such a clear unfolding of what a PDE is... Thank you so much !

[@404_pagenotfound7]

the reason why i love this channel is it can always give a “wow moment” of thinking about something in a different way

[@zvw2015]

How? How how how does this guy have the best graphics? I’ve never been able to get over how well he explains everything visually. Honestly the best on KZbin. Thank you for helping me both feel dumb AND smart ish at the same time!

[@shairotbard9890]

WOW, I could never understand what my PDE lecturer tried to explain without you! :) Thanks!

[@marklewandowski1195]

This is truly the most elegant explanation of a PDE I have ever seen. Your ability to make seemingly complex math so visually intuitive is a gift and I am just grateful to be able to experience it. Never give up on this talent and passion of yours

[@Funkymuffins123XP]

These animations are mindblowing, so much easier to visualize like this

[@mburmester719]

Your videos are awesome! The way you present math, as others pointed out already, really stand out from the procedural learning approach applied so often. It's fascinating to watch and begin to properly understand these concepts. I take my hat off to you in the most sincere manner.

[@agimcomas]

I just found this channel and I am amazed at the graphics, the explanations, everything. All I can say is, thank you.

[@Danilego]

Wow, this was a great step up from chapter 1! You really managed to simplify and make a hard topic easy to understand, plus the concept and animations of this one were particularly satisfying!

[@OrangeC7]

I really enjoy this video! One of my favorites from you so far, you really drive the point home. I especially like how far you've come with animating 3-dimensional graphs. :D

[@alecplano9563]

Your animation skills and ability to explain complex ideas is stellar.

[@benwhaley4830]

Awesome video! Loved the explanation and the animations, you made a stellar video. Great job!

[@pspicer777]

Never more clearly explained. Modern students of mathematics are so fortunate. Back in the day (yikes!!), it took me days to understand what has been outlined here in minutes. I second the notion made in the comments to do a video on the graphics tools used. Absolutely stunning.

[@mikarox2]

This is probably one of the most beautiful, satisfying videos I've ever ever seen in my lifetime. I sometimes have the wish of making KZbin videos on math or physics in my language (br portuguese), but every time I come visit this channel I feel like I'm far from ready. Thank you Sir!

[@parthasarathi1711]

This is Mathematics, art, science, poetry in one package..Mindblowing and subtle at the same time

[@howmathematicianscreatemat9226]

Great observation

[@ryanj7986]

I think just about any math class at any level could benefit from 3d animation like this. Maybe it's just my own learning style, but seeing these real-time visuals of how things change and where things go makes a HUGE difference in my understanding.

[@sclarkaz]

I got an A in PDEs and now know it was blow off. I learned more hetr than a semester in a classroom. My wife thinks I'm crazy getting hooked on math videos, but you all know why. Bravo!!

[@theredbobcat]

By far one of my favorite channels on all of KZbin! Thank you so much for engaging me even more in mathematics.

[@paulleveque2624]

can't even find the words for how good you can make people understand such hard things ! as a student in theoretical mechanical engineering, I'm just amazed and so thankful that u gave some color and visuals to the letters and white boards of my teachers at university ! thanks ! aaaaaa LOOOOOOOTTTTT

[@chawza8402]

I just realized that the smooth animation you use is really helping the viewers (like me) to understand it like the real representation of the subject. good job for all your works sir!

[@utkarshverma1]

This video was just downright awesome! I used to be intimidated by partial derivatives, but the graphs and animations helped me understand that they are quite simple. Great job 3b1b.

[@Ali-yr3eq]

The effort, intuition and love that goes behind these videos is so real and well thought and the eloquence that comes out is satisfying. It’s about time that education is revamped as rigorously and intuitively as in 3B1B... 💕

[@jisaburo198]

This channel is one of a kind. At least two or three cuts above all others attempting to illuminate math. BRAVO!

[@stephenphilbrook6239]

Greatest math videos ever! I had to watch some for a college class and now I watch them just for fun. They are so well done. You really bring math alive with your animations.

[@wolfie6175]

Man I love how much effort you are putting into this.

[@livintolearn7053]

THIS IS HONESTLY BEAUTIFUL! And your teaching skills just keep blowing my mind!!

[@sanjinred]

I have a degree in structural engineering and have spent my time in college with ODE and PDE. This is by far, the most elegant and intuitive description of PDE than any of my brilliant professors were able to give us. Your channel is brilliant.

[@nicolasthorne2162]

Your work has so much value you cant even BEGIN to imagine. THANK YOU

[@ValentineRosemary]

Learning about this isn't exactly in my major of studies but it really is helping me understand many other concepts more intuitively...thanks for the awesome video as always!!!

[@popop614]

The 3D Animation on this is amazing.

[@juandiegoorozco5531]

the effort you put into this project is stunning, thank you

[@redwoodenjoyer]

You really are the best teacher and creator here, dang! Im 2 years away from taking any classes that would cover this, but its mesmerizing on the different ideas it covers. Also its nice to learn it but not for a test. Thanks!

[@yamilessus6157]

I'm amazed for the amount of dedication I find on these videos! Keep going!

[@harjotsinghbaidwan2204]

You are really passionate with mathematical concepts , keep on going like this you are helping many of us with your great efforts :).

[@JoaquinBS92]

This video is one of the most impressive things I ever watched. Thanks for the time and effort put on this!

[@shahroozgovahi674]

This is by far the most beautiful description of a math concept that I've ever see! Well done!!

[@tushnim_5996]

The animations are getting fluider and prettier. Love it

[@AryVinicius]

this takes the visualization and understanding of DE to another level. Really good content.

[@Erik_Caballero]

You're an absolute inspiration and I love your work! I've never officially taken anything beyond calc 2 but I've been trying to figure out how to simulate fluid flow for weeks now (I'm a freshman in college and I got a position in the university's aerodynamics lab where it's my job to learn how to do this stuff; I got thrown into the deep end a bit) and your videos have explained what 3 weeks of research couldn't! I feel like there's a way to actually go about doing this now. I've never been so excited to have a general understanding of a mathematical concept in my life! You're an amazing teacher! Thank you so much!

[@GungnirStrike]

I watched your intro to calculus video before I went into calc 1 and now I'm watching this as I'm going through calc 3, I must say your videos do indeed make math fun. thanks for the great videos!

[@GKFuzzy]

in 4 years of engineering i've never trully understood the meaning of diff equations, i knew what the result was for and how to solve them but this is simply enlightening. thank you so much!!

[@tobby12347]

This is a wonderful video! It's been a few years since I've taken a class that uses partial differential equations. It all came back to me so easily thanks to your presentation.

[@kli7867]

This is one of best explanation I have heard about movement vs time. Thanks for all the graphics that pin point every small piece of equation

[@phoumint]

I can not describe the feeling I have right now. Your video is so informational. Thank you so much for all of these videos.

[@quahntasy]

Such beautiful representation of abstract ideas. Loved it.

[@Aelipse]

I don't I've ever seen such a clear and aesthetically pleasing visualisation of mathematics in my life. Absolutely gorgeous.

[@patz8995]

those animations are simply ridiculously good, thank you so much

[@sammyofer9874]

Mechanical engineering student here! I've had a math professor literally tell my class that he will make us suffer and suffering is necessary to learn. So, thank you for not making me suffer and allowing me enjoy learning about math!! :)

[@chrisoman87]

Teachers like that set me back years in learning mathematics, when you preface something as a chore you shouldn't be surprised why students aren't excited about studying the content (unless they are masochists), teachers like Grant are the opposite, his enthusiasm over the years has pushed me to explore and begin to love the math I simply put in the "chore" basket.

[@matthewe3813]

nice profile picture

[@stugranger731]

The only videos Im torn between the nagging growing urge between clicking the like button before I might forget and the inability to disengage from full screen for that tiny period of time. Your videos are always a highlight and inspiration Grant, thank you

[@david-yt4oo]

this video was AMAZING, I hope to see more videos like this one, and I'm pretty excited for the next chapter :)

[@FredCOdom]

The most incredible series I have ever seen. Remarkable in content and presentation. I may not be the oldest of your students but certainly look like I'm the oldest (74) in respect to the average of my neighbors on each side. I believe you'll get my drift. I remain mesmerized on each course and how I can connect throughly even at my age. Thanks Fred Odom

[@edwardhartz1029]

I absolutely love your attitude towards mathematics: Build intuition and get a watertight understanding before tackling problems. Keep it up!

[@mathevideos9909]

Dang, there is so much to learn from these beautiful made videos, even in the fifth year of studying maths at uni. Thank you!

[@ericb.4385]

I do really love your programming skills. Just awesome

[@Abhishek-hy8xe]

Amount of thought put in putting the things in best way in this video is amazing and it has truly made things intuitive. This is the place where I learned how to read PDE.

[@magnet2593]

No need to say how marvelous and unique your work is, I wish you extend this subject to some other cases like discussing Schrödingers wave equation. Thank you very much for your greatness.

[@williamvasudevan]

I just need to watch one of your videos, and suddenly, I there was anything that I was confused about, it immediately becomes clear to me!

[@theinvisiblearmadilloofdea6204]

I like how at 3:46, the notch goes from π to e^2 to ϕ to τ. Nice touch, and great video as always!

[@gautamsandesh6076]

One of the best animation on PDE . Hats Off to your hardwork.

[@adityakrishna4101]

This is just amazing visualisation to understand the fundamental level of DE’s. keep up the good work!

[@sanki1404]

Just WOW... Generally the Fourier equn. is derived by cartesian cube and using taylor series in books and classes..but even after that it is not as satisfying as your explanation of 'comparing it to avg of neighbouring points'... Your way of diving into complexities is very helpful and intuitive and much appreciated. Eagerly waiting for next chapter. 😀😀

[@Hodja60]

This is so well presented - I love it!

[@viper619ful]

THANK YOU SOOOO FREAKING MUCH !!!!! You have no idea how much you are changing lives. Keep doing the good work. You will have our love & support for the rest of eternity.

[@raedar7784]

how can this video be this great. i cannot thank you enough!