@@carl8703 It could also work as a way to introduce/spread your own definitions, by ”fooling” the audience they are widely accepted. You’d have to be quite confident you can convice your audience though.
"Fun little exercise" and "Ordinary differential equations textbook" are two groups of words I didn't expect to ever hear in the same sentence again now that I'm done with uni
I've spent a half-hour on this video and I'm only one minute in-I'm cross-referencing everything he says. If you get on my level, you might enjoy yourself more!
I've been a big fan of the Calvin and Hobbes comic strips since I was about ten years old. One storyline began with Calvin receiving a mysterious letter in the mail that appeared to be from some kind of secret agent -- the text consisted of individual letters clipped and pasted from magazines, the return address was nothing but a skull and crossbones, the writer advised Calvin that a coded message would soon follow -- all the tropes. The strip ends with Calvin sprinting off, yelling "This is so cool I have to go to the BATHROOM!" I found that response to be both hilarious and relatable. Calvin had no idea what was going to happen next, nor if he would even be able to figure any of it out -- he just knew it was so bizarrely exciting that he had to drop everything he was doing and hope to make sense of it just for the sheer joy of it. Anyway, that's pretty much exactly how I feel whenever I hop onto youtube and see that 3Blue1Brown has posted a new video.
I don’t know what it is about the way he speaks, but it always sounds like he’s smiling. He’s the sort of teacher that any parent hopes their child will have.
Incredible script man! My best line: Cynically, this is abuse of notation; but charitably, this is the cycle of discovery and invention. It's not just the visuals, but the script behind it is poetic. I love this guy!
God I wish I had it when I was one. I just shouted out loud asking why we weren't taught that a derivative of a vector can be expressed as a matrix multiplication. I wish I could explain how many months of second year that single thing would have saved me
I just love this intro, pointing out that the notation is initially nonsense but then receives a meaning we choose to give it. I will always remember how one of my professors introduced distributions: "Alright, we've hand-waved a definition of the Dirac delta-"function" which is "infinite" in one point and zero everywhere else. That makes no sense, a function can't do that. But here's what we're going to do: we're going to write an integral over this "function", then we're going to _cheat,_ and then we'll end up with something that makes sense in the end. Right now, this integral is nonsense. So let's instead _define_ a meaning for this integral. And that's it! We'll call it a _distribution_ instead of a function. The Dirac distribution will be meaningless outside of an integral, but completely well behaved when in an integral." Beautiful cycle between discovery and invention indeed!
Your professor is a genius 🤩. I just love those aha moments when something just clicks and which later on give you so many moments to grin while solving examples 😁
You are right. I can only regret that such a thing did not exist 40 years ago, when I needed it. Now all I have to do is admire the beauty, simplicity and clarity of this way of interpreting mathematics.
It is freely available, because it is profitable. Such videos takes a loooooot of time. Nobody would do it for free. 20 years ago I am sure he would try to make a VHS or DVD series. But there is no guarantee for selling. TH-cam is a guarantee. Patreon is a guarantee. If he would not get paid he would not do it. Money is everything. And as long as money is "enough", such videos will exist.
but it's not always trivial to download it, so you really have it, I first failed, but then found that I had to update the downloader software, as google video of some reason change their protocols so often.
I've got an exam on advanced methods in quantum mechanics and I can say: So far I got a good overview over quantum mechanics but Im still baffled by relationships.
There's a reason I have been pursuing one of these things before the other, and it's certainly not that I'm proactive to the point of engaging with difficulty first.
I teach graduate quantum mechanics to physical chemists, and I love showing them the videos you make, especially ones that visualize some of the maths relevant to QM. Please keep making them
@@andrewcihon-scott2170 I'm lecturing something called "Dynamical systems" for just one year and we've covered lots of this, characterising the fixed point of a 2D system of 1st order linear DEs by the behaviour around the fixed point of the system... And I'd never considered the relation to control theory. We need a grand unified theory of control dynamics!!!!!
I legit just learned about state space in my first introductory course to control theory, and, while I probably don't appreciate it as richly as others, it still will probably help me with intuition going forward.
A topic I've recently been learning about is Clifford Algebra, and while it usually isn't done with matrices, the various objects have matrix representations. The algebra also encodes the complex numbers. That matrix is the actual matrix representation of a+bi in Clifford Algebra, despite the fact that the the unit vectors it usually acts on look completely different in matrix form than normal unit vectors.
@@adjoint_functor If you squint, the formula for the inverse of a 2x2 matrix kinda looks like the formula for the multiplicative inverse of a complex number, z^-1 = z*/|z|; that is, the complex conjugate (the number but with the imaginary part multiplied by -1), divided by the magnitude of the complex number. If you also watched 3b1b's videos on complex numbers, the relationships are kinda neat. You can see from that formula that the magnitude of a complex number is equal to the determinant of its matrix representation, and the multiplicative inverse of a complex number multiplying to equal 1 is mirrored by the matrix representation's inverse multiplying to the identity matrix.
@ゴゴ Joji Joestar ゴゴ Additionally, for functions of multiple variables, you can take the exponential of a vector dotted into the gradient to obtain a shift in the direction of that vector For that reason, I usually remember Taylor's formula as f(x+ε)=e^(ε•∇)f(x), where ε and x are vectors
I would estimate >200 hours of work to get this video done. It would actually be nice to know how much time 3B1B actually spends on these creations. We're blessed and don't even know...
@@fzigunov ive used manim (the language he used to make his animations) a little bit and it honestly takes about an hour to make one scene so god knows how long it takes him. Propts to him for this
I just want to say that you're my favorite youtuber. You help me so much during my study in university and getting a master degree. You also reignite my interest in math, which I previously thought wasn't even possible. Words can't express how much I love your video and the way you teach math.
@@3blue1brown I came searching for the sequel. I was sure i watched it years ago, but now i realize maybe you never published it. Am i right? (By the way, terrific job. Im a long way fan of the channel, i'm currenly ill, and binge watching all your videos-again!)
I remember during my study where e to the power of a matrix confused the hell out of me. I desperately would have needed this video. Now I'm watching just out of pure interest. 😄
I did my engineering sometime ago, I remember seeing the e ^ (matrix) and I remember losing it and thinking wtf....You really have a way of explaining things so elegantly.. I hope I encounter these kind of problems in my career - Very enlightening....Thank you!
this is what i love about 3blue1brown, i have to carry out these calculations in my qm class all the time, but i have absolutely never understood the intuition behind it until now
What I love about your videos is that you might not understand parts of them at first but as you go back to them later, perhaps after gaining some knowledge, things start to click.
To the time I write this comment, Im a Mech.Eng. student and currently taking a course in dynamics. Our professor showed us a proof that if you derive a vector, the new vector is perpendicular to the first. I asked if we can determine the orientation, and he replied that he doesn't know. I'm watching this video at 3:50 a.m. suddenly it struck me - some of your equations and animations made me realize that we can determine the "natural" orientation, and even understand gradient and curl itself. And this ain't even the purpose of the video. Thanks for making math that clear, I'll even say that obvious, over and over again ❤️
@@3blue1brown joke or not, I’m still grateful that you still upload content. Your channel is a great contributor to making me pursue a career in applied mathematics.
"Often this is a function. (But whatever, functions are really just infinite-dimensional vectors)" I hope that parenthetical is a teaser for a future video on Functional Analysis.
Where was this channel when i was an undergraduate 20 years ago? The 18-24 year old students beginning their studies of higher mathematics have such amazing tools available to them now!
I don't always make the follow-ups I promise quickly (*ahem* probabilities of probabilities *ahem*), but for those willing to wait a few years I feel comfortable promising all will come in due time.
Mathematicians before e to the x extends the definitions: "We do some more theorems and we finish with math" Mathematicians after:"What if we exponentiate EVERYTHING"
I hope that kids these days can really appreciate how lucky they are to have channels like 3B1B. I can clearly remember the time when I sat in my calculus classes and the concept of matrix exponents was first used. I was too busy to write down the professor's derivation from the chalkboard to really think about it. But when I reviewed my notes back home I thought to myself: "What the hell does that even mean". Of course I figured it out eventually. But channels like this should really help today's students. Because professors rarely take the time to motivate something with examples. It is just definition -> theorem -> proof -> repeat. And of course that is important. But sometimes you miss the bigger picture.
This is beautiful. This is mostly why Dirac notation is so powerful: Dirac must have similarly reasoned that Schrodinger's Wave mechanics and Heisenberg's Matrix mechanics were essentially describing the same thing!
The summation of the wave functions on Schroedingers equation makes much more sense now than it did in college. Grant, where were you 30 years ago? ;) Thanks for making math exciting again! Want more STEM excitement? Show these videos in junior high and high school. Maybe earlier.
Finally!!! I really missed those pi-creatures (and also your voice, Grant! It is hypnotizing enough to make people stick to your videos without giving up, no matter how fighting the math in them can get sometimes😳)
i'm only in elementary school but i love this channel. It really just shows the beauty of math, and inspires me alot to learn more. Thanks for making these high quality videos, much love and respect (:
as someone who studies physics with the "just enough" math knowledge, this video is mind blowing to me, it makes me realize that I should be looking more into the math of the physics phenomenons. Physics is applied mathematics, but physics also spark ideas, problems for us to develop models, concepts that we have never thought about!
I am so excited to learn maths this way from you. I have a PhD in physics, and am more than 20 years into my career as a scientist. I do a lot of calculations to support my experiments, so I have a basic understanding of mathematics, but I feel like a fog is being wiped clean when I watch your videos. Thank you so much!
3 years later, we are still yet to receive the highly anticipated 7th part in this series on Laplace transform. Wonder if the video has now been scrapped forever..
I'm a third year physics student and by now, I have seen a fair bit of QM. It's amazing how you always manage to give new and visually intuitive ways of imagining calculations, that we do so often without further thinking about it. I wish our lectures would be taught more like this.
Grant, you are one of the most important educators in the modern world. To have your library of presentations in my pocket at all times is one of the best feelings there is. Math went from my least to to my most favored topic of study because of your content. May you continue to be an ambassador in this field!
I remember the absolute outrage of my class last semester when we saw this for the first time. Several people called it abuse. This vid makes it seem much nicer.
@@SimonClarkstone well no you need some more conditions, it needs to be an algebra over ℚ to allow multiplication by 1/n! and (unless you're satisfied with only being able to use it on nilpotent elements) you'll need a topological structure that defines convergence.
... though note that that topological structure need not be related to the real numbers, e.g. we can ignore the usual topology of ℚ (give it the discrete topology instead) and take the ring ℚ[[X]] of formal power series, then exp(X · f) and log(1 + X · f) are both well-defined for every f ∈ ℚ[[X]]
Grant is the kind of guy that can combine love and the Schrodinger equations all while raising e to the derivative operator! Btw I just wanna say that ever since you've 'disappeared' I started to slowly drift away from math. Thanks for inspiring me again! Can't wait for ur new uploads!
For me this came up in state space methods in control theory. We broke a complex higher order differential equation control problem into a set of first order differential equations, collected them into a vector, and solved the matrix exponential. The advantage was that we could completely assign the poles (roots) of the controlled dynamics of the system with matrix feedback.
The kids today have so many excellent sources available to them in a couple of clicks. Back in the day, even 10 years ago, we didn't have such clear, attractive, and deep explanations. Most people, even the ones who would get the A+ really didn't understand the concepts. They were just good at taking the test. I wish this was available to me 20 years ago when I was a freshman in college.
@@definesigint2823 Turns out I’m a creator :) on TH-cam studio, which shows analytics, no you can’t see when it happened. However, you do see percent of viewers who are still watching over the duration of the vid. So if that percentage drops, you know that part of the vid gets people to leave. Pretty useful.
Incredible. Incomparable work. As a mathematician, I can give you credit that maybe a regular suscriptor can't. I swear your channel have amazed me and you deserve some kind of great well paid recognition.
At this level, my own mathematics ability is like comparing playing marbles in the school playground to the particle physics studies conducted on the Large Hadron Collider. So why am i completely gripped by this fascinating video lecture? It's almost, i dont know, beautiful in its explanation. The way the narrator just casually but thoroughly talks about these mind blowing operations is just inspiring and to me almost hypnotic. Utterly fantabulastically brilliant video. Thank you so much for taking the time to make this incredible piece of instruction.
Oh my god I wish I'd known this years ago. They just threw this at us in QM lectures during my physics degree and not one lecturer ever bothered to explain any of this, even briefly..
Damn, every student should watch this video. No text book or professor comes close to providing such an understandable and accessible method or medium.
I like the pun @20:16 where Juliet says "it just seems like your feelings aren't REAL" When you introduced the Complex version of the solution... I just calculated @26:35 & found The exp{ d/dt } is (in a sense) an Upper Triangular Matrix compressing the whole Pascal's Triangle... Now I'm looking for its application if any. I wonder where did you find this example??
but in the expanded form you’ll need to matriculate the real part of the operator, so it works out to be more of a square-like version of Pascal’s triangle. We use it in fluid mechanics for dilations of compressible flow 🤡
This video was so lovely. A motivating thing that you could've added, together with the fact that you already said that matrix exponentiation solves linear dynamical systems, is that the latter are the first order approximations of non-linear dynamical systems around equilibrium points. You would actually have enough material on this to make a whole video, explaining linearization and e.g. normal modes/frequencies of (small) oscillations.
Great contrast on the Textbook Progression vs. Discovery Progression. I wish more lectures and textbooks would start with the discovery progression. I remember way back in High school calculus - It is especially dangerous when a teacher who doesn’t understand the subject blindly follows the textbook progression.
this channel is a gold mine. since the day I subscribed about a year ago I completely changed my views on mathematics. the creator is a very brilliant and sympethatic person. thank you so much
Wish this form of teaching/learning existed 45 years ago when I was brute forcing it.Every chapter is a revelation. I'm not one for envying others but I envy the young people who encounter these chapters during the early stages of their education. Make people better citizens of the universe. May you remain healthy and productive.
Thanks for another amazing video @3Blue1Brown! I think it could be also worth mentioning the alternative definition of the exponential of a matrix as the limit of the products (1+A/n)^n when n tends to infinity. What I like of it is: - Immediately visualizable for a given n as a simple Euler numeric integration in n steps. - Can be motivated and bring intuition from simple problems with regular numbers as continuous compound interest.
I'm Interested in the DIAGAGNOLIZATION of matrix VECTOR using Taylor expansion and DETERMINANT for infinite series and PAULI MATRIX and Clifford Algebra for solving the Schoringer Equation And HEINGBER uncertainty for QUANTUM FIELD PERBUTATION
I never attended math courses beyond those of high-school, for reasons well beyond my control. I eventually learned from some textbooks which I bought on my own. Although I never knew about the matrix as an exponent, the highlights of the solution were quite clear. The bits of information fit well and give a good picture of the line of thought.
Even stranger things have been exponentiated to. Strangest one I know: the creation and annihilation operators. It comes up in the analysis of squeezed light, such as LIGO uses. It's convenient to analyze squeezed light by changing the basis, by a Bogulyobov transform, to the sinh and cosh functions, where the arguments are the annihilation operator plus or minus the creation operator. Sinh and cosh of course are just e^x+/- e^-x. So they were exponentiating to the power of "annihilating a photon minus creating one" etc, and it actually works to describe a physical process.
I would argue that this channel is equally important to the field of mathematics than anything Fourier or Euler calculated. These big names may reach new heights in the field, but educators are are the force that drives them positively. Thank you.
I’m close to having a PhD in quantum mechanics but this video actually really increased my intuition for what matrix exponentiation means geometrically! Thanks for making it - your channel is great!
i didnt even realise (before this video) that this circular function described the R&J love story and the on and off again crush i had for a few years. tragic.
I just have to say, your videos are consistently amazing. I am taking a quantum mechanics class right now and needed to brush up on my linear algebra since it’s been a few years, I automatically knew I could watch your videos to regain an intuition and it worked. I have a much better intuition for some of the operations and ideas than others in my class because of these videos. Specifically with matrix functions, these were literally just brought up and while it looked kind of odd, my teacher mentioned he would give us some easy ways to evaluate these. I didn’t even know about this video, but now I have a visual intuition for this, in the context of quantum mechanics! 10/10, you are a god among men.
As an EE grad, yes, actually, it represents the signal shifted by 90°. When you get into Digital Signal Processing, you should see a lot of it. Think of Euler's formula and sampling the signal with different phases.
You usually do. Reactance can be modelled as imaginary resistance. E.g. Since 1/i = - i, we can see that inductive reactance is positive i and capacitive reactance is negative i, assuming you use the complex version s Xl = i.2.pi.f.L and Xc = 1/i.2.pi.f.C Anyway, plot the imaginary reactance and the real resistance and it should be obvious why impedance is calculated as it is, and the phase angle should be apparent, too.
I recall with her great fondness my first contact with this subject matter in an early edition of Hirsch and Smale’s Differential Equations, Dynamical Systems and Linear Algebra. Takes me back many years. You’re extraordinarily good at this. Truly a treasure. Thank you for these videos.
Thank you for the time and effort you took to explain these concepts. I intuitively applied them in my classes without a clear understanding of what these numbers are doing. Visualisation at a level like in your presentation is more like a peep show on what’s going on in the CPU. Finite Element Analysis and Computational Fluid Dynamics applies these concepts. Once again thank for the good work you are doing. 👏🏾👏🏾👏🏾
Publishing math content on April Fool's Day is a really good way to have your audience evaluate your work critically.
And yet this video still winds up convincing me, so if this is a joke, consider me fooled.
Naa this stuff is all legit grant doesn’t mess around.
@@carl8703 It could also work as a way to introduce/spread your own definitions, by ”fooling” the audience they are widely accepted. You’d have to be quite confident you can convice your audience though.
@@carl8703 I guess it takes a long time for him to make a video , and he wouldn't waste resources on an April fools prank.
EEEEEE
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EE
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When we needed him the most, he returned.
He Returned
What are the chances Freddy?
@@oliverlong345 Non zero...
Return of the King
My thoughts exactly.
"Fun little exercise" and "Ordinary differential equations textbook" are two groups of words I didn't expect to ever hear in the same sentence again now that I'm done with uni
I'm here for fun, I'm not in uni and I'm not a mathematician. What's wrong with me!?
Your uni prefaced the content with the words 'fun little excercise'? That's just barefaced cheek!
At least they're not partial differential equations 🤷♂️
@@elias_xp95 the same thing thats wrong with me, I guess you love learning and understanding things, and that this guy has a really nice voice.
Ordinary differential equations are already very nice. Take a look at the partial differential equations......
Juliet: "It just seems like your feelings aren't real"
Romeo: "I think your doubts are only imaginary"
What a complex relationship
@@phucminhnguyenle250 😂😂😂
love = 0 for all t is a solution (or I'm no tennis player).
"Hey e what's your favorite movie?"
*e puts on sunglasses*
the matrix
@@PMA65537 Yo is this a detective conan reference?
In the beginning: Damn this one is half an hour.
At the end: Nooooooo, it can't be over already. I want more!
I've spent a half-hour on this video and I'm only one minute in-I'm cross-referencing everything he says. If you get on my level, you might enjoy yourself more!
@cnidariantide4207 agree 💯, it was like yes this makes so much sense now 😂
I've been a big fan of the Calvin and Hobbes comic strips since I was about ten years old. One storyline began with Calvin receiving a mysterious letter in the mail that appeared to be from some kind of secret agent -- the text consisted of individual letters clipped and pasted from magazines, the return address was nothing but a skull and crossbones, the writer advised Calvin that a coded message would soon follow -- all the tropes. The strip ends with Calvin sprinting off, yelling "This is so cool I have to go to the BATHROOM!" I found that response to be both hilarious and relatable. Calvin had no idea what was going to happen next, nor if he would even be able to figure any of it out -- he just knew it was so bizarrely exciting that he had to drop everything he was doing and hope to make sense of it just for the sheer joy of it.
Anyway, that's pretty much exactly how I feel whenever I hop onto youtube and see that 3Blue1Brown has posted a new video.
It's always nice to feel like Calvin every once in a while.
Who doesn't love Calvin & Hobbes ? Lovely analogy by the way.
I made $ 6,500 profit just within 14 days of my Investment with an initial amount of $ 2000 all with the help of Mr James
Yes I'm a living testimony of Mr James trading services , he has really changed my financial status for the best
@Mary Jane how do i start trading with Mr James
There is NO person that I know that i have not recommended this beautiful channel to. You are my hero and icon. :)
same!
I don’t know what it is about the way he speaks, but it always sounds like he’s smiling. He’s the sort of teacher that any parent hopes their child will have.
Every STEM student I know is in his debt
That is so true!! I just thought about it and I can't think of anyone I know in an engeniering program that hasn't mentioned this channel
You know Schrodinger right? Have you told him already?
Incredible script man!
My best line: Cynically, this is abuse of notation; but charitably, this is the cycle of discovery and invention.
It's not just the visuals, but the script behind it is poetic. I love this guy!
As a physics student, I'm really grateful for being able to know this channel and all your intuitive videos.
God I wish I had it when I was one. I just shouted out loud asking why we weren't taught that a derivative of a vector can be expressed as a matrix multiplication. I wish I could explain how many months of second year that single thing would have saved me
Me watching this like 7 years before I need it
I just love this intro, pointing out that the notation is initially nonsense but then receives a meaning we choose to give it. I will always remember how one of my professors introduced distributions:
"Alright, we've hand-waved a definition of the Dirac delta-"function" which is "infinite" in one point and zero everywhere else. That makes no sense, a function can't do that. But here's what we're going to do: we're going to write an integral over this "function", then we're going to _cheat,_ and then we'll end up with something that makes sense in the end. Right now, this integral is nonsense. So let's instead _define_ a meaning for this integral. And that's it! We'll call it a _distribution_ instead of a function. The Dirac distribution will be meaningless outside of an integral, but completely well behaved when in an integral."
Beautiful cycle between discovery and invention indeed!
Your professor is a genius 🤩.
I just love those aha moments when something just clicks and which later on give you so many moments to grin while solving examples 😁
I just cannot be grateful enough to live in a time when this kind of content is freely available on the internet. Thank you for all your work, Grant.
You are right. I can only regret that such a thing did not exist 40 years ago, when I needed it. Now all I have to do is admire the beauty, simplicity and clarity of this way of interpreting mathematics.
+100500 man, ur so right
It is freely available, because it is profitable. Such videos takes a loooooot of time. Nobody would do it for free.
20 years ago I am sure he would try to make a VHS or DVD series. But there is no guarantee for selling.
TH-cam is a guarantee. Patreon is a guarantee.
If he would not get paid he would not do it. Money is everything. And as long as money is "enough", such videos will exist.
but it's not always trivial to download it, so you really have it, I first failed, but then found that I had to update the downloader software, as google video of some reason change their protocols so often.
"One application is relationships, the other is quantum mechanics, let's start with relationships"
Why did you go with the more difficult one first?
So that the second one would feel exponentially easier .
beat me to it
I've got an exam on advanced methods in quantum mechanics and I can say: So far I got a good overview over quantum mechanics but Im still baffled by relationships.
There's a reason I have been pursuing one of these things before the other, and it's certainly not that I'm proactive to the point of engaging with difficulty first.
Because nerds will never get a girlfriend) Just admit it, numbers will not help you with girls
I teach graduate quantum mechanics to physical chemists, and I love showing them the videos you make, especially ones that visualize some of the maths relevant to QM. Please keep making them
Heyo, would you mind telling us which videos you have found particularly relevant to QM? I’d really like to dive into them. Cheers :D
As a controls engineer, I've never considered a state-space equation in this fashion.
You blew my mind once again!
This video was great! I just started a graduate controls course, and this way of thinking about the state transition matrix really blew me away.
@@andrewcihon-scott2170 I'm lecturing something called "Dynamical systems" for just one year and we've covered lots of this, characterising the fixed point of a 2D system of 1st order linear DEs by the behaviour around the fixed point of the system... And I'd never considered the relation to control theory.
We need a grand unified theory of control dynamics!!!!!
I legit just learned about state space in my first introductory course to control theory, and, while I probably don't appreciate it as richly as others, it still will probably help me with intuition going forward.
I learned a few days ago that
a -b
b a
is a matrix isomorphic to a+bi, it's nice to see that the exponent relation also holds :)
Never thought of that. Thank you
A topic I've recently been learning about is Clifford Algebra, and while it usually isn't done with matrices, the various objects have matrix representations. The algebra also encodes the complex numbers. That matrix is the actual matrix representation of a+bi in Clifford Algebra, despite the fact that the the unit vectors it usually acts on look completely different in matrix form than normal unit vectors.
Ja
This actually looks shockingly similar to the inverse matrix formula, wonder if there’s a connection there
@@adjoint_functor If you squint, the formula for the inverse of a 2x2 matrix kinda looks like the formula for the multiplicative inverse of a complex number, z^-1 = z*/|z|; that is, the complex conjugate (the number but with the imaginary part multiplied by -1), divided by the magnitude of the complex number.
If you also watched 3b1b's videos on complex numbers, the relationships are kinda neat. You can see from that formula that the magnitude of a complex number is equal to the determinant of its matrix representation, and the multiplicative inverse of a complex number multiplying to equal 1 is mirrored by the matrix representation's inverse multiplying to the identity matrix.
It's been a long time since you uploaded. Welcome back!
Thanks! I love your symmetries video, by the way.
@@3blue1brown Wow thanks so much! Never thought you would reply!
"raise to the power of the derivative operator" YOU CAN"T JUST DROP BOMBS AND DIP
Look up something called Exponential Shift
MATH VIDEO WITH A CLIFFHANGER??? HUH
@ゴゴ Joji Joestar ゴゴ So, the MacLaurin polynomial centered on x, evaluated at (x + 1), if my mental derivation is correct...
@ゴゴ Joji Joestar ゴゴ Additionally, for functions of multiple variables, you can take the exponential of a vector dotted into the gradient to obtain a shift in the direction of that vector
For that reason, I usually remember Taylor's formula as f(x+ε)=e^(ε•∇)f(x), where ε and x are vectors
This must be how Luke Skywalker felt when he found out Vader was his father.
The king has returned. Let's hope this isn't the start of a new hiatus!
He was busy with courses in some uni, think MIT and prolly something related so makes sense. Also maybe planning and stuff too
I would estimate >200 hours of work to get this video done. It would actually be nice to know how much time 3B1B actually spends on these creations.
We're blessed and don't even know...
@@fzigunov ive used manim (the language he used to make his animations) a little bit and it honestly takes about an hour to make one scene so god knows how long it takes him. Propts to him for this
@@fzigunov He has a team for stuff like that?.
@@LimLux I think you would be surprised...
Please !!! Do the video on the Schrödinger equation: you are the only one out there, who could truely explain it and its beauty 💖💗💖
What kind of money-work do you do?
Yes, absolutely!!!
Yes℅
I love how in every video 3b1b goes from "that's utter nonsense" to "that's perfectly valid' and explains every step 0f the way
As a math professor teaching ODE this semester, let me just say: this is wonderful!
Same. I recommend both the Diff Eq playlist and the Linear Algebra playlist to my students every term.
@@samsonblack I recommend the linear algebra series to my classmates every term 😂
@@Jackisaboss1208 how tf you take linear algebra every term 😂
@@sauronstillgood6804 CE, probably
I just want to say that you're my favorite youtuber. You help me so much during my study in university and getting a master degree. You also reignite my interest in math, which I previously thought wasn't even possible. Words can't express how much I love your video and the way you teach math.
That's so kind of you, thanks. I hope you help to spread the love of math with others in your life :)
@@3blue1brown I came searching for the sequel. I was sure i watched it years ago, but now i realize maybe you never published it. Am i right? (By the way, terrific job. Im a long way fan of the channel, i'm currenly ill, and binge watching all your videos-again!)
6:35 I was fully expecting for it to go like "let's start with the simpler of the two, quantum mechanics"
I remember during my study where e to the power of a matrix confused the hell out of me. I desperately would have needed this video. Now I'm watching just out of pure interest. 😄
I did my engineering sometime ago, I remember seeing the e ^ (matrix) and I remember losing it and thinking wtf....You really have a way of explaining things so elegantly.. I hope I encounter these kind of problems in my career - Very enlightening....Thank you!
Im literally learning this in class rn and im like, oh well ig 3blue1brown will be my teacher today
this is what i love about 3blue1brown, i have to carry out these calculations in my qm class all the time, but i have absolutely never understood the intuition behind it until now
He came back when we missed him the most because that's what heroes do.
guess my father's not a hero then
@@purungo you shouldn't have written this
@@ayushdeep7900 Yes I should
@@purungo sorry bro
What I love about your videos is that you might not understand parts of them at first but as you go back to them later, perhaps after gaining some knowledge, things start to click.
We have to create a new contest besides Nobal, Fields, etc. This guy needs a medal.
To the time I write this comment, Im a Mech.Eng. student and currently taking a course in dynamics. Our professor showed us a proof that if you derive a vector, the new vector is perpendicular to the first. I asked if we can determine the orientation, and he replied that he doesn't know. I'm watching this video at 3:50 a.m. suddenly it struck me - some of your equations and animations made me realize that we can determine the "natural" orientation, and even understand gradient and curl itself. And this ain't even the purpose of the video. Thanks for making math that clear, I'll even say that obvious, over and over again ❤️
I’m highly appreciative that the matrices at 5:14 are the first 9 digits of pi lol. Great video as always, thank you Grant!!
Happy April fools!
First
The joke is that I actually upload content these days :)
@@3blue1brown joke or not, I’m still grateful that you still upload content. Your channel is a great contributor to making me pursue a career in applied mathematics.
@@3blue1brown Then maybe an "essence of probability" as a joke, hmm?
@@3blue1brown i was going to like this comment till i saw it that it had 314 likes, oh how nice things can be
"Often this is a function. (But whatever, functions are really just infinite-dimensional vectors)"
I hope that parenthetical is a teaser for a future video on Functional Analysis.
Where was this channel when i was an undergraduate 20 years ago?
The 18-24 year old students beginning their studies of higher mathematics have such amazing tools available to them now!
"Why would mathematicians and physicists be interested in torturing their poor matrices this way?"
Wait till you hear what Hilbert did to lettuces.
"torturing the MATRICES?!?" -college undergrads after spending two weeks straight calculating the inverses of 3x3 matrices by hand
Sweet, sweet revenge.
@@PMA65537 can you tell more ?
It's just "compound interest" for complex systems. It will explain linear motion, oscillations, dampening, chaos, explosions etc. Really useful.
I was waiting two years for this video! (the preview was shown in the first video about DE)
Im so happy right know! Thank you! 😍
I don't always make the follow-ups I promise quickly (*ahem* probabilities of probabilities *ahem*), but for those willing to wait a few years I feel comfortable promising all will come in due time.
@@3blue1brown yeah I've been waiting for possibilities of possibilities 😆😆
I was so sad cuz I thought the DE series was discontinued, but now I’m hyped and can’t wait for the rest it!
This is hands down one of the best channels ever
The*
@@maxwellsequation4887 They wrote it correctly
Mathematicians before e to the x extends the definitions: "We do some more theorems and we finish with math"
Mathematicians after:"What if we exponentiate EVERYTHING"
Associative algebra: *exists*
Mathematicians: "Time to define e^x on it"
They're paid to answer these questions because students ask them.
@@nolan412 They're paid to answer these questions because there's a chance someone in the future will use their work and apply it to the real world
Can't you define e^x on arbitrary rings?
@@durnsidh6483 Yes, if the ring has a metric
I hope that kids these days can really appreciate how lucky they are to have channels like 3B1B. I can clearly remember the time when I sat in my calculus classes and the concept of matrix exponents was first used. I was too busy to write down the professor's derivation from the chalkboard to really think about it. But when I reviewed my notes back home I thought to myself: "What the hell does that even mean". Of course I figured it out eventually.
But channels like this should really help today's students. Because professors rarely take the time to motivate something with examples. It is just definition -> theorem -> proof -> repeat. And of course that is important. But sometimes you miss the bigger picture.
This is beautiful. This is mostly why Dirac notation is so powerful: Dirac must have similarly reasoned that Schrodinger's Wave mechanics and Heisenberg's Matrix mechanics were essentially describing the same thing!
An entire semester's worth of my electromagnetics class in university just made sense after 20 years...
There are many such classes where that happened more or less as I was taking them because of this channel.
EM is the hardest part of physics to grasp
@@chalichaligha3234 that's not physics, that's witchcraft
The summation of the wave functions on Schroedingers equation makes much more sense now than it did in college. Grant, where were you 30 years ago? ;) Thanks for making math exciting again! Want more STEM excitement? Show these videos in junior high and high school. Maybe earlier.
@@ultimategotea but thermodynamics and stokes-navier?
Finally!!! I really missed those pi-creatures (and also your voice, Grant! It is hypnotizing enough to make people stick to your videos without giving up, no matter how fighting the math in them can get sometimes😳)
So relieved when the Romeo 'n' Juliet graph didn't trace out a heart shape
Why?
So disappointed that Romeo 'n' Juliet graph didn't trace out a heart shape :'(
But how would those differential equations look? 🤔
Dy/Dx = heart
@@pierrekilgoretrout3143 Don't feel bad! Here, have a vertical ice cream cone/pointy hat of an elf seen from behind:
20:14 "It just seems like your feelings aren't real" 😂
In fact I really love complex numbers
i'm only in elementary school but i love this channel. It really just shows the beauty of math, and inspires me alot to learn more. Thanks for making these high quality videos, much love and respect (:
as someone who studies physics with the "just enough" math knowledge, this video is mind blowing to me, it makes me realize that I should be looking more into the math of the physics phenomenons. Physics is applied mathematics, but physics also spark ideas, problems for us to develop models, concepts that we have never thought about!
This is the only channel where im practically lost from the get go but still compelled to watch till the end
I am so excited to learn maths this way from you. I have a PhD in physics, and am more than 20 years into my career as a scientist. I do a lot of calculations to support my experiments, so I have a basic understanding of mathematics, but I feel like a fog is being wiped clean when I watch your videos. Thank you so much!
3 years later, we are still yet to receive the highly anticipated 7th part in this series on Laplace transform. Wonder if the video has now been scrapped forever..
yes
I just learned the trick for 2x2 matrix eigenvalues on this channel and it's so helpful. Also, we need a next chapter soon lol.
I was like it’s been 3 months since his last upload and 6 months since his regular uploads
Things will pick up again here now. The videos might not come _super_ quickly, but there are a lot of things I'm excited about coming up this year.
@@3blue1brown excited for your new stuff. Loved all of your other content!
(Btw, have you ever considered becoming a voice for asmr videos?)
@@3blue1brown 🤩
@@3blue1brown including blockchains? ^^
@@3blue1brown yesterday I had commented on your previous community post and today here you are
This channel is one of the greatest things that has ever happended to the math explanation!
I'm a third year physics student and by now, I have seen a fair bit of QM. It's amazing how you always manage to give new and visually intuitive ways of imagining calculations, that we do so often without further thinking about it. I wish our lectures would be taught more like this.
leave physics. you are wasting your life.
@@gena8414 what isn't then wasting your life?
@@michaelking8391 cs
@@gena8414 so you can be dumber than a physicist that can code just as well as you?
@@velhacega lmaoooooooo show me a physicist who can code as well as me
without a doubt one of the best math channels on youtube, if not THE best! waiting for episode 7 btw.
Grant, you are one of the most important educators in the modern world. To have your library of presentations in my pocket at all times is one of the best feelings there is. Math went from my least to to my most favored topic of study because of your content. May you continue to be an ambassador in this field!
Never have i clicked so fast on a notification
Even me. I was watching something else. I ran to see this .
Same
me 2 :D
Same
@@hashirhussain. There is one other TH-camr that has this same effect and that is Ben Eater!
I remember the absolute outrage of my class last semester when we saw this for the first time. Several people called it abuse. This vid makes it seem much nicer.
What class was it?
Probably intro to diff eq
Works on any associative algebra, AFAICT.
@@SimonClarkstone well no you need some more conditions, it needs to be an algebra over ℚ to allow multiplication by 1/n! and (unless you're satisfied with only being able to use it on nilpotent elements) you'll need a topological structure that defines convergence.
... though note that that topological structure need not be related to the real numbers, e.g. we can ignore the usual topology of ℚ (give it the discrete topology instead) and take the ring ℚ[[X]] of formal power series, then exp(X · f) and log(1 + X · f) are both well-defined for every f ∈ ℚ[[X]]
Grant is the kind of guy that can combine love and the Schrodinger equations all while raising e to the derivative operator!
Btw I just wanna say that ever since you've 'disappeared' I started to slowly drift away from math. Thanks for inspiring me again! Can't wait for ur new uploads!
I don't even understand most of the stuff in these videos but I find it very calming and it helps me go to sleep lol
For me this came up in state space methods in control theory. We broke a complex higher order differential equation control problem into a set of first order differential equations, collected them into a vector, and solved the matrix exponential. The advantage was that we could completely assign the poles (roots) of the controlled dynamics of the system with matrix feedback.
The kids today have so many excellent sources available to them in a couple of clicks. Back in the day, even 10 years ago, we didn't have such clear, attractive, and deep explanations. Most people, even the ones who would get the A+ really didn't understand the concepts. They were just good at taking the test. I wish this was available to me 20 years ago when I was a freshman in college.
Now everyone, not just the best scientists and researchers out there, can stand on the shoulders of giants. This is what the internet is made for.
There is only one TH-cam channel where I hit the like button before I even watch it.
I've often wondered...can creators see _when_ you upvote (possibly giving some data on the moment the video was meaningful)?
@@definesigint2823 Turns out I’m a creator :) on TH-cam studio, which shows analytics, no you can’t see when it happened. However, you do see percent of viewers who are still watching over the duration of the vid. So if that percentage drops, you know that part of the vid gets people to leave. Pretty useful.
@@Mutual_Information lmao i m sure the romeo and juliet part hold the people in this video
Why the fck would you do that.
@@jn5433 lol
Incredible. Incomparable work. As a mathematician, I can give you credit that maybe a regular suscriptor can't. I swear your channel have amazed me and you deserve some kind of great well paid recognition.
I'm a Princeton grad and this is one of the best lessons in mathematics I ever experienced. You're a legend!
At this level, my own mathematics ability is like comparing playing marbles in the school playground to the particle physics studies conducted on the Large Hadron Collider. So why am i completely gripped by this fascinating video lecture? It's almost, i dont know, beautiful in its explanation. The way the narrator just casually but thoroughly talks about these mind blowing operations is just inspiring and to me almost hypnotic. Utterly fantabulastically brilliant video. Thank you so much for taking the time to make this incredible piece of instruction.
For the situation at 24:18, we get e^([[0,1],[1,0]] * t) = [[1, e^t], [e^t, 1]], so they tend to infinity at an exponential rate! Yay for true love!
Seeing the pi’s always miss each other with their heart eyes is honestly tragic 😍😭
Oh my god I wish I'd known this years ago. They just threw this at us in QM lectures during my physics degree and not one lecturer ever bothered to explain any of this, even briefly..
But, are you aware in 2021?
I was gonna make this comment
I am getting a masters degree in mathematics in 3 months and yet, I have never seen matrix being an exponent. lmao.
Damn, every student should watch this video. No text book or professor comes close to providing such an understandable and accessible method or medium.
I wish I had the necessary math background when I was taking advanced physics in college. it would have made the course a lot less painful.
haha right? At the end I was like "oh, we've basically been doing the same thing over and over with slight variations."
Never understand it fully but came for the channel
Literally me
If you don’t understand a math related thing, go see if 3blue1brown has a video on it, it might help yo- *wait a second...*
I like the pun @20:16 where Juliet says "it just seems like your feelings aren't REAL" When you introduced the Complex version of the solution...
I just calculated @26:35 & found
The exp{ d/dt } is (in a sense) an Upper Triangular Matrix compressing the whole Pascal's Triangle...
Now I'm looking for its application if any.
I wonder where did you find this example??
-ihbar d/dx is the momentum operator in QM
but in the expanded form you’ll need to matriculate the real part of the operator, so it works out to be more of a square-like version of Pascal’s triangle. We use it in fluid mechanics for dilations of compressible flow 🤡
I thank dear Zeus and the olimpians for bringing you back. You made my week with this video. Thank you 3b1b!
May your week be good too!
Your ability to explain things graphically is beyond any limits.. I would have given you a Nobel price for that, if I were able to!
This video was so lovely. A motivating thing that you could've added, together with the fact that you already said that matrix exponentiation solves linear dynamical systems, is that the latter are the first order approximations of non-linear dynamical systems around equilibrium points. You would actually have enough material on this to make a whole video, explaining linearization and e.g. normal modes/frequencies of (small) oscillations.
"Rotation in a kind of function space". Thanks, my brain is now all over the wall
It seems 4 other people laughed out loud at this comment.
My room has a new gory wallpaper as well
Great contrast on the Textbook Progression vs. Discovery Progression. I wish more lectures and textbooks would start with the discovery progression. I remember way back in High school calculus - It is especially dangerous when a teacher who doesn’t understand the subject blindly follows the textbook progression.
Girlfriend: "I want to move in"
Me: busts out a matrix
Well, that'll just introduce another variable of what it's like to live together. Better get ready for some really complicated calculations.
Scrolled for this
this channel is a gold mine. since the day I subscribed about a year ago I completely changed my views on mathematics. the creator is a very brilliant and sympethatic person. thank you so much
Wish this form of teaching/learning existed 45 years ago when I was brute forcing it.Every chapter is a revelation. I'm not one for envying others but I envy the young people who encounter these chapters during the early stages of their education.
Make people better citizens of the universe.
May you remain healthy and productive.
Notice the pi's eyes and expressions as the graph changes!! Incredible attention to detail grant! Amazing.. 10:00
"but whatever, functions are just infinite dimensional vectors" is such a nonchalant way of mentioning that offhand i love it
Gotta love power series. They make it possible to define all sorts of functions on matrixies like cosine, sine, exponentials, logarithms, etc.
Thanks for another amazing video @3Blue1Brown! I think it could be also worth mentioning the alternative definition of the exponential of a matrix as the limit of the products (1+A/n)^n when n tends to infinity. What I like of it is:
- Immediately visualizable for a given n as a simple Euler numeric integration in n steps.
- Can be motivated and bring intuition from simple problems with regular numbers as continuous compound interest.
I'm Interested in the DIAGAGNOLIZATION of matrix VECTOR using Taylor expansion and DETERMINANT for infinite series and PAULI MATRIX and Clifford Algebra for solving the Schoringer Equation And HEINGBER uncertainty for QUANTUM FIELD PERBUTATION
Thanks a lot. I'm a programmer and am making these concepts in code as we go. Really gives me another dimension of intuition.
I never attended math courses beyond those of high-school, for reasons well beyond my control. I eventually learned from some textbooks which I bought on my own. Although I never knew about the matrix as an exponent, the highlights of the solution were quite clear. The bits of information fit well and give a good picture of the line of thought.
What happened to the next episode :(
Even stranger things have been exponentiated to. Strangest one I know: the creation and annihilation operators. It comes up in the analysis of squeezed light, such as LIGO uses. It's convenient to analyze squeezed light by changing the basis, by a Bogulyobov transform, to the sinh and cosh functions, where the arguments are the annihilation operator plus or minus the creation operator. Sinh and cosh of course are just e^x+/- e^-x. So they were exponentiating to the power of "annihilating a photon minus creating one" etc, and it actually works to describe a physical process.
I would argue that this channel is equally important to the field of
mathematics than anything Fourier or Euler calculated. These big names
may reach new heights in the field, but educators are are the force that
drives them positively. Thank you.
I have been preparing for CSIR NET MATHEMATICS exam. Whenever I feel bored,I watch your videos.i m loving it... keep going...
Whenever I feel less motivated to become a mathematician, 3b1b comes back with another video
I’m close to having a PhD in quantum mechanics but this video actually really increased my intuition for what matrix exponentiation means geometrically! Thanks for making it - your channel is great!
"Its seems like your feeling aren't real"- Juliet
**laughs in complex numbers**
i didnt even realise (before this video) that this circular function described the R&J love story and the on and off again crush i had for a few years. tragic.
I guess that's why the whole wedding thing didn't work out, eh?
It is imaginary
Imagine that J(t)+iR(t), Juliet has real feelings but Romeo is faking it!
I just have to say, your videos are consistently amazing. I am taking a quantum mechanics class right now and needed to brush up on my linear algebra since it’s been a few years, I automatically knew I could watch your videos to regain an intuition and it worked. I have a much better intuition for some of the operations and ideas than others in my class because of these videos.
Specifically with matrix functions, these were literally just brought up and while it looked kind of odd, my teacher mentioned he would give us some easy ways to evaluate these. I didn’t even know about this video, but now I have a visual intuition for this, in the context of quantum mechanics! 10/10, you are a god among men.
I thought this was an April Fools joke until I realized my knowledge of mathematics is horribly meager. Another masterpiece!
Imagine if this guy uploaded on a weekly basis
If that were true I'd be concerned for his mental health
when he was started newly .. he used to upload in weekly basis ... now he is famous and more busy soooo
22:02 me, an EE student: "so you could build an oscillator with imaginary value components? :o"
As an EE grad, yes, actually, it represents the signal shifted by 90°. When you get into Digital Signal Processing, you should see a lot of it. Think of Euler's formula and sampling the signal with different phases.
You usually do. Reactance can be modelled as imaginary resistance. E.g. Since 1/i = - i, we can see that inductive reactance is positive i and capacitive reactance is negative i, assuming you use the complex version s Xl = i.2.pi.f.L and Xc = 1/i.2.pi.f.C
Anyway, plot the imaginary reactance and the real resistance and it should be obvious why impedance is calculated as it is, and the phase angle should be apparent, too.
I recall with her great fondness my first contact with this subject matter in an early edition of Hirsch and Smale’s Differential Equations, Dynamical Systems and Linear Algebra. Takes me back many years. You’re extraordinarily good at this. Truly a treasure. Thank you for these videos.
Thanks!
Thank you for the time and effort you took to explain these concepts. I intuitively applied them in my classes without a clear understanding of what these numbers are doing. Visualisation at a level like in your presentation is more like a peep show on what’s going on in the CPU. Finite Element Analysis and Computational Fluid Dynamics applies these concepts. Once again thank for the good work you are doing. 👏🏾👏🏾👏🏾