It begins! Differential equations, chapter 1

Well, the rate of change for theta is theta-dot. The arrow represent rates of change. So if you're higher up in the diagram, that means theta-dot is higher, which means the rate of change for theta (the horizontal position) is higher. Does that make sense?

3blue1brown

tnx for the video, i finally have the stuff to start learning the differential equations. But I've a question: in 14:18 you say that the more we're up in the state diagram, the more the dot is going to move on the right direction. Well, I'm not sure I understood...why?

This video was particularly good! And timely too. I'm taking Introduction to Differential Equations in the summer and Introduction to Numerical Analysis in the fall. I loved the programming section. I know it isn't particularly applicable for all videos, but when you can do a quick demonstration in python I think you should. Of course using your judgement on when it's best to include one or not.

Awesome video! Differential equations were something complicated and incomprehensible to me, but thanks for your explanations, it became much easier for me to understand them. Thanks a lot!

VitAnyaNaked

Thanks for the kind words. As to the arrow colors, I wanted to draw a distinction between the physical velocity of the pendulum, and the rate of change for the phase-space point.

3blue1brown

Thank you for teaching calculus in a digestible way. I've long been frustrated by the non-intuitive way calculus was taught to me in engineering school, and your approach addresses the issues. I've got a question though about the color-coding of the arrows and variables that appear circa 12:30-13:00 in the video. It appears that what's coded in yellow on the center plot (phase space) is coded by an orange arrow on the left plot (pendulum). I was confused by the pendulum arrow not being yellow, as theta dot and the pendulum velocity arrow seem to be equivalent, but... are they?

I recently overcame a conceptual hurdle and was able to solve a system of coupled springs: <a href="http://bit.ly/2uIVVpU" rel="nofollow noopener" target="_blank">http://bit.ly/2uIVVpU</a> (links to a gif on github). It's a real bummer about that loss of well-posedness problem with the double pendulum though. Thanks for being a constant inspiration!

kos

Grant, I'm so jealous! I wish I had this video around to explain how to intuit differential equations back when I was struggling with them over a year ago.

kos

Neat, I'll take a look. Thanks for sharing.

3blue1brown

There will be several other videos first, but don't worry, at least 1 or 2 on Laplace transforms are coming.

3blue1brown

Thanks Burt! I hope you enjoy the rest :)

3blue1brown

The python code at the end was a welcomed surprise

This is a great beginning on a great series of animations. I have begun looking at Eulerian and Lagrangian Descriptions in Fluid Mechanics with relation to ocean chemistry. I found this video from 1968- John L Lumley PSU and Cornell : <a href="https://www.youtube.com/watch?v=mdN8OOkx2ko" rel="nofollow noopener" target="_blank">https://www.youtube.com/watch?v=mdN8OOkx2ko</a>

That was fabulous! I think you should take two charges for this one. Thank you so much for this. I can't wait to see where this series goes.

Burt Humburg

Excellent as always! Very excited for Laplace transforms!

Navagram

For me the dialog is a bit fast. The overall pace could be slower. Maybe it's me, but it seems faster than some of the others.

Thanks!

3blue1brown

Best of luck with the exam. I don't know how helpful the intro framing video will be, but hopefully the series to come can be helpful for more students like you.

3blue1brown

As long as theta_double_dot is computed before both, it seems correct to me that that should update before theta_dot. Otherwise, when you're updating theta, it will be based on the changed theta_dot.

3blue1brown

So are you :)

3blue1brown

Thanks!

3blue1brown

Thanks Jacob. I hope you enjoy the heat equation one. I don't think it's actually *that* heavy a topic when you break down what's really being said, but we'll see. PDEs, in general, can be super daunting (for me at least) but I like how this feels graspable, both in terms of reading what it's saying and finding its solution.

3blue1brown

Ah yes, a lovely twist indeed, made all the more mind-warping in the part where he bends that cylinder such that it's z-coordinate correspond to energy. It might be worth mentioning, but there are a lot of things worth mentioning, you know?

3blue1brown

Glad you enjoyed it, thanks!

3blue1brown

I was running a little late for a schedule I had in mind, so this one didn't have the same early access. Keep an eye out soon for the heat equation video, though.

3blue1brown

Hi Sascha, Good to know on the frames. As to the numerical methods, my hope was that showing directly just how inaccurate it could give some indication that more sophisticated machinery would be welcomed, and it is something I plan to elaborate on in future videos. What I wanted was the simplest thing that could be shown explicitly without occupying too much time in an already long intro video. I also wanted to avoid just mentioning other methods by name without going into them at the time, but I think you right calling it out specifically as Euler's likely would have been for the better.

3blue1brown

thank you very mutch for this fantastic video, I will write a exam about differential equations next week, so this video came out at the exact right moment.

This video is fantastic, thank you! In particular the part dedicated to the phase space is clear: it is not easy to get the idea of what someone is looking for in it. Also the animations are amazing :) I have a question: in the last part of the video, when you implement the function theta you put theta before theta_dot: might it be inverted? Shall we first compute the derivative of theta and then theta?

You beautiful human being

I like the cheeky inclusion of tau at the position 2Pi and -2Pi on the graph. Great video!

This is ambitious even for you, Grant. You're going straight to the Essence, I believe. But Fourier series PDE solutions in the second video!? I'll come with a bowl of popcorn and a bowl of skepticism, but ready to be blown away. By the way, this video was lovely, holding it's own against some of your best stand-alone work.

Jacob Mirra

Really nice work putting together an intuitive intro to phase space and its utility! In the Strogatz book, I recall a really insightful further twist of wrapping the pendulum phase space around a cylinder to emphasize that theta and theta + 2Pi represent the same physical position even though they look like distinct points in phase space. I'm wondering if future videos could animate that sort of twist. As a beginning student, it was an epiphany for me and later really helped with thinking about polar coordinates, boundary conditions in quantum mechanics, etc... Thanks Grant!

I've been studying differential equations for a year or so and to be honest, I've been struggling to get to the heart of it, and with this one video I now feel that I have a better intuitive grasp of them now than I ever did before. It just clicks. Your approach is just so illuminating for someone who thinks visually. Sometimes I am just so overwhelmed with joy at the beauty in the underlying systems you describe that I'm brought to tears. I'm really looking forward to the rest of this series :) You are an amazing teacher Grant. Thank you!

J

Oh and one more thing: at around 16:20, the grid suddenly stops and it’s rather jarring. I’m also curious whether I just missed the early access part of this video or whether there wasn’t one?

Sascha Baer

That's the goal at least, I'm glad you enjoyed!

3blue1brown

Great video! Two little things though: I’ve found the flashing outlines around the boxes you show occasionally to be rather distracting. Further, I feel like it would’ve been nice to mention that the numerical method you used a) has a name (explicit euler) and is b) both very crude (e.g. it doesn’t preserve invariants and fails horribly on some differential equations, possibly not even converging at all) and far from the only possiblity. I don’t know if it’s in the plans, but an overview of some of the more popular numerical methods would be awesome (like, idk, explicit and implicit euler, trapezoidal or implicit midpoint rule and perhaps a bit on Runge-Kutta and Splitting methods)

Sascha Baer

This is amazing so far Grant, thank you man

Your videos make topics which are usually very daunting, extremely approachable and interesting! Thanks a lot, Grant :)