Laplace equation

By popular demand, here’s a video of me solving a PDE! Here I find a nontrivial solution of Laplace’s equation Delta u = 0 by turning it into an ODE. At the same time I’m finding a formula of the derivative of |x| in multiple dimensions. Enjoy!

[drpeyam]

‘Like’ this comment if you want to see a derivation of the solution of Poisson’s equation.

UPDATE: Thanks for your likes, the sequel is now up on:


Also in 19:10, it should be r/n-2 not r times n-2. Thanks Bonnardot Philippe for noticing!

[TheMauror]

Yes! Please do more PDEs!!!! And solve that last eq you wrote!

[TheMauror]

I'm taking a course on PDE's and I just can't stop watching this playlist about PDE's hahaha please do more Dr Peyam!

[fountainovaphilosopher]

Didn't understand a thing I love it.

[franklinbolanos]

Dr. Peyam, thanks for explainning interesting topics and greeting from Ecuador :D

[Koisheep]

Me: A 2ND ORDER LINEAR EQ WITH NO CONSTANT COEFFICIENTS
Me, seconds later: No wait it's separable ok crisis avoided

[loicetienne]

Fantastic. I guess there are other videos showing (examples, intuition) why Laplace's equation is important/interesting. In particular, I believe that there is a connection with complex analysis (Dirichlet' principle, Riemann mapping theorem, Cauchy-Riemann equations, harmonic functions), it would be nice to see some videos about these topics. Thank you so much.

[davidkwon]

Finally, I understand what’s going on here after passing the half of this semester.
I love mathematics but it need a great time to understand her because there is a big gap between us.

[ozzyfromspace]

In your summary, you put the radial part as N>=3, but it also works for N=1 😊. Thanks for the awesome video Dr. Peyam! I watched your video a few times and now it makes sense.

[jameswilson]

Laplace's equation! My favorite equation! Thank you!

[jozsefgurzo]

Omfg that was soo long. I mean not your video, that was the most entertaining thing in my day. So thank you! But I felt a bit unsatisfied, because although I could believe the rotation stuff in the beginning, I felt I need a proof for this, and so I prove it... Finally... At some point there were three sums in my equation and it seemed pretty awkward... But I finally did it and I feel so good now. The whole thing is complete now in my head. The pieces are all together! So thank you for this "homework" you (technically) suggested!

[quaternions]

Hi Dr Peyam, at 19:10 when you integrate V' the -(N-2) should be over r and not in front of it, have a nice day !

[emanuellandeholm]

The U( x ) = V( |x| ) really threw me into a loop because I wasn't quite listening and | x | resembled 1 ⨯ 1 to my tired brain.

It's amazing to me to think that functions satisfying the Laplacian condition also satisfy the conditions in any rotation. Simultaneously, it kind of makes sense because no matter how you rotate an object, its geometry/curvature is still the same. Rotations just change your POV, they never extend/contract.

Also, the notion of radial functions f( |x| ), that's a power tool in itself. Can we extend this "radiality" to any norm in a vector space, and would that be productive?

In any case: Thanks Dr. Peyam. You made me spend a couple of hours on research, which is all good. :)

[RalphDratman]

It was difficult to see what you were writing over on the far right side of the blackboard, and also on the top left.

More substantively, this video was interesting, and I enjoyed watching it. Thank you very much for your video work.

However, I need to see an example application of this fundamental solution to something tangible like a physics problem. Without an application, for me it is just algebra and calculus, which obviously you enjoy, and for which you have great talent.

For those of us who do not have much talent for algebra, and always try to use geometry and visualization instead, the presentation you made does not give immediate insight to the (stunningly important) Laplace's equation.

[carlosmonte]

Great stuff. In engineering classes they usually tell, "well if you solve this thing you actually get this thing..." Nice to see why. Maybe unrelated topic, but how can you know when you can write the infinite series solution as a combination of simple functions? Have seen in these pdes sometimes pop weird integrals with no formula like integral of sin/x or gaussian monsters

[mihaly]

Nice video peyam! You should do the integral from -inf to inf of cos(x)/(1+x^2), using feynman’s technique I can’t quite figure it out

[naeemakhtar]

Thank you for this video.i understood the gist of fundamental solution of Laplace's and poison equations.

[rafaelmarques]

God damn PDES are a whole new creature... I love it

[tomatrix]

Loved it. This is some tedious stuff sometimes seeming hopeless but a lovely result

[MrRyanroberson]

also try E.i. "et inventa" for 'to be found', just like I.e. 'id est' and Etc 'et cetera'
you could be the first to coin this very useful acronym that more relevantly translates to "to be determined"