Laplace Equation

Fundamental Solution of Laplace's equation in 2 dimensions
In this video, I derive the Laplacian in polar coordinates, and use this to find the fundamental solution of Laplace's equation in 2 dimensions, simply by looking for radial solutions. Enjoy!
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[@drpeyam]

Rotation Invariance: th-cam.com/video/r_BvzJkRQVc/w-d-xo.html

[@tylershepard4269]

This is a VERY important equation in electromagnetics and it comes right out of Maxwell’s equations. In fact, under certain assumptions it is possible to transform the Helmholtz equation into Laplace’s equation. Another thing that is very important in electromagnetics is conformal transformations and the fact that lap-Laplace’s equation holds under the new coordinate system. Electromagnetism is just so amazing and I have a deep passion for it. Your videos couple (no pun intended) very nicely with my subject of study.

[@jayjayf9699]

I was learning about partial differential equations then you made this video

[@SahilSharma-ev1hu]

I wish you would have been my professor.. I would have felt blessed

[@thethug1946]

Great teaching technique my friend, you really know how to paint a picture

[@dhunt6618]

Ah, nice! It seems like "all" math can be canceled out and reduced to 1 is equal to 1, unless you made a misteak :) Thanks for the video!

[@marouaniAymen]

Thanks again for your video, this last video did remind me solving the Hydrogen Atom in Quantum Mechanics course

[@The_Professor123]

Very cool, If i remember correctly it can describe steady radial groundwater flow to a well

[@ezras7997]

That’s pretty awesome

[@weimondo]

Could you please discuss the Laplace equation on a flat torus or some other domain? I'm interested in knowing how the domain affects the fundamental solution and properties such as rotation invariance. Thanks.

[@matrixstuff3512]

it makes me so happy that you dont use nabla squared

[@theproofessayist8441]

Dr. Peyam if you don't mind my asking what camera (and its specs - particularly pixel resolution) are you using to record your dope videos?

[@drpeyam]

Just my iPhone x

[@davidkwon1872]

2:45 X = rcos(theta) => dx/dr = cos(theta) , How dr/ dx = cos(theta) ??

[@dm_saj1119]

Iove u dr

[@franciscobarrera8966]

Nice video 😄

[@dm_saj1119]

Are you sure baby 😒

[@tiagomotacrispim5018]

Nice:)

[@kqp1998gyy]

Delicious! 🌷

[@annali9577]

I love u 3000

[@jonasdaverio9369]

Isn't any bilinear function f(x, y)=Ax+By a solution of Lapl(f)=0? So you would miss them think the solution must be invariant under rotation...

[@drpeyam]

Still invariant under rotation, if you rotate a linear function you still get a linear function

[@jonasdaverio9369]

@@drpeyam But not the same... I mean, your
formula (A*ln(r)+B) surely doesn't encapture linear functions. Why is that so?

[@nathanisbored]

@@jonasdaverio9369 (A*lnr + B) isnt meant to capture all solutions, from what i understand.

[@drpeyam]

I never said those were all the solutions

[@jonasdaverio9369]

@@drpeyam Well, indeed, I think I was a bit confused and did not really think of what it meant to be rotation invariant. But I think it might have been a bit confusing when you went from this invariance to expressing the solution only in term of r. You said indeed that there was no implication, but I was a bit confused nevertheless. Ain't saying it's your fault

[@jamesbra4410]

Can you do a video solving this using complex analysis?

[@drpeyam]

How? Unless you just take any holomorphic function and take the real and imaginary parts. But that’s just in 2D

[@jamesbra4410]

@@drpeyam Yeah like 2d fluid flow or the stream function.

[@agardy6679]

Now do it in toroidal coordinates.

[@tomkerruish2982]

Okay, now do it for the variables z = x + iy and z* = x - iy, showing that u is the sum of a holomorphic function and an antiholomorphic function.

[@knivesoutcatchdamouse2137]

Ooh, I'm going to try this right now! Thanks for the idea!