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!

[tylershepard]

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.

[thethug]

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

[jayjayf]

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

[marouaniAymen]

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

[SahilSharma-evhu]

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

[dhunt]

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!

[Hobbit]

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

[davidkwon]

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

[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.

[theproofessayist]

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?

[jonasdaverio]

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...

[matrixstuff]

it makes me so happy that you dont use nabla squared

[jamesbra]

Can you do a video solving this using complex analysis?

[tomkerruish]

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.