Harmonic Functions

If the Laplacian of a function is zero everywhere, it is called Harmonic. Harmonic functions arise all the time in physics, capturing a certain notion of "stability", whenever one point in space is influenced by its neighbors.

[TheAwesomoe]

I really am glad that Khan Academy was built, and especially expanding. It truly is a great tool.

[KydroxHD]

3BLUE1BROWN?!?!?!? I am so excited right now

[ziadyoussef]

can't express how happy I am seeing people like khan and his team just trying to teach people in a great way without willing to get money but instead of taking money they give time and knowledge, thank you

[kellywshere]

3BLUE1BROWN!!! Best math explantions on the web! You provide TRUE insight that I have not seen anywhere else.

[jagandwarampudi]

Man, Why are you still here? You should be a professor at Stanford may be. You deserve that. You're the best teacher ever...

[joeyquiet]

I know his voice, this guy is my hero, learned a lot from this guy

[monisha]

Suggestion: 5:57 it would be good to say that the `magnitude` of the neighbours is less than the magnitude of the point, instead of `less than` since we are also dealing with negative numbers here.

[rahulmukherjee]

In the book called introduction to electrodynamic by D.J. Griffiths the same intuitive explanation of laplacian is given.
I want to know why the term 'Harmonic' is used for such kind of functions which follow that rule of averages.
Can anyone share some views why the term harmonic is used

[danielchoi]

Best explanation on harmonic functions... very intuitive. thank you for the video

[SomeOfOthers]

the computation is:
partialf/partialx = e^x*sin(y) partial^2f/partialx^2 = e^*sin(y) (e^x is it's own derivative, y is a constant)
partialf/partialy =(e^x*cos(y)) partial^2f/partialy^2 = -(e^x*sin(y) ) (e^x is a constant, sin(y) --> cos(y) --> -sin(y))

e^*sin(y) - e^*sin(y) = 0

[theomommsen]

Thank you so much! This video is absolutely brilliant and has made me understand the Laplacian so much better!

[joluju]

This great video was too short. Still wondering what "landscapes" of harmonic functions have in common (when it's not a plane). I understand there must be no "top of hills", nor "bottom of depressions", but does it mean the landscape is full of saddles ?
Since this guy loves geometric interpretations, I wished he had gone further.
Also, why is the word "harmonic" used ? I feel there is much more to understand here.

[avichein]

I understand the explanation, but still don't understand why it's called "harmonic"? What does harmonic have to with anything ? Can someone please explain..

[MechanicalEI]

3B1B, Great video. Its's always cool to find you here! Thanks Khan Academy for doing this and uploading!

[AshirwadPradhan]

This guy is X-GOD of Mathematics! Keep up your insanely good work, 3B1B!!!

[beppenonantola]

He is too good.. Such a great way of explaining math!

[tomaszgruszka]

It might be an incredibly dumb question, but it just got me thinking. Since on average, every point in the neighborhood-circle has exactly the same „height”, no matter where, can we actually calculate this average height for the entire function?

[sharifsircar]

Did need to watch it twice and look up laplace, but finally understood it

[laurahoughton]

I wonder whether you could imagine the surface as consisting of an infinite set of equally spaced strings emanating out from the central point. Some go up, some go down, but any one that goes up must be matched in height from the centre by one going down. That would yield an average height equivalent to the height of the centre.

[dominicellis]

Does the harmonic series have anything to do with harmonic functions?