Dirac Delta

Dirac Delta Definition

In this video, I define the Dirac Delta functional, and show that it is strictly speaking not a function. Along the way, I show that for infinite dimensions, a vector space is not necessarily isomorphic to its dual space. Enjoy!

[johannesh]

Well, in Physics we use this very often, also in combination with Fourier series, and very much like a function. But of course we keep in mind that it is actually a short pulse with short -> 0

[imgayasheck]

Dr peyam you make me love dual spaces because they are very versatile

[stayawayfrommrrogers]

I would totally enjoy a series on distributions.

[johannesh]

Man, functions of functions are more and more fun the more of your videos I see about them!

[smg]

U just solved a physics problem. Two slit experiment: - when wave passes through a slit it must slow to virtually zero so energy goes sideways as fast as u can't obviously blink an eye at, to relieve pressure. However slit has walls so side bounce gangs up between sides then once through other side it has forward + sideways movement

Cheers mate

[tylershepard]

This is very neat. In electrical engineering we study impulse responses a lot because it contains one frequency. In fact summing all cosine overtures at the same amplitude results in a periodic dirac delta “functions.” Even though it’s quite useful for this reason, you never see an actual impulse. A voltage will follow a very sharp Gaussian envelope instead. Even more so, that’s how ultra wide and works. We transmit sharp Gaussian envelopes to transmit over a wide variety of frequencies. I know this was long, but it’s amazing how useful it is!

[jaronfeld]

This one was seriously amazing. I loved the Dual Space work a lot. I work in Riemann Geometry and Sub-Manifolds so I love the linear algebra+analysis combinations. This was fun!

[dgrandlapinblanc]

Hard but always interesting. Thank you very much.

[Tomaplen]

everyone: Yeah i know its not a function its obvious etc etc..






only Dr Peyam: Here is why booom

[정대영-le]

Thank you for this video. Now I got some points of distribution and how linear algebra becomes powerful.

[RalphDratman]

This is so much fun! I have never seen such joy in connection with mathematics. Thank you Peyam.

[kevin]

Wonderful video! I have always hated this "function" when it was introduced in my physic classes, but after I learn more about math, I see that its nature is actually that of a functional. This made me more comfortable when thinking of Dirac Delta. Your video gives me a clear understanding of what it is, at least in its basic definition. Thanks a lot.
Just some quick note, at 17:27 you talk about taking open interval to deal with the problem if the chosen x being ±1. I wouldn't suggest that since it might give rise to situations that you don't get a finite number after integration, e.g. g(x) = 1/(x-1). You may just assume that x, while not being 0, is also not ±1. You can deal with that with continuity, like what you were going to do in 20:15.

[Jose_Hunters_EWF_Remixes]

Would a less restrictive condition on f hat suffice? For example, could we consider the case of measurable functions on [-1, 1]? My impression would be "yes".

[Jose_Hunters_EWF_Remixes]

So, in the finite dimensional case, V and V* are isomorphic as vector spaces? Aren't all n-dimensional vector spaces isomorphic (as vector spaces over the same field) to all other n-dimensional vector spaces?

[Bombelus]

20:20 - On the left side you've just proved that "d != f hat", but on right side showing what "d = f hat" looks like. I take it that the latter "f hat" is different from the former.
Also, isn't the latter "f hat" just a non-continuous function (which happens zero everywhere but at zero)?

[mehdisi]

Very nice video. Thank you dr.peyam so much. If possible, make a video about distributions.

[LobsterInSuit]

Very nice video, I hope you keep talking about functional analysis!
It would be interesting to touch some elements like tempered distributions and Schwartz spaces!

[cameronspalding]

With the delta function: can it only be said to belong to L1

[ShubhayanKabir]

Idk why but I just start by liking the video. 😅

[imgayasheck]

I'm just going to say this before watching: MEASURE