The Dirac Delta 'Function': How to model an impulse or infinite spike

In this introduction to the Dirac Delta Function we'll see how we can deal with something happening instantaneously like a hammer hit. We will model this impulse with a 'function' that is infinite at one point and zero everywhere else. But such a thing isn't really a function! Sometimes we called it a functional or a generalized function. Regardless, it is defined by its pleasing properties such as what happens when you integrate the dirac delta function multiplied by another function. It can also be thought of as the derivative of the step function. In our next video we will study the Laplace Transform of the Dirac Delta Function and see how we can use it when studying differential equations.
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[@NumbToons]

Came here to hear the pronunciation of 'Dirac', forgot about it middway, ended up watching the whole video. Then realised No utterance of 'Dirac' but still, very bounding explanation.

[@kittyblossom7342]

Wow! You explained it really well. I've been trying to understand this Dirac Delta Function for about an hour now, and couldn't find any resources that explains it visually. Your explanation is just what I needed. Thank you.

[@homervalizadeh5315]

It's not even about distribution theory, but I found here a right place to learn about distribution theory. You're awesome in your job.

[@interest21stcentury74]

Thank You, Dr!!!! A wonderful and clear explanation

[@disasterslight]

I can' thank you enough for this video. We were doing generalized functions rigorously, and I was so lost. This explanation just made so much sense. Please consider making some rigorous advanced math videos! We need you!

[@sumitkumarsahoo7601]

your vector calculus helped me too much, very very grateful to you

[@SHAHHUSSAIN]

Exceptional performance❤❤

[@naman4067]

Nyc comment 👍

[@ahmedtoufahi5198]

The neatest explanation one can find on youtube ! Thanks

[@subhadipkarmakar2841]

Thank you Sir for explaining these concepts 🌻

[@asnakeseyoum3109]

I watched your linear algebra.It help me a lot. Thanks!

[@tylerellison1584]

Thanks! Your content has helped me with conceptualizing many ideas presented in my EE courses.

[@bernardlemay8563]

Thank you so much for that clear explanation.

[@ThefamousMrcroissant]

Excellent little explanation.

[@ayanbhowmick5427]

I'm Doing My MS in Chemical Science at Indian Association for Cultivation of Science , We have a course on Mathematical and Computational Chemistry , this video really helped me a lot.

[@trexbattle]

I’m doing blow at penn state

[@DineshGurijala]

@@trexbattle lolll

[@porit1023]

I didn't even know there is a mean value theorem for integrals. Thank you so much!

[@aayushgautam4919]

Amazing lecture and figures!

[@taylorb2162]

cool video. i remember first encountering this function in the context of point charges in EM theory.

[@tariqc5760]

Great video and explanation! 👌

[@merijnvandewater1029]

yessss, awesome video !

[@imashnake_7151]

so underrated...

[@continnum6540]

Thanks a lot sir 🔥🔥🔥

[@RadoslavFicko]

The Dirac distribution is the Fourier transform of unity and a special case of convolution, where A*f=g, g(x)=d(x-y). f(y)dy , if we imagine the gravitational interaction as a function of g(x) and the electromagnetic interaction as a function of f(y), then these forces (i.e. the lines of force) only interact when x is equal to y ( the Dirac impulse).

[@RaedMohsen]

At 2:39, What if I want to dive deeper into the meaning of delta function ? Any recommendation for resources.

[@rupakkhatiwada5229]

Thank you

[@sourabhchoudhary77]

Good explanation

[@philip7039]

You are actually a legend

[@DrTrefor]

Thank you!

[@mozayn2378]

Sir That was some Good stufff

[@wilurbean]

Do you have any good videos where a deeper dive is taken, not exactly mathy proofs but still deeper than this? I'm taking a theoretical methods class (physics based, math for physics.... so we're covering geometrical vectors, special functions, odes, pdes, matrices, bases, and more in 1 semester. The expectation isn't really "prove this for all cases" but still much more abstract than merely execution of a Laplace. An example hw was to derive a generalized laplacian operator for a non-orthogonal coordinate system which was defined in Cartesian and then use it on that generalized system... so videos at the oddly specified level?

[@RichardJohnson_dydx]

Thank you. Once again, your videos are getting me through engineering school.

[@ashutoshjha225]

Nailed it

[@annothree7881]

great video :)

[@Scene1654]

Great video as always! Could you elaborate on why the delta function isn't considered a function? At about 2:24 you mention that it's because the value at x=a isn't a number, but neither is x=0 for the function 1/x, so what's the difference between the two? Thanks!

[@thharrimw]

Late reply but when defining 1/x you exclude 0 from the domain

[@leonhardeuler8457]

thank you :)

[@jenkinsj9224]

the math God!

[@physicslover1950]

Sir I have a humble request please make a video on tensor calculus and vector calculus of curves and vector calculus of vector field. I have a question in your video of difference between single and multivariate calculus you told us about rectangular prisms using two integral signs one with respect to x and the other with respect to y means integrating the function with respect to x and y at the same time. Here can we do the reverse means derivative of a multivariate function both with respect to x and y at the same time which we call mixed second order partial derivative? Why dont we call this total or complete derivative as you called the multivariate integral the total or complete integral and not second order complete integral? What is total or complete derivative? What does dz/dxdy tells us it surely doesn't tells us about the tangent plane. Then what is this and why it is not in practical use? One example is z= 2x²y +4xy then dz/dxdy or dz/dxy = 2x+4 .. Are dz/dxdy and dz/dxy same? If not then which one of this is total derivative and which is complete derivative and in which direction it is pointed? I have heard somewhere that it tells about the torsion of geodetic . This sounds very unintuitive. Sir please make a detailed video on this about this misconception. Please sir, you are the best.

[@TheRevAlokSingh]

The delta is in fact a function if you use the hyperreals. One model of it is a gaussian. It cannot be given as a real function because both its domain and range are not standard (infinitesimal and unlimited)

[@leonginear123]

Doesn't the Mean Value Theorem only apply to continuous functions?

[@zlatanbrekke6538]

On an continous interval

[@wilurbean]

"Step-function... what are you doing back there..!" 😲😲😈😈

[@anboli9067]

🤩

[@hekk_tech5975]

I don't understand why did someone decide that the area of a delta function is one? The same way one could take a rectangle with an area of 2 or above and shrink it to almost zero, the amplitude would jump also to infinity right? so why 1 and not 2 or any other number???

[@DrTrefor]

It’s just a convention, but a decent one as it means when you integrate you aren’t stretching other things by a number, just multiplying by 1

[@user-qs1tp1ll9i]

This is just like the unit circle has a radius with 1, that is simply a convention. Besides, the Dirac function has a lot to do with quantum physics

[@jhmaths]

Hi Trefor, I think you could be a great teacher but unfortunately I can't tolerate this thing that seems fashionable at the moment (especially among UK news reporters) of speeding up and slowing down speech for no apparent reason.

[@PresCalvinCoolidge]

2:00 then a miracle happens as epsillon goes to zero, delta goes to infinity, but the integral stays = 1.

[@wolfelkan8183]

So it's the derivative of functions that don't have a derivative.

[@soccerbels7947]

You are like saying that the ice creams expired and so they don't exist....... For an function to be differentiable it must be continuous, a flat line have no change in time........

[@reynaldomesmo]

666 likes!

[@ktporousktmedia8021]

Your description of Delta function is not correct. The integral of Delta function is Zero, not 1 by Lebesgue integral theory, because Delta function takes infinity at t=a, and takes Zero except t=a on the board. You do not understand the elementary rule in mathematics, that is, the change of the operation "limit epsilon-->Zero" and "taking integral" is not permitted. I guess that you do not know Distribution theory and Lebesgue integral theory.

[@mikovanderheijden3730]

am not doing any math related study and have never done one. i'm a social worker. how did i get here and my brain hurts.

[@DrTrefor]

Ah the KZbin algorithm at its finest

[@soccerbels7947]

My brains cheers for such amazing content

[@Mathin3D]

Gbage. What was the doctorate for???