Demystifying the Dirac Delta - #SoME2

In this video, I explain what the Dirac delta REALLY is - and no, it's not a function, at least in the usual sense!

I always felt unsatisfied by the informal definition of the Dirac delta given in physics or engineering courses, but when I finally learned the rigorous definition, it seemed disconnected to the informal one and not very enlightening. I decided to make this video to connect the two definitions and show how intuition and rigor can (and should!) coexist.

Technical footnote: Sometimes the Dirac delta is presented as a distribution or "generalized function", i.e. a continuous linear functional on the space of smooth compactly supported functions. Distribution theory is very powerful, and the Dirac delta appears when trying to make sense of differentiation of non-differentiable functions, or solving differential equations in a "weak" sense. Despite this, distributions can get quite technical (even the existence of smooth compactly supported functions is a bit technical), so I felt it best to avoid covering them in this video.

Submitted as part of the Summer of Math Exposition 2 (#SoME2) contest.

Timestamps:
00:00 - Introduction
00:25 - Informal Definition
1:45 - Measures
3:08 - The Dirac measure
3:58 - Integration with respect to measures
4:53 - Explaining the sifting property
5:35 - Why infinite at zero?
7:10 - Linear functionals
8:45 - A rigorous definition

[JaGWiREE]

Underrated & underappreciated video. A must watch for anyone planning on going down the measure theoretic probability route.

[ejovo]

I've been a little confused while learning about distribution theory and how the dirac delta function is defined through that lens, so it was really interesting and informative to see how a different branch of mathematics adds intuition and creates a more complete understanding of this object.

Thank you for this great video.

[claudestan]

As a former engineering student I have never thought of the Dirac function as anything more than a normalized ideal pulse input. It is really nice to know about the math it originated from. Great video.

[billy-cgqq]

Summer of math expedition #some2 is the GOAT of Youtube content, hashtags, collabs, or whatever this awesome thing is. Wow!!

[aminzahedi]

As a physics student whose had way more than his share of suffering from major mathematical itchiness and headache when it comes to the Dirac delta function, I really appreciate your effort and enjoyed the simplicity of the presentation and also think I actually did learn a number of new things; namely, the Riesz-Markov-Kakutani thing. Truth is, even with my preliminary trainings in real analysis, abstract and linear algebra, and even some rudimentary knowledge of basic topology and measure theory, this particular subject is way too advanced to be readily grasped, even by the expert folk with a healthy background in math. I also watched a lot of content under #SoME2 to understand how “renormalization” works and learn its basic principles, but unfortunately it’s one of those difficult, inaccessible materials. Nevertheless, kudos on a great video and hope your channel grows BIG👍🏻❤️

[quasaree]

This was the clearest interpretation of measure theory I've seen in a while.

I feel like your video could really use a better thumbnail to get more views.

[johnchessant]

The delta function can also be defined as the limit of Gaussians with mean 0, as variance -> 0.

[PowerhouseCell]

This was really well done! I can't believe I just found your channel — as a video creator myself, I understand how much time this must have taken. Liked and subscribed 💛

[wowZhenek]

Thank you for this video. It honestly made quite a few of my holes in math knowledge filled. Not just the dirac delta function. I wish you success in this youtube adventure.

[Phito_Sch]

I quit my carrer in formal mathematics last year, i remember concepts from this video in a course i took on the second year. there's always so much to learn in math i will always love it for this, as for me i will be shifting gears into electrical engineering next year i recon it suits me better and truth be told the formalisms and rigor got the best of me but you keep at it shit pays off, or so I've been told.

[mayatrash]

This was absolutely fantastic. As somebody doing Many-Body QFT with interactions I always just took it as a given and even though I did some measure theory but never Functional Analysis I was always surprised and bothered by some integrals (for example a rigorous calculation of a particle-hole bubble) and the relationship of things like the Heaviside step “function” and the Dirac “function”.

This helped a lot for my intuition and I hope you will gain subscribers soon, your way of explaining is superb

[JCisHere]

I've been (ab)using the delta function quite a bit lately. Great Video 👌, interpreting delta in the context of measure theory demystifies quite a few things

[franciscomacedo]

THANK YOU FOR MENTIONING THE RIESZ MARKOV KAKUTANI THEOREM
I have been having analytical mechanics and have been messing around with variational principles, and I've always wondered why we're extremizing one type of functional (namely one where we are integrating) and now I know why! Thank you, I searched everywhere for this!

[wn]

The Dirac Delta is absolutely a function. It's a set of ordered pairs such that if (a, b) and (a, c) are elements, then b=c. That's the definition of a function, and it fulfills it.

[RB-ewlo]

Excellent video, enjoyed every second of it! Also didn’t know the Riesz-Markov-Kakatai theorem, so cool!

[pra.]

Awesome, and a great intro to measure theory too

[JGHFunRun]

So I‘ve actually been thinking about the idea of integrating with dx as an input and not just as the final term, and the Dirac delta seems like it’d be 1/dx at 0 and 0 everywhere else based on the introduction. This would satisfy both properties (no I didn’t accidentally recreate the Dirac delta I just realized how something I was thinking about could be used to create something like it)

[erik]

Yessss buddy this is the best SoME2 video so far, though I might be biased as a physicist.

[reformierteapologetik]

I love this and really appreciate your explanation.
Do you have any sources that explains this further by any chance?

[tuongnguyen]

Oh my god can you do more on measure theory, this is a great video. I did not appreciate measure theory (as an engineer) until I watch your video