The Shadowy World of Umbral Calculus

An introduction to a famously enigmatic area of math, for calculus students of all levels
↓ Info and Timestamps ↓
In this video we use a primer on discrete calculus to motivate an exploration of the idea you saw in the thumbnail, deriving 2 summation methods along the way!
This was made for the 2022 Summer of Math Exposition (#SoME2), an annual jam encouraging new creators like me to make content like this. It's been a really satisfying project to complete, and I'll definitely be making more in the future!
Discord Server: / discord
Patreon: / supware
0:00 Intro
1:28 Forward Differences
2:49 Summation
5:59 Falling Powers
7:23 Umbral Calculus
9:53 Stirling Numbers
11:39 Umbral Exponentials
13:27 Newton's Forward Difference Formula
14:31 Thanks for watching!
Music:
Chris Haugen - Glacier
Chris Haugen - Northern Lights
#umbral #calculus
Corrections:
7:13 denominator should be (x+1)(x+2)...(x+n+1) - Jose V
13:13 and so ϕ(1/x) = 1/(x+1) - Jose V
13:22 ϕe^-x is undefined, and the other "curious equation" shown means that implicitly applying ϕ to a circle gives a logarithmic spiral - Discord community

[@al.7744]

You explained the topics enough to understand what was going on and showed barely enough for us to be intrigued and interested in this without us getting really spoiled or you being tiresome. I am fully convinced to at least attempt and learn more from these fields eventually because of this video. I cannot help but praise you

[@krhino42]

“By rearranging the question, we get the answer.” Imma use that

[@fabiant.2485]

In the sci-fi rouge like rpg "Caves of Qud" dark calculus is a forbidden field of mathematics, because it's study opens a path into a transcending layer of reality inhabited by an infinite ocean of psionic minds... After watching this i am impressed by how accurate to real life the devs made their lore.

[@jayst]

I could feel my Glimmer rise after watching this

[@elkinmontoya9640]

There is so much mystique in this area. I feel like there is a mystery that is just lurking, waiting to be discovered. I see little tidbits of group theory conjugation, analytical combinatorics, probability density functions, so many paths begging to be traversed. From a personal point-of-view, so many potential application to physics

[@TheRationalPi]

Oooh, this seems like it could have lots of utility in digital audio processing, since you're regularly moving between the discrete and continuous domains.

[@Supware]

Interesting, I'd love to see more practical applications of this thing

[@OdedSpectralDrori]

brilliant direction. time to see how these transforms would help analyze some filter and the fourier transform

[@Supware]

@@OdedSpectralDrori don’t quote me on this but it seems the Laplace Transform is VERY relevant here :p

[@OdedSpectralDrori]

@@Supware this will make a fine quote

[@Supware]

@@OdedSpectralDrori what have I done

[@lexinwonderland5741]

this is SO much better than the wiki page. It left me fascinated (even moreso than Dr. Michael Penn's talks on the matter), and honestly someone should REALLY add the homomorphism between discrete and infinitesimal calculus you described here to the wiki AT THE MINIMUM. thank you so much for the contribution to math education!!

[@Supware]

Wow haha thank you! Guess I've no choice but to keep it up :)

[@pauselab5569]

it's already there but not much on it. like 3 line paragraph style

[@vladimirmakarov9849]

Absolutely fantastic video. The Newton difference formula derivation was simply amazing, i used it before but never knew where it came from and this was just the cherry on top. Can't wait for the follow up!

[@johnchessant3012]

Very nice! The example of umbral calculus on the Wiki page is pretty cool too, it relates to Bernoulli polynomials B_n(x) which satisfy the identity B_n(x) = (B + x)^n, i.e. B_n(x) = sum (n,k) B_k(x) x^(n-k). And you can actually simplify some proofs of identities involving the Bernoulli polynomials by doing "calculus" with this umbral notation.

[@Supware]

They’ll be appearing in the follow-up!

[@corbinsmith5015]

I love umbral calculus and generating functions. Ive been reading George Boole’s book on the calculus of finite differences, and I really appreciate videos like these which make the ideas more accessible to the general public

[@user-ou3mp2lz2o]

I've been struggling with abstract algebra and your video presents a perfect example for why isomorphisms are useful! Really appreciate it!

[@Supware]

Great to hear!!

[@MirMarksman]

Rad! The familiar-but-different feeling makes this feel almost like a math dream

[@Supware]

After working for a while on a second video I think this might actually be a major vibe I wanna aim for haha

[@alpers.2123]

Is this similarities related to group theorem?

[@Supware]

@@alpers.2123 phi is a homomorphism between discrete and classical calc

[@dyllanusher1379]

Definitely looking forward to another video, thank you for sharing these mind blowing ideas!

[@minimath5882]

This was great, I look forward to seeing more in the future!

[@georgecote]

awesome quality! I'll be very happy to see more of that :) good luck!!

[@KoHaN7]

This was an incredible video. The way in which you merged everything together was mesmerising! I can't wait to se the follow up and some more exceptional quality of work and educational content. Bravo!

[@pr4_kp]

Incredible! I hope you do more on this topic - really got me thinking :)

[@riccardofiori828]

This is one of the best math videos i've ever seen! Ideas are presented so neatly and however so mind-blowing. You earn a subscriber, I'm hoping for more videos like this!

[@timkohl9831]

This is something I've been curious about for a long time, thank you. This is very well made.

[@MultiAblee]

Super interesting Stuff! I like how categorically you can see in this topic that calculus itself is a limit of this discrete version! The exposition was super easy to follow, love it.

[@dcterr1]

Wow, this is fascinating! I never learned much about discrete calculus before, but you've definitely whetted my appetite! Great job!

[@orirrr8]

Extremely clear, insightful and interesting exposition!

[@discreet_boson]

One of the best SoMe vids yet! I literally just learnt about the binomial theorem and summation, intriguing to see it can also be expressed using discrete calculus

[@ficakcblack1090]

This video is so amazing, it blew my mind, please continue making these

[@jonipaliares5475]

Really interesting topic and amazing explanation! I had never heard of Umbral Calculus up until now. I'm loving this summer of math exposition, I'm learning about so many new interesting things!

[@mcdudelydoo3116]

I gotta admit, this is one of my favorite videos of the SoME2 this year. This intrigued me so much and you explained it pretty straightforward even though I didn't completely understand everything on the first viewing. This year's SoME really gave us some banger math videos, can't wait for next year!

[@SirTravelMuffin]

Really solid pacing, and definitely leaves me with a lot of curiosity for the subject!

[@NoNTr1v1aL]

Absolutely amazing video! Subscribed.

[@Catt0]

This was beautiful, I had never heard of this branch of calculus before but I'm absolutely in love with it. Your sub count honestly shocked me, this was such high quality.

[@jaafars.mahdawi6911]

Simply fabulous! More of these, please, sir!

[@miklosbognar1449]

this is probably one of my favorite math videos on youtube, well done

[@vnever9078]

Thank you Supware for introducing me to this beautiful world of Umbral Calculus!

[@adamantmist9394]

Great video about a topic I didn't really know anything about. Surprised of seeing stirling numbers too, when I learned about them they were shed by a completely different light and these kind of connections are what make math so interesting. Looking forward for a follow-up! And I hope this video gets blessed by the algorithm just like other SOME2 videos have been. Thanks and have a great day.

[@paulkohl9267]

Loved it! Please, by all means, more on this topic.

[@ColeCoug]

Great explanations and animations for a topic that I have never heard of before!! This video makes me want to learn more about it

[@tricanico]

Wow, that was so cool. Thanks for posting this.

[@HaliPuppeh]

This was really interesting. Thank you for explaining things slow enough that I could keep up. So many videos on advanced topics go too fast and I get really lost.

[@pedroth3]

Great explanation of this concept! Congrats and thank you

[@LebronJ0]

What a legend only one ad in the beginning . Your so damn underrated

[@gianpierocea]

Incredibly well done, this is exactly my kind of thing. Thank you :)!

[@djamondaxuzm4712]

Thank you for the intro, really professional and keeps everyone on the same page

[@kodirovsshik]

This was just WOW I always had the idea and the basics of the discrete calculus calmly sit somewhere in the back of my mind with me knowing that is it like "somewhat" related to the canon calculus, but now that i see this video, i realize that OH MAN is it something completely different!! I would surely love to see more content on some more advanced stuff on this topic, top interesting stuff

[@diogeneslaertius3365]

Thank you for the video! I spent like 45 minutes going over the video, writing everything down, checking. It was a great experience. Please make more videos like this one, including the follow-up video on the Umbral Calculus.

[@EC-pf1tr]

This was awesome. I feel like I finally have a window into why I leaned everything that I did and how everything is connected from combinatorics to sums and differences and the discrete, to the continuous, to linear algebra, to complex numbers

[@RSLT]

Very Interesting! Great Job. Well and easy explained!

[@akio-the-lazzycatto]

Wow, that is pne of THE BEST videos I've seen! I am impressed! This is magic in real life!

[@michelangelowebb4362]

I hope you continue with this videos. It it amazing work. Thanky you man !

[@whitestonejazz]

this video was a trip. crazy to think this was never mentioned in any calculus classes. Very cool, thanks!

[@robharwood3538]

Great video! I had been exposed to Umbral Calculus a wee bit, but not much stuck with me; and I had also learned a little bit about doing finite sums (a la Newton) using falling and rising powers. But the two different parts of my brain didn't make the connection between the two topics until this video. Thanks! This new (for me) connection has already illuminated a lot of things I had been confused/stumped/stuck about before, and I can't wait to put this into practice!

[@Orionhart]

Totally worth the wait!

[@nagoshi01]

Please keep making these!!

[@meccamiles7816]

Excellent video. Thanks for sharing.

[@eduardoabreu78]

Fantastic, you Sir did a Great Work!

[@georgegaumond2733]

Brilliant calculus video!

[@funnydog7817]

I’ve been shown another area of mathematics that peaks my interest, and has given me a decent view into the essence of it! Thank you, when it’s a drag it’s always better learning something new, and maybe finding some meaning within it.

[@user-ef8kc4rv7n]

Just in case you didn't know, it's "pique" in the phrase to "pique one's interest"

[@lookupverazhou8599]

@@user-ef8kc4rv7n Like Piqueachu.

[@pawebielinski4903]

Excellent video. I'd love to see more on the topic!

[@notjerrett]

Definitely looking forward to that followup!

[@firefox7857]

3:01 I've never seen that explanation for the fundamental theorem of calculus... it seems so simple now.

[@angelmendez-rivera351]

Well, unfortunately, the equation shown on screen is not actually what the fundamental theorem of calculus is or says.

[@rosettaroberts8053]

Oh wow, this is really cool! I've played around with the umbral operator before without realizing what it was. I think the most recent time I used it is when I was converting a formula for factorial moments into a formula for non-central moments a few weeks ago.

[@Supware]

Oh nice! I'd love to know if it has a name or symbol, I somehow haven't come across either yet

[@Fenristation]

Nice stuff! Thanks for sharing

[@symbolspangaea]

This is a gem!

[@proxagonal5954]

Very high quality vid! I once read the Wikipedia page on discrete calculus, and the conclusion I came up with after reading for a bit was that it was dumb people calculus for dumb babies, and also that it was boring and dumb. But this was actually pretty interesting! The video & graphics quality here was great, loved the visualizations and I would've loved it if you had even more graphing and illustrations, especially in the later parts of the video. I'm looking forward to your next video!!

[@Supware]

Thanks! Illustrations are certainly gonna be an interesting challenge in the next one...

[@Briekout]

@@Supware what are you demonstrating with sir? MathCad ?

[@Supware]

@@Briekout Manim

[@alpers.2123]

It is calculus for engineers lol

[@proxagonal5954]

@@alpers.2123 Don't laugh at engineers man. They cool

[@PamSesheta]

This has reminded me of something from a calculus class near the end. This was amazing, first encounter with phi operator and now I wanna know more because this seems really handy for dsp

[@ILSCDF]

Wow, i definitly want more of this

[@GelidGanef]

This hit the sweet spot for me in that it's perfectly intuitive that this should work, but how it works and why blows my mind.

[@Luan-bs1vp]

Awesome video!

[@kuppersrocky6834]

wow, great video, thanks!!

[@MCLooyverse]

Absolutely wonderful. Something about conjugation (q * x * q^(-1)) makes me happy every time it comes up (it comes up a lot).

[@ooos2989]

Out of all of the videos from SOME2, this one was the most eye-opening. Looking forward to SOME3!

[@maxime2949]

I don't usually comment on videos but such quality from a channel with 254 subscribers amazes me. This video's topic is really well chosen as it is understandable and complex and your explanations make it a great time !

[@Supware]

Thanks, means a lot!

[@SnapThority]

The maths animations are really good, I appreciate the effort you put into them!

[@Supware]

Thanks, and sorry for only just noticing this comment haha! The animations were made possible by 3B1B's Python library. Great software but takes some setting up

[@diegohcsantos]

I really want to see more advanced discrete calculus stuuf!!! Thank you so much for the high quality video!!

[@Supware]

I'm planning some shorter videos on summation by parts and summation using complex numbers :)

[@TranquilSeaOfMath]

Really nice presentation .

[@randomz5890]

Amazing work here. Grant will definitely take notice to this! I generally love videos explaining more niche areas of maths, so I unfortunately have to be a bit selfish here and ask that you keep making more videos like this because they're amazing!

[@Supware]

Working on it!

[@randomz5890]

@@Supware awesome, I should expect you to be two steps ahead 😄!

[@luiz00estilo]

Just leaving a comment to help with the algorithm. This video was _enlightening_ . Great work man!

[@arongil]

Agreed, the algorithm needs to know this video is top quality! Leaving this comment for the same reason :)

[@haroldbn6816]

This was just beautiful 😍😍

[@CCequalPi]

This was great thank you

[@mrtfttkhv]

Great video!! Keep up

[@benjaminojeda8094]

Woooow, this is incredible

[@sejr8053]

Awesome video 😍

[@paulchelou3596]

Excellent, thank you very much 🙂

[@SeanGhaeli]

fantastic video

[@TheHonestFellow]

This is exactly what I’ve been looking for.

[@orktv4673]

The way that forward difference operator is defined strongly reminds me of Dirac's derivation of the gamma matrices.

[@devincustodio2258]

This was amazing! Would love to see more umbral calc videos. I've also been getting into surreal numbers. Would be cool to see your treatment on that subject. :)

[@Supware]

Thanks! I just started on my second umbral calc video :) Oh man, surreal numbers haven't been on my radar for such a long time! RIP Conway. I'll see what I can do :p though there are a few topics I wanna talk about already

[@apm77]

There's a pretty good intro to surreal numbers at kzbin.info/www/bejne/kIrNZYGhmqyDmq8 but I would definitely present some things differently, especially early on. In particular I'd lay the foundations (up to the definitions of star, half, and up/down) using a simpler game than Hackenbush, one specially constructed to provide a 1:1 mapping between mechanism and formalism. Just draw trees with red and blue branches, put a counter (monkey) at the bottom of each tree, and specify that the red player must take a monkey up a red branch, while the blue player must take a monkey up a blue branch. If you can't move a monkey, you lose.

[@aBigBadWolf]

that was so cool!

[@pamdemonia]

Please do more!

[@kkski4817]

I like this video a lot

[@Supware]

Hey thanks! I installed a de-esser for the next video, hopefully that'll do it :)

[@striga314]

This is so cooool. Which books or other texts could you recommend about this topic?

[@Fru1tyy]

Now this is super intriguing 😳

[@davidblauyoutube]

Nice video! It seems that category theory produces some deep connections between continuous and discrete calculus.

[@lachlanperrier2851]

Make more that was great!!

[@cassandrasinclair8722]

Just found this, and your channel. This is the wildest math I have seen in a long time! I wonder if this has applications elsewhere.

[@Supware]

I think "the wildest math I have seen in a long time" is what I'll be going for from now on hehe :) I think you'll enjoy the second video..!

[@cassandrasinclair8722]

@@Supware looking forward to it! :D

[@therealjordiano]

The part at 7:35 when you introduced the main idea I near enough leapt out of my chair and yelled 'holy shit!' from the bottom of my lungs (not rly but still), amazing stuff and very nice explanation

[@emanuellandeholm5657]

Nice video!

[@bobbyking2490]

Wonderful!

[@kiiometric]

Oh my god my brain is tickling, this is beautiful

[@starkissed5795]

Please do more of these! 👀

[@Supware]

Will do :)

[@chennebicken372]

I love this video. You did a very great job at showing all these relevant connections 🤩

[@alejrandom6592]

This is beautiful

[@micuhh]

3b1b's manim is an incredible tool for stuff like this. calculus and its practicality go hand in hand, so being able to visualise stuff in calculus is incredibly important to understanding it

[@Supware]

It's such a great tool! Even for simple things like manipulating equations, it makes everything so much faster and easier to follow If I were using paper or a whiteboard instead this video would be 2 or 3 times as long for sure