Div, Grad, and Curl: Vector Calculus Building Blocks for PDEs [Divergence, Gradient, and Curl]

[@EfficientEnergyTransformations]

The best explanation of these concepts, from the point of view of physics and physical problems, I have seen in Oliver Heaviside's "Electrical Papers". Many people, in science do not know that it was Heaviside who came up with the vector form (in the way we know them today) of the Maxwell's famous 20 equations. Maxwell wrote his theory in quaternions, but it was highly misunderstood for more then 20 years, and it was Heaviside who basically single-handedly morphed, end extended, that theory in what currently is known today as classical electrodynamics. The truly amazing fast in all this is that Heaviside had only a high school degree! It is a true testament what an intelligent mind can achieve, even without being thought in formal educational system, ones focused on an intellectually challenging and exiting problem.

[@kiabtoomlauj6249]

That's interesting.... someone with only a high school formal education could extended or expanded on Maxwell's equation. I think it's a bit easier, if you're born in the late 1800s to the early 1900s. From the 1800s to the early/mid 1900-ish, I think really smart individuals were able to learn, large on their own (or with only high school or undergraduate levels of formal education), deep mathematics. Ramanujan was one & the late Freeman Dyson was the other, both rare talents. I believe it was said, even while Dyson was alive a couple years back, that he was the "last professional mathematician" to never have formally earned a PhD in mathematics. I think part of the story was that he was too eccentric, to want to spend years under someone who's going to tell him to think or do things he didn't want to; it could also be that his formal studies (for an advanced degree, for example), was "interrupted" in the late 1930s to mid 40s, when he started working with giants like Feynman, Hans Bethe, Fermi, Teller, Lawrence, Fucs, Seaborg, Oppenheimer et al in the Manhattan Project & related Pentagon works.... I believe it was Freeman Dyson who recounted, later on in life, that among that exceptional group, the soft spoken, ever-courteous Hans Bethe --- with a thick German accent --- was the one with seemingly photographic memory who didn't need to refer to tables or notes or books on the various numbers (atomic weight, melting point, boiling point, density, electronegativity, ionization, thermal conductivity, etc.) of the various major elements they're working on.... because he had all those numbers in his head!

[@dalisabe62]

Think of all original mathematicians who formulated the knowledge we know today. What schools did they attend and what teachers did they have to help them come up with their fantastic exploration and knowledge? Schools are great when they serve as a motivator to learn and a supplement to hard work and creativity. When there is no interest, no hard work or no creativity, schools are just another wasteful spending. Deeper understanding of any subject requires curiosity, further reading and much contemplation. I kept my textbooks and visit them once every while to reformulate what I have learned and put it in a different perspective. I love it when I see things the second round differently and more broadly or abstractly. Mathematics specifically is a subject that is full of amusement and applicability. The same subject could be generalized and applied in so many novel ways as deeper understanding achieved.

[@mdsaddamhossain3565]

After I have completed my msc in Chemistry and more then after that silly thing I got something intuitive approach to calculus, imagination with calculus is Great 👍

[@willthecat3861]

It's not that simple... at least for Heaviside's contribution to vector calculus. Yes, he became to strongly go against quaternions (which IMO are easier to understand, if you already know vector calc.) ... but, Heaviside wasn't the only one trying to explicate Maxwell's equations, by using vector calculus. Heaviside was... for a time... forgotten... and he made many important contributions to eletro-magnetics... but the history of his work is sometimes popularized incorrectly.

[@coreymonsta7505]

The best explanation from someone who’s already seen it before kind of comment lol

[@thegil-martingetaway8804]

That del the funky homosapien reference floored me haha, great video professor thank you for the insight

[@Eigensteve]

Glad you liked it!

[@krewmaga]

Few weeks ago I have no clue nabla, gradient or divergence are even things. Now I need it on my first semester of college and I'm grateful channels like this exist

[@karthik_nt.]

we need this in 12th grade in India for Jee Exam

[@ricksan5583]

It all make sense when it is applied to physics, electromagnetism.

[@lioneloddo]

Ces vidéos sont incroyablement pédagogiques ! On ne perd pas son temps. On a le sentiment d'apprendre quelque chose de vraiment utile et de durable dans le temps.

[@amalshadin]

So this is what Harrison Wells does when he's not in the flash.

[@CodyChen-s8d]

Underrated comment 😂

[@raymondigbana737]

Naah...this is goated fr😂

[@junaidahmed9063]

😂

[@Jonasuniaccount]

i thought the same xD

[@FilipeSilva1]

You are fantastic. I hated learning this in university and am absolutely astonished with how interested it can be when related with a physical sense/purpose

[@liboyan7010]

Dr Brunton is also a great professor for teaching....

[@Toto-cm5ux]

As a computer scientist without training in mathematics, I understood your explanation! Thank you very much !

[@mdsaddamhossain3565]

Thanks for the people like you who present mathematics in meaningful way❤

[@ianfinley89]

Shout-out to Nabla, the Funky Homosapien! The greatest rapper to convert a scalar field to a vector field!

[@mathman1475]

Thanks Professor Brunton. My graduate advisor who was from the Czech Republic used Nabal when I was in graduate school. The term confused me at first until I realized what he was talking about. He was a great professor like you. I owe much of my knowledge of fluid mechanics to him.

[@rajanalexander4949]

Excellent explanation, thank you! Extra points for "nabla the funky homosapien".

[@diveintoengineering6089]

Thank you and I´m waiting to see the future lectures of this series! "Motivation to WHY we are learning this". That´s the key. Best wishes from Chile, South America.

[@01adarshkumar8]

That was one of the most interesting and conceptual class i have ever had for Engineering Mathematics... Wowww.... Thanks a lot sir!!!

[@Eigensteve]

You're welcome! Thanks for watching :)

[@ashafaghi]

🌺 excellent explanation 👏👏👏

[@MrBeasClayton]

love you for very celarity 1:33 partial, partial X- Y Direction its clear all the things

[@physixtential]

This is the first of your videos I've been able to follow start to end. Nice clear and concise explanation of the 3.

[@nouralabbass7055]

There aren't enough words to describe how much I'm grateful to you ❤️

[@Eigensteve]

Thanks for watching!

[@joaopedrorocha5693]

Hi. A small caveat that's useful to remember is that the definition of grad, div and curl that Steve has shown is the definition in rectangular coordinates. There are general definitions that are valid for any coordinate system (such as spherical or cylindrical coordinates). Once i've got stuck on a problem because of that ... was using spherical coordinates and applying the divergence definition for cartesian coordinates.

[@harry3004]

Our teach was supposed to cover this in a 1.5 hour lecture, and he never got around to solving any practice problems for our reference. I'm glad KZbin exists

[@nharshithreddy8445]

In my observations I think this guy is writing with his right hand So he is observing and writing in the reversed direction And that's amazing 👏

[@remyassier1758]

one of the best explanations I've heard thank you so much

[@oguzzaydin]

I love your passion on these subjects, sir. This content is really educational and exciting. I'm looking forward to the other parts.

[@paxdriver]

"Lll-literally" looking forward lol 😜 jk, my sentiments echoed. Love these videos

[@ishangoinyambo6523]

This is so cool thank you again sir! Looking forward to more videos on this.. I was doing data foundations and I wish I had started with this intuitive understanding of vector calculus

[@davidstrid348]

This was brilliant as a student who often gets stuck on specifics I feel like this was especially illuminating after having loved my first algebra courses at university and struggling badly with one and several variable calculus.

[@GoodLife-ru8di]

How I wish I had a Professor like you during my undergrad. Amazing presentation! Thank you.

[@rydinorwin]

Steve is just the best, I love these, I am so grateful. I work in Acoustics and I am determined to get a deeper level of understanding. You make it so clear!

[@ZahraAmini-c6w]

The best explanation i’ve seen so far

[@Mhd-sv]

But I didn’t get anything 😪

[@semihcanakca1762]

Great explanations as always. Looking forward to the next episodes!!

[@kilelevo3097]

By far the best explanation of the nabla operator that I've found during the university studies

[@ricksan5583]

I have heard a good one with which I have enough, I will compare this one another day. Got to go.

[@sandrotomazmartins5257]

Wonderful content! Thanks for posting it. Greetings from Brazil.

[@danielhoven570]

Excellent! I'm trying to rebuild my proficiency in this stuff before going back to school, literally perfect timing!

[@fak892]

Could have used this about 30 years ago. Fantastic explanation.

[@Eigensteve]

Thanks!

[@fak892]

@@Eigensteve so to get slightly more specific, I'd say that I don't recall one prof who made any effort to explain the application of the tool. Just, this is the tool, this is how you do the mathy part, and this is the answer. I didn't make physical sense of any of this until many years after and had to do it on my own.

[@laurentiubucur9586]

I studied THIS in 1976-1980 Marine Institute Constanta Romania, lots of maths and I can visulize Why/How Universe expands and How/Why NABLA X DELTA B( x, y,z,t) induces on a perpendicular Plane the Electrical Field [NablaX DELTA E(x,y,z,t)]. It is amazing that we actually can understand CuanticFluctuations being EnergyTensorialNablaDistribution(GQGAD, DIV, ROT) applied on a EnergyField, whichever it is, is all about MathematicalCalculus=i.e. INFORMATION: ENERGY IS ULTIMATELY INFORMATION we name it GOD, Consiuness, TheObserver, TheExistent= thoughts are fluctuations, time being the GRAD of DIV and ROT of ENERGY distribution( gradient of tensorial NABLA* DELTA (ENERGY) + NABLA X DELTA (ENERGY)

[@timcussins]

Love the fantastic layout, and quality of presentation on all these videos, great work! The deltron 3030 reference, I did _not_ see coming though. Just.. wow :D

[@Eigensteve]

Thanks! And good catch on the reference -- that just goes to show your excellent and refined taste ;)

[@mr.cutback3642]

Literally just the video i needed 20 minutes before my wave mechanics quiz Thank you

[@ShreyaAdhikari-m1n]

This lecture is literally so cool

[@anarisnoveanu6778]

I did not know how much I did not know before watching this video :')

[@njathifrancis]

Thank you for explaining what my professor couldn't

[@juleswombat5309]

Awesome - I still retain my copy of William Hayt book on Engineering Electromagnetics after 40 years because his use of Div, Grad and Curl simplified Electromagnetics into a thing of beauty. Coulomb's and Gauss's laws are much simpler to remember and apply through the use of Div ,Grad and Curl

[@rza_ramezanii]

Beautifully presented Dr. Brunton.

[@memetb5796]

Man, I love these videos. Thank you for your time.

[@Mathews-y1b]

this is really amazing 😊😊. It makes me wanna cry. The way it was being taught before I found this video was like learning some foreign complicated language, I ended up hating it, but you have changed that, thank you soooooo much.

[@chandradeepkumar7750]

Thanks a lot professor. These things haunted me & you have literally served them in a plate. Many many thanks.

[@hrperformance]

This made the whole thing much less scary and now I'm really looking forward to getting stuck in! Thanks!

[@anandteerthrparvatikar5359]

One of the Best lectures, definitely TED quality

[@urigroismancfdinstitutonew4650]

Always inspiring the way you explain things. Nabla is the greek word for harp. Hamilton introduce the name together with the i, j, k notation (originally for quaternions). Maxwell was the first to use it extensively, and generalizing it to other systems of coordinates.

[@Eigensteve]

This is so cool -- I love learning new things from the comments. Thanks!

[@FadkinsDiet]

The nabla symbol is used with different meanings in a few other contexts, like theoretical computer science. Del specifically refers to the nabla symbol used in vector calculus.

[@robertschlesinger1342]

Excellent overview and review.

[@MrMaceuan]

Can't wait for upcoming videos. Thank you very much. Badly need these videos.

[@mrr5183]

Cool stuff professor, looking forward to more.

[@ashanatroy9575]

That's pretty easy!! Wow! Fascinating presentation. Huge respect from Bangladesh ❤

[@CarlosRodriguez-mx2xy]

Simply the BEST !!! Thank you very much.

[@arvindp551]

Charged up for this series.

[@lazarossidiropoulos1431]

Amazing and simple explanation. Thank you very much sir.

[@Eigensteve]

Thank you for your kind words :)

[@dsds-rj9rg]

I did NOT expect a comment about Deltron 3030 or Del the Funky Homosapien in a vector calc video haha. Also, thanks for all your material Steve (book and videos)... your videos are some of the best math material (if not the best) on KZbin.

[@neosporran]

This is gold. I wish I'd had this when I was first studying differential equations.

[@martinsanchez-hw4fi]

Thank you for the content you make.

[@__________________________6910]

hummm thank you sir

[@dewashyadubey9829]

He is an awesome teacher!!! Loved it❤

[@aritrabhattacharyya2374]

I wish I had had this kind of learning during college days...❤ From India

[@MrPablolael]

I really enjoy this kind of videos, thank you so much for your time Professor Brunton

[@fabricetshinangi5042]

Thanks Steve, can't wait to see the futures lectures.

[@mikexbox1]

Found this channel from your ME565 lectures years ago and i am pleasantly surprised to see you uploading these topics that i always wish i understood more at this moment! Thank you for the great explanations!!

[@coder8i]

Great start! Clear and to the point.

[@kevinsimba5687]

If actually we can have lecturers liie this we can absolutely pass exams with ease❤❤😂great job my math tutor😊

[@KasnazanM]

Thank you so much

[@Thejus_5511]

Waiting for the upcoming lectures, Really nice and clean

[@BingtheLizard]

Great start; I'm really keen to see the next episode. It's a shame that in formal education we are often made to learn concepts in isolation from their applications. It's many years after my engineering degree(s) now and I'm keen to solidify a deeper intuition for the application of this.

[@__hajar__]

I'm not even an English speaker I study in frensh and my mother tongue is Arabic,but this is the only video that actually helped me somewhat undrestand

[@zivhellman2806]

To answer your question of sources of the names del versus nabla: del from ancient Greek, nabla from ancient Hebrew

[@Eigensteve]

Very cool, thanks!

[@juniorcyans2988]

I love your demystification so much! Things are not that difficult originally but those people like to make them look like a mystery to show off how smart they themselves are😅

[@vanessawertheim]

Thank you for this video! The concept of div, grad, and curl became clear to me now ^_^

[@mathjitsuteacher]

Very nice introduction to the key concepts of vector calculus. Would it be possible for you to talk a little bit about the definition of these operators in general coordinates?

[@Eigensteve]

Great point. Let me think about this... might take a little while before I'm back in the studio, but I like this idea.

[@VijayGS-mw9tb]

Dear Professor, that's indeed very well explained, thanks a lot. Looking forward to more educational videos on such topics.

[@joanr.g.1723]

I love your channel, Steve. Good work.

[@fredericoamigo]

Really good communication style and well explained! Thank you so much for this! This really helped me se thing’s clearer. Keep up the good work!

[@akhilpatel2886]

Thanks for making things easy to understand sir

[@vahidgharaee3518]

It took me back years ago. thanks, great job

[@rafaelrodriguez3338]

I am so stoked for this series!!!! thank you

[@chychychyp]

Now I got to know why grad has N factors! Thank you so much!

[@manny_k2988]

And just like that it clicked! Great explanation

[@CuriousMindUnplugged]

Wowww. When an engineer teaches mathematics ❤️❤️❤️ I'm learning for the love of physics.

[@muhammedbakr7770]

Many thanks for u .this vedio was very benifitial for me

[@jones1351]

I remember the days of learning the math without the motivation. Thanks for demystifying.

[@Eigensteve]

Thanks for watching!

[@SRIMANTASANTRA]

very good and lovely class, thanks Professor Steve

[@mahmoudzemzami3570]

Your explanations are fantastic.

[@HimanshuSharma-jh6ki]

Gem of a video , great work 🙏

[@SergeyB1995]

An excellent video as always, professor! I am wondering, what is the current plan for PDE/vector calculus series? From my prospective, it would be very interesting to cover some of numerical methods for PDEs briefly (such as Galerkin method). Another idea would be to connect this series with control systems and do some model reduction methods for PDEs, i.e. in order to obtain a finite-dimensional dynamic model that approximates dynamics of outputs of interest as good as possible. This last one is particularly interesting, as it allows construction of observers/controllers for PDEs with usual methods for finite-dimensional systems, that are much more popular and much better developed as of now.

[@Eigensteve]

Thanks for the kind note! I would also like to make videos on these topics, and have some planned, but they might take a little while.

[@HIeuTran-vw9lq]

the lecture is very easy for understanding

[@SUMON041]

Love your class,,,, respect from Bangladesh 🇧🇩🇧🇩🇧🇩

[@Eigensteve]

Thank you!

[@ArafatSafwan]

koto batch?

[@srivatsabhargavajagarlapud2274]

Thanks a lot Prof. Steve for helping us vividly see the connections between two very important fields ! As an undergraduate a decade ago, I did see this connection implicitly made when we learnt electromagnetics and wave propagation... Very excited to find a lecture series explicitly making this connection! Gold mine!!! :) Hope to revisit some examples from electromagnetics and wave propagation too (Maxwell's equations). Eagerly waiting for the rest of the lectures... Actually hate to wait for more on this... :)

[@Naveen_Anan]

Nice presentation !!!!

[@vijaykumars5041]

Really nice man. looking forward for next series of lectures

[@WilliamDye-willdye]

4:30 ("I don't know what the history of del vs. nabla is..."). The Wikipedia entry "Nabla_symbol" discusses the origin. Broadly speaking, "del" won out because it saves a syllable and we use it a lot.

[@edemkumah5248]

I'm looking forward to this

[@ba2tripleO]

The content is so consistent that the instructor even wears the same outfit for all the lectures 😅

[@TejKiran]

Thank you so much for this series.

[@chazfenner910]

I cant wait for the next videos!

[@stevelamprou]

Steve Brunton, you are the God of Div, Grad, Curl and All That!

[@MohammadAlshahrani]

Amazing ...

[@AJoe-ze6go]

Great video - loved it.

[@curtpiazza1688]

Very well explained! Thank you! 😊

[@ashutoshsingh-et7vm]

Great professor burton nice lecture.Eagerly waiting for other videos on Lagrangian coherent structure

[@michaelpotter3418]

Superb. Thank you!