The Most Important Math Formula For Understanding Physics

Math is the language of physics, and there's one formula that you'll encounter time and again in your study of physics: the Taylor series. Get the notes for free here: courses.physicswithelliot.com...
If I had to name one mathematical theorem that's the most essential for understanding physics, it would be Taylor's formula. It shows up in virtually everything we do in physics. In this video, I'll teach you the basics of Taylor series, and show you a few applications in different corners of physics. In the first part, we'll go through the math, including how to write Taylor's formula in a particularly beautiful way that also makes the generalization to the multi-variable case straightforward. In the second half, we'll look at three physics applications. First, we'll discuss how to linearize complicated F = ma equations to simplify them in special limits. Then we'll learn to take the non-relativistic limit of Einstein's energy formula, and see how it's related to the fine-structure correction to the binding energy of a hydrogen atom. And third we'll look at the definition of the momentum operator in quantum mechanics.
All about pendulums: • Everything You Need To...
Taylor expanding the potential energy and discovering the harmonic oscillator: • To Master Physics, Fir...
Dimensional analysis and the binding energy of hydrogen: • How to Succeed at Phys...
Defining the momentum operator in quantum mechanics: • The Symmetry at the He...
Tutoring inquiries: www.physicswithelliot.com/tut...
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About physics help room videos:
These are intro-level physics videos aimed at students taking their first physics classes. In each video, I'll teach you the fundamentals of a particular physics topic you're likely to meet in your first classes on mechanics and electromagnetism.
About me:
I’m Dr. Elliot Schneider. I love physics, and I want to help others learn (and learn to love) physics, too. Whether you’re a beginner just starting out with your physics studies, a more advanced student, or a lifelong learner, I hope you’ll find resources here that enable you to deepen your understanding of the laws of nature. For more cool physics stuff, visit me at www.physicswithelliot.com.

[@whovikrantsingh]

I'm a Physics Major Undergraduate student, and I can say that this is the best video on Taylor Series that I've seen on KZbin. This explanation stands out!

[@PhysicswithElliot]

So glad to hear, Vikrant!

[@abhisheksoni9774]

Hey, from which university?

[@whovikrantsingh]

@@abhisheksoni9774 St. Xavier's College, Ahmedabad

[@pingpong_]

@@PhysicswithElliot Hello Elliot, why do we need to express Taylor Series into something with exp(ε*d/dx), if we still take sum of derivations for solving Taylor Series?

[@visheshkashyap8517]

Me who has nothing to do with this still this video came when i searched for taylor swifts song love story.

[@Phi1618033]

When I first studied calculus, and I got to the chapter on Taylor series, I thought, "What heck is this and why am I learning it?" But all these many years later I am now asking myself, "Why don't teachers emphasize that pretty much every non-linear equation in every field that uses mathematical models (physics, engineering, economics, computer science, etc.) is calculated via its Taylor expansion, so that students understand how important and useful it is?"

[@rotondabooden7475]

Hello everyone I'm just tuning into this channel trying to get ready for my MCAT exam and boy let me tell you that I'm a little lost😢

[@joelcurtis562]

Honestly, every physics course should open with a whole lecture on Taylor series approximation. We literally could not do physics as we know it without it! Instead, it's one of those topics that is hugely important but somehow gets overlooked in early education. At least it was for me. Anyway, great video. I especially like the clarification about the momentum operator as generator of translations really just being a case of Taylor expansion. It is usually presented far less clearly, so that connection isn't obvious.

[@PhysicswithElliot]

Glad it helped Joel!

[@jaw0449]

Every physics class I had from undergrad to grad had a brief overview of Taylor Series somewhere along the way

[@abhishankpaul]

We had 2-3 lectures on Taylor and McLaurin series when we started off. We were asked to remember all the things said and the method to use them till our de@th.

[@ShadowZZZ]

Wow that was beautiful. I already knew the Taylor series, how it's used to show that einstein relativistic energy can boil down to classical kinetic energy, and how it's used to make physics problems easier by approximations like small angle in pendulums, but I didn't how you can formulate it in such a short way like f(x+ε)=exp(ε*d/dx)f(x). I have also always wondered why the momentum operator in QM is exactly the way it is, even though I've used it a lot in calculations, so thank you.

[@PhysicswithElliot]

Thanks Shadow!

[@anonymousl5150]

Einstein was not responsible for the formula, in fact all his papers on mass equivalence were wrong.

[@Trenz0]

I hated covering this section and the divergence theorem when I took calculus. These sorts of videos are beyond invaluable and I'm a little jealous of the students today. All this efficient learning can only propel humanity's understanding of the physical world

[@punditgi]

Superb explanation. Absolutely brilliant! I'm another old man trying to learn new hard subjects before I shuffle off this mortal coil. Your videos are invaluable in helping me bridge the gap! 😃

[@kwccoin3115]

Same here

[@sirwinston2368]

Same here. I am 62 as of this writing. BSChE in the 80's. Hope to get a degree in physics when I retire in 21 months... so I'm trying to get a head start. I will need it. Great videos Elliot. Thank you. Wish we had KZbin when I was in college.

[@ull893]

​@@sirwinston2368 best of luck❤️🙏✌️

[@sietsebuijsman8523]

That was fascinating! I already knew about the translation operator exp(ε*d/dx), and why f(x + ε) = exp(ε*d/dx)f(x), but it never occurred to me that this could be used as a way to express the Taylor expansion in such a compact way. Thank you!

[@PhysicswithElliot]

Glad it helped Sietse!

[@rickcarroll]

What a great video lecture. I picked up so much stuff that made sense of a lot of my previous math's and physics reading. I am an elderly man who has time to relearn my old schooling and I am impressed by your approach on how to imparting knowledge at more than a general level. I love relearning and gaining new and very interesting facts about Math's and Physic's . Keep up this great work, there are many of us out here who just love this stuff. Cheers.

[@PhysicswithElliot]

I'm so glad it's helping, Rick!

[@chilledvibes99]

I cant actually believe i have only just found this channel, easily the most clear description of hard concepts and smooth animation.

[@abhay_cs]

Beautiful video. The Taylor expansion for sin(x) were actually known to medieval Indian mathematicians. Some now call it the Taylor-Madhava series, where Madhava is the Indian mathematician from the 14th Century. Almost 300 years before Newton.

[@voidisyinyangvoidisyinyang885]

yes and Abraham Seidenberg proved the Indian scientists knew the Pythagorean Theorem thousands of years before Pythagoras! You gotta make those chariot wheels precise. haha.

[@ankidokolo]

I always thought about Taylor's expansion as being a magnifying lens, the higher terms you use, the more detailed and closer to reality the view will be. These are really very nice and in-depth videos/lessons. Keep up the good work.

[@PhysicswithElliot]

Thanks Ankido!

[@wisnuape]

Wow that's a great analogy, thanks !

[@davidharper1152]

OMG... I was trying to explain this issue about teaching Taylor Series to a Parent/friend at my son's school... Great Subject... Great Video

[@dwaynep6174]

I'm a first year undergrad majoring in Physics and next semester I am taking my first physics class. Thank you for uploading these and giving me a taste of whats to come. I cannot wait to start and learn the nature of our universe

[@PhysicswithElliot]

Thanks Dwayne! Excited for you!

[@asadwrites786]

The best explaination and derivation on earth

[@BillFilson]

Wonderful explanation. I'm going to watch this again. I will be pressing pause regularly to consolidate each step in the logic. More explanation is not needed, just a little time to think about each step. Thanks for the marvelous explanation of the mathematical derivatives of physical equations.

[@sirwinston2368]

I studied the Taylor series in calculus back in 1980 . In about 19 months I am headed back to my alma mater to get a degree in Physics (gotta have something to do in retirement). I have been watching a lot of videos and lectures to get prepped because it's been so long since I did anything with "upper-level" math. Your videos are great Dr. Schneider. I still want to see something about Dirac. Thank you. sw (BSChE, PE)

[@sudheerbright]

It brings me to tears when I watch something so beautiful.

[@sweetpotatoambassador]

Great video as always, Elliot. The way that you simplify mysterious topics by pointing out deep connections to simpler ideas if amazing.

[@eyedontunderstand8326]

Love your channel Dr. Sneider! Please, continue to put up more. They are enjoyable, insightful, and a great resource for people who don't have the depth of knowledge of physics as you do. I would go as far to say that even for those that do, few are able to articulate their knowledge and understanding well to others, especially to those who are not at the same level of understanding. Thank you!

[@thiennhantranle6682]

I just came across your channel and I was amazed by your content! Thanks for providing these such quality lectures.

[@penz1as]

I've been hunting for a good, intuitive explanation of Taylor's formula. Goes without saying that this is the best that I have come across, but also, I ended up understanding so much more than just that. This is excellent stuff, so many "oh damn" moments.

[@voidisyinyangvoidisyinyang885]

intuitive to me would be like the Fourier series - the more extensions you add then the more precise it fits the geometric function.

[@rarichie3829]

Great video as always! One of my favorite applications of Taylor's formula is in nonlinear optics, where one expands the optical response of a material in powers of the incoming electric field, leading to all sorts of interesting processes.

[@PhysicswithElliot]

Interesting!

[@dlrmfemilianolako8]

Elliot this is beautiful ! Its the best way to transform a function to another function ( polonomyal ) that help us to derive or integrate it with more easy way

[@BENJAM0IN]

I´m learning QM right now and never saw this way of expressing Taylors´s formula, thanks for the video!

[@Jaylooker]

The compact formula of the Taylor series at 14:02 looks similar the generators S: z -> -1/z and T: z -> z + 1 of the modular group. The del operator formulation at 16:55 could also be considered in relation to Möbius transformations. A theta functions could be taken as the Taylor series expansion of a polynomial P(x) and its lattice with the standard basis be taken to be described by P(x). Note that every lattice can be assigned a theta function. This theta function would give a 1/2 weight modular form. Theta functions also satisfy transforms of Θ(z + 1, t) = Θ(z) and the very similar Θ(z + t, t) = exp(-2πiz) exp(-πit) Θ(z, t). Theta functions also happen to describe a wave functions in Chern-Simons theory though I don’t understand it that well.

[@MrOvipare]

Very well put! I've been using Taylor series for years now, but your presentation was very insightful, especially the bit about the translation operator. Cheers!

[@mathe.dominio4765]

💙🙏

[@wisnuape]

Great content as always. This is the first time i see such a compact formulation of Taylor series. Thanks !

[@vaidyanath3098]

Amazing video, Elliot. I liked the way of explaining taylor series and its importance in physics.

[@brendanfan3245]

it is just amazing! Thanks for your deep understanding of both physics and math!

[@anthonyheak3479]

Good explanation with a clear voice, awesome!

[@muhammadibrahim-nj3vy]

Omg this is very clear to understand. Thanks Elliot!

[@abhishankpaul]

Nice bootstrapping. We are using Taylor series expansion to compactly write the definition of Taylor series expansion

[@saudyassin5352]

Superb video, thank you so much for sharing your knowledge with such enthusiasm.

[@charleshudson5330]

Brilliant! Bravo! The best explanation and demo of Taylor I've ever seen. Especially graphic - searingly so - was the segment from 6:30 to 6:40. That said it all. (When I was a physics major, at Columbia, in the sixties, we didn't have videos at all - just a graph on a page full of equations. You had to use your imagination all the way down. P.S. The highlight of my physics career was an afternoon spent at lunch and in conversation with David Bohm, in London, when he was at Birkbeck U.)

[@PhysicswithElliot]

Thank you Charles! I love how technology makes it possible to teach things in new ways!

[@user-hh2mu5le2k]

Hi Elliot, my name is Joseph. It’s incredible how you seem to make the exact videos I want at the right time. Please continue making these amazing videos and spread the physics! Thank you very much

[@PhysicswithElliot]

Glad it helped Joseph!

[@nashs.4206]

This is an incredibly underrated channel. Amazing quality. Thank you Elliot :)

[@PhysicswithElliot]

Thanks Nash!

[@gerardsagliocca6292]

Great insight into this darn difficult topics. We all appreciate this insight which eventually leads to stronger understanding of math and of physics. Greater understanding often leads to less memorization of a bunch of formulas and facts. My only tiny request/wish would be to somehow slow the speed down a tiny amount. It seems as if somebody is standing behind you prodding you to hurry up so you can finish faster due to some self imposed time requirement you or your producer may have. If I were your Film Director, I would find ways to slow you down a bit and take normal breaths. So that in the end , a better product would be produced so that general viewers/students could enjoy the experience more. Good luck on your goals in producing these valuable educational series. PS At this point, I am now reminded of coming across the Convolution integrals. They would have an asterisk between two functions and then the author(s) would assume every reader knew what this was about ! I have seen no textbook that has adequately explained this rather straightforward general idea but nobody ever shows in diagrams what is actually taking place and also they never tell you why people use it. When I read electrical engineering texts that contained Convolution verbiage, I also froze up because of that forbidden character somebody starting using, the asterisk inside of an integral setting. Hope you can assist in this..

[@emilytownsend4983]

I have no idea how I got here. One minute I was watching videos of Patti lupone singing don’t cry for me Argentina. A few clicks later I landed here. 🤷‍♀️

[@mitrikoudsi8060]

Great video! Thank you so much Elliot!

[@michaelbaum6796]

This is a beautiful video on Taylor Series. Thanks a lot👍

[@agapamebandikot9997]

you are honestly so underrated! Your content is so useful for physics enthusiasts like myself

[@gok_dogan]

9:52 Yes Exactly I run into sin(x)=x for small angle approximation. After years I come across again and intuitive behind that approximation.

[@eugenebroni3501]

I'm enjoying this stuff..Thanks very much Elliot

[@jaimeduncan6167]

This basic series is great, very well made. I will made a suggestion for all the teachers: someone I normally do before going into proofs is do an example with a polynomial ,one student will invent the polynomial and give the derivatives evaluated on the point as requested and another will construct the Taylor series, it’s nice for the students to see that it will return exactly the secret polynomial. If they is no time , I can do it myself on the board.

[@mathe.dominio4765]

💙🙏

[@Infinium]

Great video, very informative! Thanks :)

[@anAlokDubey]

Finally I understood the basics of Taylor series. 😊 Thankyou

[@Quasar.Chaser]

13:51 I'm in awe. Wow. Goodness me, absolutely mind blown

[@astorina]

Superb and respect for the deep work

[@Bhuvan_MS]

11:00 so that's how e^x formula is derived. I have seen some manipulations using cos x and sin x series to derive it, but this is far more easy and elegant to do!

[@jatinsoni1979]

As always What a great video lecture. I picked up so much stuff.

[@PhysicswithElliot]

So glad, Jatin!

[@deeper1993]

Very cool video. It would be nice to show next regarding the radius of convergence. Not all functions can be well approximated no matter how many derivatives are considered, for e.g. the logarithmic function.

[@nagymikhail5851]

Amazing ...love the way you linked the three subjects together... math is not a mental gymnastic far from physical reality after all...

[@lucidity910]

Great video. Thanks for making it.

[@Justasong55]

Besssssssssssssstttttttttttttttt video ever........i seen in my life on Tylor series...❤❤❤❤❤

[@Definitelynoone69]

This helped me a lot THANK YOU

[@nickwisely2581]

Hi Elliot! I really enjoy your video very much! Honestly, I never saw the very compact notation of Taylor’s Series like you do. Even at First, I thought it will be just a fancier way to represent the Taylor Series. I’m waiting patiently until you explain how it correlates with the momentum operator in Quantum Physics. I must say that it’s very mind blowing. I never had this feeling before when I’m watching another physics or math video. I really enjoy the story very much. Please do more video about Physics and Math. Love to see your next video.

[@PhysicswithElliot]

Thank you Nick!

[@lover123777]

Superb video, thanks

[@xuantruongduong8438]

Perfect explanation!

[@tumisomalema5301]

Thank you, this has been made simple.

[@owen7185]

Another awesome video Elliott

[@PhysicswithElliot]

Thanks Brian!

[@meetghelani5222]

This is priceless, thanks a lot!

[@matveyshishov]

At 3:35, what makes the bear think "including many more powers of X" will lead to more precise function values, and even over a wider range? And why exactly powers of X and not, say, trig functions? Everything after that point is understandable, but that leap of faith is really the blocker.

[@fireclub493]

Wow! Excellent video!

[@CesarAlvarez-pl5sm]

This is great!! keep it up

[@ssym2]

Thanks Dr. Elliot!

[@deepakjanardhanan7394]

I have been waiting for a long time. I used to check daily for your new uploads.😊

[@PhysicswithElliot]

This one was really a 2-for-1!

[@hrperformance]

I feel like I understand polynomials as a whole better, as a result of this video. I loved watching it and found it incredibly useful. The notes are really good to use in tandem with the video, and very much appreciated. I can't get over how clever and simple the technique is to get a Taylor series. And on top of that, how useful it has been for us as a species. This stuff really makes me appreciate the power of maths. I feel really privileged I can study this subject 😊 That was an awesome video 👍🏽

[@tombouie]

Thks I;ve been trying to figure this out for years.

[@JuanGonzalez-dy1jb]

Best explanation ever!

[@ThePiMan0903]

Thanks Elliot!

[@lj_ljh]

Dang this was elegant! Thank you for this man

[@PhysicswithElliot]

Thanks LJ!

[@aritrabhattacharyya2374]

Awesome teaching sir ❤from India

[@noahwilliams9231]

Absolutely amazing

[@sanjayd411]

Outstanding! 👍

[@NemanjaNiketic]

great work!

[@sergiolucas38]

Great video, impecable :)

[@PhysicswithElliot]

Thanks Sergio!

[@alphalunamare]

It's a long time ago now but I recall undergraduate Maths being full of formulae which although very useful were a tad mystifying as to their 'magical' nature. This presentation from a Physicist's view point shows the path and the reasons for 'finding' such equations. As such it would seem to me that this approach should be utilised by Mathematicians when attempting to inculcate 'magic' without wands in the minds of new undergraduates. In the UK we had/have 'Pure Mathematics', 'Applied Mathematics' and 'Pure and Applied Mathematics' at High School but we do not have anything foundational like this video series which gives understanding rather than a calculus tool kit that most never open again. Tools are useless unless you know what they are for how to use them. I would hope that these presentations achieve a wider audience.

[@Gshi-pn9tb]

wow!amazing!thanks for your video!😍😍🥰🥰🥰

[@maurocruz1824]

Missing your videos. Keep going!

[@danv8718]

Superb explanation.

[@PhysicswithElliot]

Thanks Dan!

[@curtpiazza1688]

Most interesting presentation!

[@PhysicswithElliot]

Thanks Curt!

[@no_one6749]

Great work

[@remomagalhaes4707]

Really nice indeed!!

[@AMADEOSAM]

Very good, thanks

[@kabandajamir9844]

So real and nice thanks

[@teslapark3406]

Are the equations correct in this KZbin video, kzbin.info/www/bejne/iKmXnYGfnbSHarc equations in the description. The audio isn't that good but the math and what's being said in the video should be discussed.

[@eriktempelman2097]

World-class. Well done!

[@PhysicswithElliot]

Thanks Erik!

[@wesrobertson8753]

Watched and loved this. Went to give a well deserved sub and whaddya know. I already was xD

[@rosskious7084]

Who knew in Cal 1, when they were teaching linear approximations, that they would step up the game in the taylor series? Math builds step by step . When you get to green's theorem you need to basically be a master of every math discipline beneath it. Geometry, Algebra, Trigonometry then basically ALL of calculus underneath it. Parametric equations, partial derivatives, line integrals, polar math, double integrals ( a weakness in turning the region into the points of integration could cause massive issues) . In of itself green's theorem is very, very easy and straight forward, but the fact that basically ANY weakness underneath it will come to light makes it a killer for some. Never learn math to just pass a test if advanced math is in your future.

[@devamjani8041]

Great video, but if you ask me , I would say the most important theorem to understand physics is the Noether's theorem.

[@nafishsarwar2077]

Dear Dr. Eliot, Can you please provide me few more terms for the correction of Bohr energy formula that includes 137.036? Thank you in advance

[@RobinHillyard]

OMG, that's the first time I've understood where the fine structure constant comes from! And, the momentum operator in quantum mechanism! Beautiful :) Incidentally, 1685 was a good year for births: not only did we get Taylor, but also Bach and Handel.

[@PhysicswithElliot]

Glad it helped Robin!

[@manaoharsam4211]

Yes I the content is great. But maybe too fast for beginners. You need some background to understand things like momentum operator etc. But I liked how you showed as you capture more terms you got better approximation of any general function . The graphical representation meant a lot to me. Real good video.

[@gerardooviedo4145]

Me encantan las matemática ya que soy Ingeniero

[@AkashPandey-lz2rj]

Sir please make more videos on lagrangian field equation I am from india I have recently able to see your videos you are god of physics I learned a lot please continue field explanation

[@mathe.dominio4765]

💙🙏

[@pedrovpa1]

Taylor Series is the engineer's best friend

[@passionateinspiration9186]

Great content

[@ghides24]

Excellent.

[@lifehope4201]

hey Elliot....!! tell your mom that your son is just awesome at visualizing maths . love from India

[@johnchristian5027]

This is a great video