The Most Important Math Formula For Understanding Physics

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.

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.

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[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 YouTube. This explanation stands out!

[Phi]

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?"

[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! 😃

[Trenz]

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

[joelcurtis]

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.

[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.

[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.

[sietsebuijsman]

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!

[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.

[chilledvibes]

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

[dwaynep]

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

[rarichie]

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.

[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.

[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.

[davidharper]

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

[sirwinston]

I studied the Taylor series in calculus back in 1980 <sigh>. 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)

[asadwrites]

The best explaination and derivation on earth

[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.

[penzas]

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.

[BENJAMIN]

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