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 RUclips. This explanation stands out!
@@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?
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?"
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
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.
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.
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.
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! 😃
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 RUclips when I was in college.
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!
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.
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)
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.
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
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.
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.
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.
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.
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.
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.)
ive always loved math. my dad taught me to love math as a chore but he didnt expect me to internalize it, for some reason. i asked him why, he said that "it'll be useful for understanding higher physics when you prep for difficult exams concerning physics and maths". i never understood his logic until now. i feel like i can love physics the same way i do maths, for the first time. im grateful for this channel. thank you. such an incredible little gem. and thanks dad. you know next to nothing about science but simultaneously know everything. because true science is just writing stuff down for later, not knowing what it truly means.
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.
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
At 6:50, you claim “If we know all the derivatives of a smooth function at a single point, we can reconstruct the rest of the function everywhere else.”. Technically speaking, there exists real valued functions which are smooth everywhere (meaning you can take any derivative of the function at any point and get a value) but which are analytic nowhere (meaning the taylor series of that function taken from any point doesn’t equal the original function near the starting point except at the point itself). You can find an example of this kind of function on the wikipedia article titled “Non-analytic smooth function” under the section “A smooth function which is nowhere real analytic”.
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!
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!
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
I have never learned physics, but here I am, taking notes and learning from RUclips videos. Physics is cool and fun and I want to pack my brain full of it (and my notebooks)!
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!
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.
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..
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.
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.
Beautiful stuff man. Great example with the sin(x)=x approximation. I didn't quite understand where that approximation came from early on in my physics journey, but makes sense now.
Dive into single-variable calculus (derivatives & integrals of elementary functions), and Vector Algebra (addition/subtraction, dot products & cross products). These are some of the first mathematics featured in Introductory Physics. Learning to apply physical principles to a situation to formulate a reasonable mathematical representation (model), and conjecturing how the system should behave based on the physical principles are the "hardest part" of Physics. Learning the mathematical techniques to analyze the implications of the model is the other "hard part".
This was enlightening and relatively easy to follow for my rusted brain, however I can't see where the transformation comes from at 11:40 when he takes a different origin and includes x_0. Why is the "new x" x_0 times x-x_0? I'm probably overlooking something very stupid, please help!
6:15 I think by including all the terms u cant predict the function at all points. It will give better and better approximation only near the point of concern. Please check! I doubt the simulation where the function is getting matched with increasing the number of terms.
13:22 I could follow the whole video but at this point its not clear to me, why the "substitution" is valid for all functions if we just plug in the information we have for e^x. May someone please help to understand?
I know what (d/dx)f(x) is but I don't understand what kind of animal exp(e*d/dx)f(x) is. In my mind (d/dx) always requires a function to take the derivative of but here's none. Or is it exp(e*(d/dx)f(x)) so the exponential of the first derivative of f at x times e? e = epsilon
Now that you pointed it out, I see e lurking there in Taylor's Formula. I suppose the next level is to starting plugging in complex numbers for x and then the trigonometric functions, via Eulelrs formula, staring showing themselves too.
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.
0:35 This is not true since there are smooth functions which are not analytic like e^-1/x (near the origin the Taylor coefficients are all zero, but the function is not identically zero).
Probably you mean something like e^-1/x^2, since e^-1/x isn't smooth, but yes, I'm looking here at the well-behaved functions that we typically need for physics purposes
@@PhysicswithElliotYou are absolutely right! I was too lazy to write out the whole piecewise-defined function as I thought I had made my point clear. I apologise for not completing the definition. Here is a more precise one of the function I had in mind: It is identically zero for x0. People use e^-(1/x^2) most often when giving an example of a smooth non-analytic function, so I wanted to give a lesser-known one which decays slower than that one. Great video btw! Keep 'em coming!
It's funny how I just want to know the basic formulas for physics and now I'm learning this 😂. I had a calculus class but we were unfortunately unable to get to the taylor's series because of how hard it got. If our teacher wasn't as kind and nice to us, we will surely be learning this. Anyways thanks for the video!
Great vid! I don't fully understand the way that the nth derivative was equated to the first derivative (d/dx) raised to the nth power (12:43): Is (d/dx)^n just notational shorthand for the nth derivative just like f^(n)(x), and if so then how is it treated the same way as epsilon?
Thanks! (d/dx)^n just means d/dx d/dx ... d/dx, n times, so that it takes the nth derivative of whatever's sitting to the right of it. And e^{d/dx} is defined by its Taylor series expansion, where each term is such a power of d/dx.
Physics is easy!! All you need to know are the Harmonic Oscillator, the Two-body Central Force Problem, add the Taylor Series and everything else is perturbation theory. TA DA!!
I have a general question. The local information at a point allows us to extract the complete function. Is there some more fundamental math for this to read?
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. 🤷♀️
A small correction, the taylorseries does not work for all smooth maps. The functions on which it works are called analytical. (This is actually an important distinction when working on manifolds)
Doesn't it still work within a small neighborhood of the origin? If I understand correctly, analytical functions can be well approximated over its entire domain by knowing the value of the function at a single point while this only works at "small" values of epsilon for non analytical functions, right? Also: don't most differential equations involve analytical functions (exp, sin, cos) anyways? (My notions of differential equations are pretty rusty, so I'm not sure).
@@misterroboto1 That is not completely correct. There are smooth functions who cant be approximated by their taylorseries even in small neighbourhoods. Also, about your second question: most differential equations dont really involve analytical functions, it are just the ones we write down do most of the time. (this is an important distinction)
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.
d/dx is an operator, so yes in that sense it does work that way since you can just pull the constant epsilon out and perform the "operation", which here is differentiation.
i thought you were referencing another part of the video, in that case you can just forget what i said about the epsilon but the operator part is still valid.
@@forgor7180 so, d/dx is an operation and this operation applied n times is (d/dx)^n. but this is just compositions of d/dx, not exponentiation. it is not the same as 2^n, right?
@@forgor7180 I haven't yet gone back for a re-listen, but I do remember him saying somewhere the phrase "...at least notationally..." or its equivalent...
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 👍🏽
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.
Really confused about the fact that enough derivative at a single point can approximate the whole function... anyone having an intuitive explanation for that? Great video!
When dealing with real valued functions, it only works on very nice functions. Just about every real valued function doesn’t obey this rule, but just about every function you encounter in school is a very nice function (also known as an analytic function). There exists functions where at any point on the function you can take any derivative you want, but the Taylor series of that function at any given point doesn’t match the original function anywhere near the starting point except at the starting point itself. You can find a description of such a function in the wikipedia article titled “Non-analytic smooth function” under the section “A smooth function which is nowhere real analytic”.
Imagine you want to figure out the position of an object rolling up a hill at the time t = 1 second. If you know nothing about the position of the object, then we can approximate the position of the object with f(t) = 0, the simplest polynomial. If, however, we are given the position of the object at t = 0, say f(0) = 3 meters, we can assume the object will not have had much time to move in a single second, so we’ll approximate our function with f(t) = 3. Our guess for the object’s position went from 0 (a random guess really) to 3 (slightly better guess). The issue with this guess is that it assumes the object isn’t moving. If we knew both the objects position and how fast that object was moving at t = 0, we could better predict the objects position at t = 1. If the speed of the object at t = 0 is 0.5 m/s, then our new guess as to what the position function is f(t) = 0.5t + 3. This gives us the approximation f(1) = 3.5 meters. This function agrees with our functions position and speed at t = 0, but is the simplest function to do so. The issue with this approximation is that it assumes the object isn’t accelerating or decelerating at t = 0. If we knew the position, speed, and acceleration of the object at t = 0, then our approximation should be even more accurate. If the acceleration of the object at t = 0 is -2 m/(s^2), then our new approximation of the function becomes f(t) = -t^2 + 0.5t + 3. At t = 1, our new approximation is f(1) = 2.5 m/(s^2). This function agrees with the position, speed, and acceleration of our function at t = 0, and is the simplest function to do so. The issue with this approximation is that it assumes the object’s acceleration never changes. If we knew the jerk (i.e. how fast the acceleration is changing) of the function at t = 0, our guess as to what the object’s position will be at t = 1 should be more accurate. Tldr the idea is that the more information you know about the behavior of an object’s movement and position at some starting point, the better you are able to predict where that object will be in the future (and where it was in the past). If you knew everything about the behavior of the object’s movement and position at some starting point, you should be able to perfectly predict the behavior of the object’s movement and position at any given time (at least within a close enough region to the starting point). In my previous post (above), I show you where to find an example where this intuition collapses, and for just about every real function out there, this intuition collapses. However, for just about every function you encounter in school, this intuition works perfectly.
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
Can you do a vid on entanglement entropy and the ecological crisis from increased gravitational entropy since Roger Penrose points out that the entropy of matter is the opposite gravitational entropy? thanks
This is a very interesting video. Thanks a lot. However, there is a very strange thing in the way you name derivatives of order greater than one. This is not the first time I hear this, I've heard it on several earlier occassions, all from USA academic world. You name f'' 'ef double prime,' f''' 'ef triple prime,' and so on. In European countries, derivatives are named in what translates to first, second, third, etc., for f', f'', f''', etc, using superscript Roman numerals. A derivative of order 17, thus, uses the superscript XVII and is called seventeenth. Prime meaning first, I convene that 'ef prime' is the right name of the first derivative. But 'double prime'?, 'triple prime'? Double first and triple first? It's like your teacher didn't understand Roman superscript ordinal numerals and invented those funny expressions. Or her own teacher did. Do you know why an angle such as 3⁰ 2' 1" is read 'three degrees, two minutes, one second'? It was originally this way: 'three degrees, two prime-minute degrees, one second-minute degree,' wherein minute meant little, prime-minute and second-minute expressing the littleness of those subunits. If you wanted, you could have used additional subunits with Roman numerals III, IV, V, etc., to be called third, fourth, fifth[-minute degrees]. Roman numerals as ordinals were extremely common in everyday and in scientific use until the 20th. century, not so much nowadays; Americans have forgotten their naming convention, at least in what respects to derivatives.
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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 RUclips. This explanation stands out!
So glad to hear, Vikrant!
Hey, from which university?
@@abhisheksoni9774 St. Xavier's College, Ahmedabad
@@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?
Me who has nothing to do with this still this video came when i searched for taylor swifts song love story.
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?"
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😢
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
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.
I'm so glad it's helping, Rick!
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.
Glad it helped Joel!
Every physics class I had from undergrad to grad had a brief overview of Taylor Series somewhere along the way
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.
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! 😃
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 RUclips when I was in college.
@@sirwinston2368 best of luck❤️🙏✌️
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!
Glad it helped Sietse!
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.
Thanks Shadow!
Einstein was not responsible for the formula, in fact all his papers on mass equivalence were wrong.
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)
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.
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
Thanks Dwayne! Excited for you!
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.
Interesting!
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.
I cant actually believe i have only just found this channel, easily the most clear description of hard concepts and smooth animation.
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
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.
Thanks Ankido!
Wow that's a great analogy, thanks !
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.
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.
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.)
Thank you Charles! I love how technology makes it possible to teach things in new ways!
The best explaination and derivation on earth
ive always loved math. my dad taught me to love math as a chore but he didnt expect me to internalize it, for some reason. i asked him why, he said that "it'll be useful for understanding higher physics when you prep for difficult exams concerning physics and maths". i never understood his logic until now. i feel like i can love physics the same way i do maths, for the first time.
im grateful for this channel. thank you. such an incredible little gem. and thanks dad. you know next to nothing about science but simultaneously know everything. because true science is just writing stuff down for later, not knowing what it truly means.
I´m learning QM right now and never saw this way of expressing Taylors´s formula, thanks for the video!
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.
intuitive to me would be like the Fourier series - the more extensions you add then the more precise it fits the geometric function.
Good explanation with a clear voice, awesome!
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
Great video as always, Elliot. The way that you simplify mysterious topics by pointing out deep connections to simpler ideas if amazing.
At 6:50, you claim “If we know all the derivatives of a smooth function at a single point, we can reconstruct the rest of the function everywhere else.”. Technically speaking, there exists real valued functions which are smooth everywhere (meaning you can take any derivative of the function at any point and get a value) but which are analytic nowhere (meaning the taylor series of that function taken from any point doesn’t equal the original function near the starting point except at the point itself). You can find an example of this kind of function on the wikipedia article titled “Non-analytic smooth function” under the section “A smooth function which is nowhere real analytic”.
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!
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.
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!
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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
Glad it helped Joseph!
I have never learned physics, but here I am, taking notes and learning from RUclips videos. Physics is cool and fun and I want to pack my brain full of it (and my notebooks)!
I just came across your channel and I was amazed by your content! Thanks for providing these such quality lectures.
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!
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.
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It brings me to tears when I watch something so beautiful.
This is an incredibly underrated channel. Amazing quality. Thank you Elliot :)
Thanks Nash!
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..
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.
13:51 I'm in awe. Wow. Goodness me, absolutely mind blown
Amazing video, Elliot. I liked the way of explaining taylor series and its importance in physics.
Great content as always. This is the first time i see such a compact formulation of Taylor series. Thanks !
it is just amazing! Thanks for your deep understanding of both physics and math!
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.
Beautiful stuff man. Great example with the sin(x)=x approximation. I didn't quite understand where that approximation came from early on in my physics journey, but makes sense now.
What are some basic equations for people new to physics
Dive into single-variable calculus (derivatives & integrals of elementary functions), and Vector Algebra (addition/subtraction, dot products & cross products). These are some of the first mathematics featured in Introductory Physics.
Learning to apply physical principles to a situation to formulate a reasonable mathematical representation (model), and conjecturing how the system should behave based on the physical principles are the "hardest part" of Physics. Learning the mathematical techniques to analyze the implications of the model is the other "hard part".
you are honestly so underrated! Your content is so useful for physics enthusiasts like myself
This was enlightening and relatively easy to follow for my rusted brain, however I can't see where the transformation comes from at 11:40 when he takes a different origin and includes x_0. Why is the "new x" x_0 times x-x_0? I'm probably overlooking something very stupid, please help!
6:15 I think by including all the terms u cant predict the function at all points. It will give better and better approximation only near the point of concern. Please check! I doubt the simulation where the function is getting matched with increasing the number of terms.
13:22 I could follow the whole video but at this point its not clear to me, why the "substitution" is valid for all functions if we just plug in the information we have for e^x.
May someone please help to understand?
I have been waiting for a long time. I used to check daily for your new uploads.😊
This one was really a 2-for-1!
I know what (d/dx)f(x) is but I don't understand what kind of animal exp(e*d/dx)f(x) is. In my mind (d/dx) always requires a function to take the derivative of but here's none. Or is it exp(e*(d/dx)f(x)) so the exponential of the first derivative of f at x times e?
e = epsilon
Nice bootstrapping. We are using Taylor series expansion to compactly write the definition of Taylor series expansion
Omg this is very clear to understand. Thanks Elliot!
can someone explain what happen in 5:10 ? the whole c2 and c0 thing. I understand after differntiation, the contant dissapeared.
Understanding higher- order math can be described by this statement: People (i.e., human beings) are stupid, but not everyone is an idiot.
Now that you pointed it out, I see e lurking there in Taylor's Formula. I suppose the next level is to starting plugging in complex numbers for x and then the trigonometric functions, via Eulelrs formula, staring showing themselves too.
Superb video, thank you so much for sharing your knowledge with such enthusiasm.
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.
As always What a great video lecture. I picked up so much stuff.
So glad, Jatin!
Amazing ...love the way you linked the three subjects together... math is not a mental gymnastic far from physical reality after all...
0:35 This is not true since there are smooth functions which are not analytic like e^-1/x (near the origin the Taylor coefficients are all zero, but the function is not identically zero).
Yeah, should've said "analytic"
Probably you mean something like e^-1/x^2, since e^-1/x isn't smooth, but yes, I'm looking here at the well-behaved functions that we typically need for physics purposes
@@PhysicswithElliotYou are absolutely right! I was too lazy to write out the whole piecewise-defined function as I thought I had made my point clear. I apologise for not completing the definition. Here is a more precise one of the function I had in mind:
It is identically zero for x0.
People use e^-(1/x^2) most often when giving an example of a smooth non-analytic function, so I wanted to give a lesser-known one which decays slower than that one.
Great video btw! Keep 'em coming!
@@NoNTr1v1aL Ah yes that makes more sense!
It's funny how I just want to know the basic formulas for physics and now I'm learning this 😂. I had a calculus class but we were unfortunately unable to get to the taylor's series because of how hard it got. If our teacher wasn't as kind and nice to us, we will surely be learning this. Anyways thanks for the video!
You cannot do calculus without good algebra. Starting with a solid basis of algebra is really the fundamentals of applied physics.
Finally I understood the basics of Taylor series. 😊 Thankyou
Besssssssssssssstttttttttttttttt video ever........i seen in my life on Tylor series...❤❤❤❤❤
Elliot: explains math.
Me: 23:02 p is small :'(
This is a beautiful video on Taylor Series. Thanks a lot👍
Great vid! I don't fully understand the way that the nth derivative was equated to the first derivative (d/dx) raised to the nth power (12:43): Is (d/dx)^n just notational shorthand for the nth derivative just like f^(n)(x), and if so then how is it treated the same way as epsilon?
Thanks! (d/dx)^n just means d/dx d/dx ... d/dx, n times, so that it takes the nth derivative of whatever's sitting to the right of it. And e^{d/dx} is defined by its Taylor series expansion, where each term is such a power of d/dx.
Physics is easy!!
All you need to know are the Harmonic Oscillator, the Two-body Central Force Problem, add the Taylor Series and everything else is perturbation theory.
TA DA!!
I have a general question. The local information at a point allows us to extract the complete function. Is there some more fundamental math for this to read?
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. 🤷♀️
A small correction, the taylorseries does not work for all smooth maps. The functions on which it works are called analytical. (This is actually an important distinction when working on manifolds)
Yep!
Doesn't it still work within a small neighborhood of the origin? If I understand correctly, analytical functions can be well approximated over its entire domain by knowing the value of the function at a single point while this only works at "small" values of epsilon for non analytical functions, right? Also: don't most differential equations involve analytical functions (exp, sin, cos) anyways? (My notions of differential equations are pretty rusty, so I'm not sure).
@@misterroboto1 That is not completely correct. There are smooth functions who cant be approximated by their taylorseries even in small neighbourhoods.
Also, about your second question: most differential equations dont really involve analytical functions, it are just the ones we write down do most of the time. (this is an important distinction)
@@yannickgullentops6857 Oh ok. Thanks for the info!
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.
Thank you Nick!
Great video! Thank you so much Elliot!
In physics, it seems like they never kept more than the first two terms, then pretended the function being approximated was linear.
I didnt get anything 3:05 onwards, like whats a derivative? What is a prime of 0?
This assumes you know calculus
@vaclav222 oh
Another awesome video Elliott
Thanks Brian!
12:46 are we just multiplying d/dx's ? do higher order derivatives really work that way?
d/dx is an operator, so yes in that sense it does work that way since you can just pull the constant epsilon out and perform the "operation", which here is differentiation.
i thought you were referencing another part of the video, in that case you can just forget what i said about the epsilon but the operator part is still valid.
@@forgor7180 so, d/dx is an operation and this operation applied n times is (d/dx)^n. but this is just compositions of d/dx, not exponentiation. it is not the same as 2^n, right?
@@behzat8489 yes . I guess its just confusing notation.
@@forgor7180 I haven't yet gone back for a re-listen, but I do remember him saying somewhere the phrase "...at least notationally..." or its equivalent...
I don't recall seeing Taylor series in general relativity
This guy is the greatest RUclipsr I have eve seen
Glad you liked it Noah!
hey Elliot....!! tell your mom that your son is just awesome at visualizing maths . love from India
I'm enjoying this stuff..Thanks very much Elliot
math and physics can be our guard or predictors i cant allow that.
Superb and respect for the deep work
Watched and loved this. Went to give a well deserved sub and whaddya know. I already was xD
19:51
Dang this was elegant! Thank you for this man
Thanks LJ!
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 👍🏽
I would suggest if you want to do physics start with basic algebra.
Me before clicking this video: please let it be the harmonic oscillator please don't say it's Taylor's series please
Can i cry .this lecture made me feel than i can throw away my engineering degree and still be fine.😢
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.
This is priceless, thanks a lot!
Perfect explanation!
Great video. Thanks for making it.
Awesome teaching sir ❤from India
Really confused about the fact that enough derivative at a single point can approximate the whole function... anyone having an intuitive explanation for that? Great video!
When dealing with real valued functions, it only works on very nice functions. Just about every real valued function doesn’t obey this rule, but just about every function you encounter in school is a very nice function (also known as an analytic function). There exists functions where at any point on the function you can take any derivative you want, but the Taylor series of that function at any given point doesn’t match the original function anywhere near the starting point except at the starting point itself. You can find a description of such a function in the wikipedia article titled “Non-analytic smooth function” under the section “A smooth function which is nowhere real analytic”.
Imagine you want to figure out the position of an object rolling up a hill at the time t = 1 second. If you know nothing about the position of the object, then we can approximate the position of the object with f(t) = 0, the simplest polynomial. If, however, we are given the position of the object at t = 0, say f(0) = 3 meters, we can assume the object will not have had much time to move in a single second, so we’ll approximate our function with f(t) = 3. Our guess for the object’s position went from 0 (a random guess really) to 3 (slightly better guess). The issue with this guess is that it assumes the object isn’t moving. If we knew both the objects position and how fast that object was moving at t = 0, we could better predict the objects position at t = 1. If the speed of the object at t = 0 is 0.5 m/s, then our new guess as to what the position function is f(t) = 0.5t + 3. This gives us the approximation f(1) = 3.5 meters. This function agrees with our functions position and speed at t = 0, but is the simplest function to do so.
The issue with this approximation is that it assumes the object isn’t accelerating or decelerating at t = 0. If we knew the position, speed, and acceleration of the object at t = 0, then our approximation should be even more accurate. If the acceleration of the object at t = 0 is -2 m/(s^2), then our new approximation of the function becomes f(t) = -t^2 + 0.5t + 3. At t = 1, our new approximation is f(1) = 2.5 m/(s^2). This function agrees with the position, speed, and acceleration of our function at t = 0, and is the simplest function to do so. The issue with this approximation is that it assumes the object’s acceleration never changes. If we knew the jerk (i.e. how fast the acceleration is changing) of the function at t = 0, our guess as to what the object’s position will be at t = 1 should be more accurate.
Tldr the idea is that the more information you know about the behavior of an object’s movement and position at some starting point, the better you are able to predict where that object will be in the future (and where it was in the past). If you knew everything about the behavior of the object’s movement and position at some starting point, you should be able to perfectly predict the behavior of the object’s movement and position at any given time (at least within a close enough region to the starting point). In my previous post (above), I show you where to find an example where this intuition collapses, and for just about every real function out there, this intuition collapses. However, for just about every function you encounter in school, this intuition works perfectly.
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
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Can you do a vid on entanglement entropy and the ecological crisis from increased gravitational entropy since Roger Penrose points out that the entropy of matter is the opposite gravitational entropy? thanks
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
This is a very interesting video. Thanks a lot.
However, there is a very strange thing in the way you name derivatives of order greater than one. This is not the first time I hear this, I've heard it on several earlier occassions, all from USA academic world. You name f'' 'ef double prime,' f''' 'ef triple prime,' and so on. In European countries, derivatives are named in what translates to first, second, third, etc., for f', f'', f''', etc, using superscript Roman numerals. A derivative of order 17, thus, uses the superscript XVII and is called seventeenth. Prime meaning first, I convene that 'ef prime' is the right name of the first derivative. But 'double prime'?, 'triple prime'? Double first and triple first? It's like your teacher didn't understand Roman superscript ordinal numerals and invented those funny expressions. Or her own teacher did.
Do you know why an angle such as 3⁰ 2' 1" is read 'three degrees, two minutes, one second'? It was originally this way: 'three degrees, two prime-minute degrees, one second-minute degree,' wherein minute meant little, prime-minute and second-minute expressing the littleness of those subunits. If you wanted, you could have used additional subunits with Roman numerals III, IV, V, etc., to be called third, fourth, fifth[-minute degrees]. Roman numerals as ordinals were extremely common in everyday and in scientific use until the 20th. century, not so much nowadays; Americans have forgotten their naming convention, at least in what respects to derivatives.