What does the second derivative actually do in math and physics?

[@quantumsensechannel]

Hi everyone! A quick note: At 7:55 and onwards, there should be a vector sign over the input of the function: f(vector{x}), since now whenever we are talking about 3 dimensions, the input to the function is a coordinate in 3D space. Apologies for any mild confusion! I remember I used to dislike when my professors would lazily forget to write vector symbols - but years later it seems I have become what I once despised, whoops. Hope you all enjoyed the video! -QuantumSense

[@luke.perkin.inventor]

It's a great video, but perhaps the visual and conceptual leap from 1D, a line plotted on a 2D graph, to a 3D scalar field was slightly glossed over? You covered it with the leap from charge density to scalar field potential but maybe just one more slide and line would have smoothed it over :-)

[@raajnivas2550]

​@user-ky5dy5hl4dAgree with you. If I may suggest: Intuition is a guide to imagination of how the reality exists. Imagination is each person's view, and when we all concur using the precision of mathematics, then we are realigning our imagination to reality with precision. And when we accept internally this as TRUE, it becomes our intuitive perception, and an almost perfected view of reality. Then we take another step forward. It is why mathematics is precise, but Intuition is still learning based on existing knowledge.

[@NewCalculus]

@user-ky5dy5hl4d Intuition is what idiots use. Look up that word!

[@NewCalculus]

Mainstream mathematics academics have never understood calculus. Only after I came along did they start writing: f(x+h)-f(x) = \int_x^{x+h} f'(x) dx It's not true at all that the second derivative represents an arithmetic mean ("average value" is meaningless nonsense). In the above equation (which is derived in one step from mean value theorem), ( \int_x^{x+h} f'(x) dx ) / h is the arithmetic mean. Similarly, f ' (x+h)-f ' (x) = \int_x^{x+h} f ' ' (x) dx implies that f ' ' (c) is the arithmetic mean of all the ordinates of the function f ' ' (x) in the interval (x, x+h). www.academia.edu/81300370/Mainstream_mathematics_academics_are_arrogant_and_incorrigible_ignoramuses_The_mean_value_theorem_IS_the_fundamental_theorem_of_calculus

[@Mal1234567]

That's okay. I always go around in a state of mild confusion.

[@tiagoreisalves4480]

Return of the King

[@Orillians]

yes sirrrr W

[@Nzargnalphabet]

The two towers >:)

[@vinniepeterss]

😂😂

[@user-vq3lk]

Yes🎉🎉🎉

[@ayuballena8217]

yep

[@sumedh-girish]

You're back! Edit: Changed the course of history from talking about his back, to the fact that he is back. You are welcome.

[@charlie_0823]

I never understood why there was all this talk in my classes about the second derivative/laplacian being related to an average value, but no actual calculation/explanation was ever provided. Thank you so much for doing god’s work! 🙏

[@jaw0449]

You did an entire physics degree without being shown? Not even in QM? Huh

[@NormanWasHere452]

@@jaw0449 I'm in the same boat actually

[@jaw0449]

@@NormanWasHere452 you should go to your profs and and ask for derivations, then. That, or they’re expecting you to do the derivations on your own. No physics program should ever just give formulas (unless freshman courses)

[@krupt5995]

LET'S GO DUDE. I got an 9/10 in quantum mechanics I thanks to you

[@aquaishcyan]

how it's only been an hour since the vid's upload

[@lux5164]

@@aquaishcyanother videos

[@squidwarg]

nice profile pic

[@krupt5995]

@@squidwarg you too

[@av_naturasmr]

got 9.1/10 so was up?

[@jacoblampmatthiessen9862]

Thank you! For this very clear and intuitive explanation. This view really helps seeing the very deep philosophical connection to notions and axioms of locality in mathematical models. And it also makes the connections between wave equations and continuity equations very intuitive! ❤

[@MrFtriana]

Great! The Schrödinger equation is postulated in many texts and one form to derivate it is using the path integral formalism, but you give a good argument about why it have the form that we know.

[@unebonnevie]

In college/university, professors/TAs regurgitate a lot without much explanation! Thank God there is KZbin and channels like this!

[@damienthorne861]

I couldn't agree more

[@mattcorrigan6639]

If you read all the book and do more problems outside what is assigned, this kind of intuition will come. Your professors can't force you to think

[@TheFireBrozTFB]

As a physics major, you are carrying my ass through QM and modern physics. Cheers! You’re amazing!!

[@bobross9332]

I think that Feynmann was talking about the Cauchy integral theorem. He stated he didn't need to know the center value just the value on the exterior ball.. that is exactly the Cauchy integral theorem -- you average the surface of the ball and you have the center value

[@sarveshpadav2881]

The video content was quite insightful! Thanks for the upload. I hope you'll continue to do so in the future.

[@VegaOfficiaI]

You're content has really motivated into learning more about quantum mechanics and physics. Keep doing what you're doing!

[@scarlet0017]

free education for a guy like me who can't pursue physics due to the conflict in Manipur and now here in hyderabad getting a free education for ba course hahaha

[@Isma3el]

@@sidheart7447Moron.

[@gregrice1354]

You know MIT and other universities offer all courses as open source/free online, right? You clearly have web access and desire to learn.

[@Iameverywhereinfinite]

U don't have to do anything with education, u all have to do is a propaganda.Those who are funding u, will leave u useless after sometime.

[@GarrettChance]

My discord friend had to leave Manipur as well, I pray for you all. It’s stupid senseless violence, same story that has happened a thousand times before all over the world, lil details change but it’s the same group identity issue.

[@harnoorsingh1015]

Hope you’re well now

[@pekorasfuturehusband]

YOU’RE BACK!!! This is what we’ve all been waiting for, welcome back king 🙏🏻

[@constilad006]

Welcome back bro

[@DevRajyaguru-lx8pi]

This is a very fundamental concept to understand laplace equation in potential flows also. I was just binge watching this video few days back. just two days before, my teacher introduced the laplace equation and asked the meaning of it. I was really able to explain reasonably well and he was happy. Thanks to you, very well put video. My best wishes for your future!

[@arunsevakule]

This is one of the finest educational videos I've ever come across! Please never stop making them!!

[@ajejebrazor4936]

Thank you! What a great video! Multiple insights and new visualisations.

[@LucasVieira-ob6fx]

I've already read about how Laplacian can be interpreted as the difference between a point and the average of its vicinity, but your visuals nicely complement that picture. Nice work!

[@erikhicks6184]

I think that's true if all second derivatives. After all, that's all a laplacian is. If I remember correctly, with scalars there is only one meaningful second derivative, but for vectors, 3 can be formed by permitting curl, div, and grad.

[@slixeee]

HE'S BACKKKK

[@kimchi_taco]

The heat equation is twice differentiated in space and once differentiated in time because it accurately captures the dynamics of averaging over spacetime. Twice differentiating in space can be intuitively explained by Feynman's ball average approach. The rate of change towards the average is represented by the Laplacian. I believe that the single differentiation in time is due to the fact that heat changes are only affected by the past. Since the present is not affected by the future, only the rate of change in one direction is considered in time, resulting in a single differentiation.

[@larrywildman4381]

Look at "a treatise on electricity and magnetism" by Maxwell, vol I, pag 29 .... not Feynman's approach. It was well known before Feynman.

[@TurinBeats]

Honestly I hate math, mostly because I was forced to cram formulas to pass exams. But this video opened my eyes to the practicality of it, now I love math a little bit more. So thank you, currently binge watching your playlist on Math for QT.

[@erikhicks6184]

You were forced?

[@anjanavabiswas8835]

Funny enough for me it is the reverse. I like math a lot but I really hated physics because I couldn't grasp it. Physics felt more arbitrary and formulaic than math.

[@logician1234]

Excelent video, it really gave me a new perspective on the second derivative. I wonder why the third, and other higher order derivatives are so rare in physics compared to the first and second...

[@trufflefur]

I wrote a comment explaining how I understand the derivatives. Well, I suggest you to read my comment but I will give you a hint: f(x)=x around x=0 is useful, one side is negative, the other positive, f(x)=x² around x=0 is useful because it shows only positive and f(x)=-x² only negative, but f(x)=x³ is flat around x=0, very little useful information and when going outside the surroundings of 0 it pretty much behaves as x being one side positive and the other one negative, item with the following functions powers of x.

[@pluton_7139]

THE KING HIMSELF RETURNED! (thx for good video btw)

[@capreal26]

What an elegant and simple explanation of the second derivative. Subscribed.

[@jyothinath9820]

6:36 this is what we were taught to find the maximum or minimum value of a function

[@TheYoutubeFreak]

Hi, I found your channel just yesterday. I did check out all your videos. I don't know how to express my love and respect towards you. I'm an undergrad student from Bangladesh. I am really interested in quantum computing. I want to learn more. And your channel seems to be a great resource for people like me. Keep up good work.

[@simpletn]

Just what I need when it’s 4am

[@jwbqu1u889]

I never knew that second derivative was so helpful in 3d quantum mechanics kudos to you ❤❤

[@4th_wall511]

Niceee love the intuition behind the first derivative not contributing towards the curvature i took numerical approximations but this connected some dots

[@nDreaw12]

Nice Video as always!

[@delta07-507]

I am 56 and that was a trip to memory lane... Good video

[@Johnsenfr]

Phantastic video! Never saw such a clean and straight-forward explanation on the 2nd derivative!

[@govindanraman4301]

Superb crystal clear concept on second derivatives and its evolution with examples from physics 🙏❤ congratulations sir🙏🌹

[@supernovap.c.-jk]

O my Allah,,,what an explanation! Thank you brother for your hard work. the people who represents physics in a meaningful way, i respect. May Allah grant you.

[@mzg147]

Loved the video! You are really an amazing presenter. One thing that I *will* bite the bullet for is calling Laplacian *the real* second derivative in 3 dimensions. The full second derivative is really a bilinear form, also represented as the 3x3 matrix (hessian) of all possible second order partial derivatives, which the laplacian is just the trace of. There are other second order differential operators that you could get from it.

[@vikrantsingh6001]

really glad you returned , i was really fed by watching your videos on repeat , finally some new content

[@DeJay7]

Ah, another Feynman enthusiast, I see! Really, he was just an incredible person, every person who ever had the chance to be taught by him was blessed. And, of course, great video, and very much needed for a lot of people who passionately care about these things.

[@mitchellhayman381]

Most physicists admire Feynman second to only Newton himself. He represents the joyful genius and the spirit of scientific curiosity

[@larrywildman4381]

As I said in another comment, I saw the same concept in "a treatise on electricity and magnetism" by Maxwell, vol I, pag 29 .... so, I don't think was a feynman's idea.

[@hetmanfoko]

That's what I least expected. Thank you.

[@AtomicQuanta]

Yo I just have to say as a physics major when you mentioned the wave function evolution over time and how as the particle moves outwards the curvature flattens as the second derivative was just an awesome light bulb switch in my brain for me so thank you sir!

[@frankmanismyname1147]

No way. I actually understood everything. Thank you man

[@SethTheOrigin]

This is an great video. I have a BSc in Mathematics, and I never knew about this

[@NormanWasHere452]

This is such a great video, can't believe I've never looked at the second derivative like this. I'll definitely go and watch your series on quantum!

[@imPyroHD]

Fantastic upload, maybe a series on second quantization in the future like your first one on QM?

[@adriangroeneveld9341]

As a mech. engineering student I was busy enough learning just how to apply calculus rules to solve problems, but it's good to be able to see an explanation of math. fundamentals like this. There were some problems which could only be solved numerically l remember. If we had time we'd write a program for those in Pascal.

[@Danielle-ew1el]

your narrative style is absolutely captivating!

[@viktorvegh7842]

We need more channels like this! Subscribed

[@rudypieplenbosch6752]

Never heard this way of thinking about the 2nd derivative, provides great insigt, thank you.

[@piyushkaushik4154]

Bro . Although this is a channel for quantum maths but pls do cover such micro but nuanced and important topics of math as well . Like topics of calculus - I think concept of limits and meaning of it's formulas is part of an abstract section of mathematics (at earlier levels of maths ofcourse. otherwise higher theoretical maths is nothing but abstract). Take up other such concepts from calculus , complex numbers like topics . Great video ofcourse. Subscribed ur channel . Cheers. 🎉🎉

[@Primarch-Arlian]

I now know what happens when I 《f》around and find out. Thank you!

[@shuvro6358]

As a 15 year old.All of this looks so cool!

[@tylerboulware6510]

Very cool! I was thinking about how to think about the first derivative in this way and I'm thinking that it's like the average of the points on the positive side minus the average of the points in the negative side. I haven't done the analysis in the same way to verify that but I do really like this alternate way of thinking about derivatives.

[@cyberpilgrim1]

Certainly gives an insight into what we thought was beyond us. This would help in lots of things_ perhaps even without being aware. The kind of logic is fuzzy but inherently available given a lead like this. You don’t need to fully grasp. In fact nobody fully grasp anything as all these are ultimately pointers. U r close to Eisenstein but not exactly there. Thank you genius

[@foobar-xh5gs]

I can't grasp the physics part coz lack of relating knowledge, but the second derivative part really amazed me, didn't think about how it related with average.

[@Emirp37]

Nice bro , that was actually great (also inspired me to create a video on some qm topic ) Thanks bro Keep making these type of videos

[@ShredEngineerPhD]

This entire video is absolute gold. Thanks bro.

[@stevewhitt9109]

Great video. Heisenberg's Uncertainty Principle explained with KE. Thanks

[@Raphoo-doodles]

Wowow so much calculus lore!!!😳😳😳 Great video ❤️❤️

[@ShredEngineerPhD]

14:50 The laplacian ∇²f is negative for positive crests of the wavefunction, so the wavefunction f is changed in the direction of a negative crest. According to the heat equation, this change scales with the wavenumber k = E / ħc. So higher energy -> higher wavenumber -> higher rate of oscillation.

[@blisard2648]

mate youve killed this video! Such a complex idea explained so concisely

[@bronzeplayer3930]

Got a 2/10 on my second QM problem set. Ended with a 100% on the final and just pulled a 100 on a QM2 midterm! Would love more advanced quantum, but you gave me such a good basis :D

[@bjornragnarsson8692]

The next step is second quantization - redefining the non-relativistic fixed particle mode to a framework capable of analyzing relativistic many body systems in which the number of particles in a system are no longer fixed. There are quite a few approaches to this, the most common and most utilized framework being quantum field theories appropriate for the different types of fundamental interactions and particle properties. Extending to the Fock space - the Hilbert space completion of the symmetric and antisymmetric tensors in the tensor powers of a single particle Hilbert space is standard to incorporate creation and annihilation operators of quantum states that change the eigenvalues of the number operator by one, analogous to the quantum harmonic oscillator. Something that becomes more important in QFTs. You may have already been introduced to some of the fundamental aspects of this approach, as the natural extension beyond a Junior/Senior undergraduate QM course is the introduction of different QFTs, with particular emphasis on QED.

[@tanvirhossainfahim7025]

Please make more and more videos on Physics and Math.❤️

[@parkerstroh6586]

1 minute in and I’ve already liked and subbed!

[@akademesanctuary1361]

Where the first derivative is a tangent telling you the rate of change like the shift in change of state. The second is secant, a measure of curvature. In Hooke's law it focuses value from the field into the spring. If you are talking energy from the field subject to weak mixing, that angle applies to the secant to establish the focus of position=mass. Equilibrium for a set is defined by its curvature.

[@mn-lc7em]

Ancient greeks have used intuition and reasons If you know space ecuation, the first der is speed and second is acceleration. So second der is the dynamic of the function. Your demionstration is excelent model. Cong.

[@_cyantist]

10 mins ago? welcome back!

[@darshildhameliya3619]

Can you make a series of videos on various interpretations of QM? I have read the Helgoland and I love how Carlo has described the relational interpretation, would love understand the intuitions behind other interpretations!

[@Downlead]

Wow, a new video after 9 months. I miss you Bro..

[@raajnivas2550]

Thanks for the simplified version of seeing QP

[@prathameshpatil0810]

Great Video! Appreciate the effort you take in explaining all these things to enthusiasts! Must have been a lot of effort in the editing as well, Could you please tell me which tool/platform do you use to edit videos like these with equations and numbers flowing around the screen? I would love to create something similar very soon!

[@fuffalump]

Nice idea about the average on the ball! But must correct the misleading idea in the QM part - localized particles in position is equivalent to large uncertainty in conjugate (momentum) space, like you said. But this does not translate to necessarily large kinetic energy. The equivalence principle is for the mean of the distribution, and this would be the "classical" kinetic energy of the particle, which does not change due to variance. This explanation was a stretch, but you could explain this exactly with the diffusion equation, which the Schrodinger equation is just a specific case of :)

[@123string4]

Interesting! But why then would the gaussian spread out over time? Uncertainty principle means when the particle is localized in space it has access to higher momentum states. You're correct that the mean of the distribution is the classical momentum but you can't separate KE from momentum so I believe that yes indeed it's possible for the particle to have a large KE do to the uncertainty principle.

[@RezaJavadzadeh]

wow i just found gold(en content) in this channel! thank you so much keep making more this is amazing

[@varunahlawat169]

Bro what have you made! Beautiful!

[@justinx6498]

Awesome video dude. Please what software did you use to make the animations?

[@asifalamgir5135]

Our Quantum Sensei is here!!!

[@KindCaptain-ou4yh]

That's fire man! Great lesson

[@gengormacsgo3647]

Hope there‘s a lot more to come from your channel! Love your work!

[@k.chriscaldwell4141]

The rate of change of the rate of change. Example: The rate that the acceleration of something is changing.

[@JUNGELMAN2012]

I feel so proud of being able to follow your lecture!

[@ophthojooeileyecirclehisha4917]

thank you so much for your science, hard work and generosity

[@larianton1008]

wow, what, an upload? big fan

[@meaningfulmind]

Even for heat equation, this is the most intuitive tool I've ever used to understand the temperature distribution. What a great explanation. I was wondering how you could understand the Newton's second law using this though.

[@kadabrium]

if the distance an object has travelled in the past dt is less than the distance it will travel in the next dt, it means the object is acccelerating

[@MikeMagTech]

Excellent video. Thank you!

[@alexsere3061]

Man, I was afraid that you were gonna forget about the heat equation. Using this reasoning it just means "the temperature at a point wants to approximate that of the surrounding points", as in a cold point surrounded by hotter points will get hotter. I think it is the absolute best example of this, because once you explain it like that it becomes trivial.

[@whatitmeans]

nice video: I think the big question for a folowing video is this one: How this "averaging" intuition of the 2nd derivative is related to the "aceleration" intuition of the 2nd derivative when time is the studied variable?

[@Project_x09]

Great work man :) don't stop to make videos its really helpful !!

[@lofturbjarni9274]

Glad you're back.

[@Pjbfrrr]

Omg the legend is back😭👏

[@sp0_od597]

4:35 I always thought that arround = indout. Perhaps they are equal

[@giornogiovanna5222]

I tried to interpret the heat equation and here's what I've got: So, if ∆f>f(x0), it means that heat will go to the x0 And if df/dt>∆f (meaning the change in temperature over time is greater than the difference of temperatures), we will have body heating So, if the change in temperature excesses the difference of temperatures in a neighbourhood of a point, then the body will be heating, because heat will go to this neighbourhood faster than it will distribut I hope I got it right

[@tincantank5174]

I truly wish I knew what he was talking about. We only got up to IROC in high school, so he’s describing a topic that i haven’t even been introduced to.

[@johncgibson4720]

The Heisenberg intuition with a simple second derivative is very good.

[@EagerLearner23]

This reminds me if my Numerical Analysis class in undergrad...good times!

[@shortmoviemedia7068]

You are changing the world ♾️

[@algorithminc.8850]

Nice coverage of topic. Thanks. Subscribed. Cheers

[@Damn-Age]

Welcome back, on world quantum day!

[@ominollo]

Interesting take 🙂 The video from Feynman, which one is it? Or what was his lecture about?

[@wirelessboogie]

Thanks for the great explanation! You won't get any further in maths if you don't have an intuition for its laws and theoremes, which makes your video especially useful. Shame most manuals in maths don't have this policy being overly formulaic at the cost of intuition. P.S. I'm only slightly confused by you wishing us a quantum day, a superposition of which two states is it supposed to be? haha!

[@JonnyMath]

Yessss!!!!🤩🤩🤩🤩 These are the BEST videos ever!!!🤩🤩🤩🤩🤩🤩

[@paulodallacosta1062]

Sensacional!! Fascinante!!! Congratulations from Brazil