Hi friends, thanks for watching! For more videos about classical physics, check out this playlist: ruclips.net/p/PLOlz9q28K2e7UlSbJIwYTtR77CLmV5_3z And as always, let me know what other topics to cover in future videos :)
I disagree. It is no less impossible than any other function that has an infinite value when given a finite input. f(x) = 1/(x-a) is also "impossible" in this universe. The space of functions is simply the power set of the space of real numbers, and in those terms, the Dirac delta function is perfectly reasonable. My guess is that it bothers people because it has 2 properties people don't feel comfortable with: 1. The aformentioned infinte value for a finite input. 2. It is not analytical. Neither is the Lambert W function. A bit of set theory and number theory can go a long way in feeling more comfortable with some of the math used in physics.
Physics student Dating profile: corrects people by saying the delta function is a distribution. Loves correcting people. Also doesn’t know what a distribution is.
How do you deal with studying physics when it gets boring and frustrating and when math gets so stupid and ridiculous? Thanks and hope to hear from you.
@@leif1075 I struggle with the same, but watching videos from channels like this (Up And Atom, veritasium, The Science Asylum etc) helps me with my motivation since the "zoom" a bit out and helps more with the intuition rather than diving deep into the math and pure physics as they do in the lectures. This brings the fun and curiosity back to the physic and helps me to see the bigger picture, at least for me
This is like sitting in my 2nd year Electrical Engineering lectures again! 😅 We engineers use the dirac-delta function as an approximation of physically sampling a signal. Then we string a bunch of time-shifted dirac-deltas - we call this a *dirac comb* - to get a mathematical approximation of a discrete sampled analog signal. My brain exploded in those lectures ... 🤪
A Dirac comb was used in physics first as an approximation of the electric potential set up by an array of 1D atoms. Basically a solid. It's use in this context is limited but servers as a primer for learning Block states and Bloch's theorem.
As someone who has studied Signal Processing (Electrical and Computer Engineering), this video is a great explanation of a concept that wasn't so clear back then. Thank You, Parth! - Eduardo
Never will forget learning the Dirac delta function in differential equations. The brilliance of it hit me right away, particularly when considering how an impulse affects some model. It has the mathematical effect of multiplying by one, but it sets things in motion.
Nice way to explain the Delta function. I thought that this function is used only in Quantum Mechanics- done a short (and not so easy) course on QM (by HC Verma ji). But you have shown it can be used to evalauate Impulse- which is more familiar. Thanks Parth for this nice video. Started seeing your videos recently and liked them Keep going!!
The dirac-delta function is not exclusive to Quantum Mechanics at all! Electrical Engineers use the dirac-delta function all the time. My fundamentals courses, plus digital control systems and signal processing courses made heavy use of the dirac-delta function.
@@pauligrossinoz okay man I got your excitement. You are electrical engineer. To sum up it's used when you have very large amount of force for very short interval of time. According to what I studied in my mathematical physics class.
@@i2keepitrealInreseach - the point of the Dirac delta function is that it's integral is 1, so we use it as an idealised sampler. For discretised system inputs we treat the continuous input function as the result of multiplying it by a sequence of equally spaced Dirac delta functions called the Dirac comb.
Parth G, thanks. RUclips, in its infinite algorithmic wisdom, recommended this video for me. Your bluntness about its curious combination of utility and inaccessibility in experimental physics helped pull together some thoughts I've been mulling over for months about the interplay of absolute conservation and spacetime localization in quantum physics.
Great explanation Parth, I find myself coming back to your channel for more and more things. You explain things very clearly, without any baggage. Cheers!
Coming from a math background I remember the first time I got introduced to distributions and it seemd a quite natural extension of what was going on (even in number theory measures are defined as continuous maps in the dual space). A lot of friends with a physics background seemed quite surprised (and sometimes unhappy) with distribution definition. I don't really get what's the big deal with shaping intuition with abstract concepts rather than obtusely frame everything with old unsuited concepts. Modern algebraic geometry was literally born by a mind fuck interpretation of space using ring spectra, enstabilishing a parallelism between geometrical intuitiion and abstract algebra behaviour. Embrace abstractness, people, it's all good.
I always found the definition of distributions very intuitive. Especially in the context of physical measurement. You use test functions with finite support because you can't measure anything on one mathematical point. The notion of convergence of distribution sequence is somewhat less intuitive, but every property still has a good motivation.
Nice job describing this. It's certainly a challenging concept to explain and you did a great job. Too many applications in engineering to cover in a short video so the engineers in the house will likely continue to list and discuss them. For me the elegance and broad application are a thing of beauty.
Hello P. This is so apropo for me. I'm a radio amateur and I'm trying to understand SDR (software defined radio). I have realised I must understand DSP (digital signal processing) first. It's a rabbit hole indeed! The undersanding of impulses is fundamental to signal processing and it's comforting to know that the Dirac function has it's roots truly imbedded in physics! I like to think of the Dirac function as clapping my hands together (whack!) but the outcome isn't intuative because of the infinite number of sinusoids so created (thank Fourier). Keep 'em coming...
Thank you so much for your help, I've spent weeks struggling with the concept of this Dirac, but your video turns everything easy. I hope you will make videos about Fourier Series
@04:10,Shouldn't sin(x) be OUTSIDE the integral? Because otherwise, integral of sin(x) is a cosine function with a different value at (x-a)!!!!! Also, should we not be integrating by parts here?
Understandably he didn’t go into the detail, but the delta acts like an identity in this context, so indeed the value of that integral is sin(a). But that comes from the fact that the delta is defined as a distribution and that integration in this context is not regular Riemann integration.
The deeper understanding of physics, the easier you can explain it. That is what you are doing now. All the topic you are choosing is fundamental and essential, and also of course, interesting.
"Theory of distributions" is a subdiscipline of mathematics, unrelated to the homonymous probability distributions. In theory of distributions, a distribution is a bounded functional on some space of nice functions (called test functions in this context). Distributions are more general than measures: Dirac delta is both a measure and a distribution, the derivative of Dirac delta is still a distribution, but not a measure anymore. While the theory is quite technical, it helped me to have a much deeper understanding of some derivations done in electrodynamics and QM.
I would say bollux but that would be impolite and upset the other nut cases who have answered your post. A distribution is generally considered to be continuous. The dirac delta function is discontinuous by definition. Analysis 101.
@@alphalunamare That's not accurate. A distribution in the context which is relevant for the dirac delta (ie the functional analysis definition) is commonly defined to be a continuous linear functional on the space of test functions (smooth functions R to R with compact support + some topology). It is given by the „evaluation at 0“ map. However, the „continuous“ here refers to the test function you put in, ie if you change your test function a little bit, your distribution should assign only a slightly different value. So it's a little more complicated and certainly not just „Analysis 101“.
All these years, been using Dirac Delta function to shred complicated problems yet I never understood it's mathematical interpretation until I saw this video and the connection was made. ❤️
Another use of this function is in probability theory. If the probability density function is described as the Dirac delta, it means that the quantity that it is related to is deterministic, and takes the value that the delta is shifted. Cool video!
That one is good for convolution, to calculate reverb, when you take a sample of the sound in a particular place. Lots of harmonics, just clap your hands. Get a nice impulse recording. Hello matrices, transforms, linear stuff you know. Best for computers. You can apply the reverb of the acoustics to your sound. Useful when you have to redub and you need that same reverb and it's the real one, you need to simulate it. And this is too good for the task. Pure Math xd d³x/dt³, so Mathematics are so useful. So feared. Why?
To understand the area being 1. Start with a square with height and width both 1. The area of this figure is 1*1=1. Now divide the width by some factor t and multiply the height with the same factor. Now still (1/t)*(1*t) =1. Take the limit of t to infinity and you have our situation with infinite height and zero width!
Shoutout to the course I took on Signal Processing. The professor drilled into our heads that anyone who mistook the Dirac pulse with a unitary pulse would essentially receive selta of a nonzero value for our grades
This is great for getting a physical intuition of what the delta distribution is doing. I would just add that delta is defined (uniquely, by Schwartz kernel theorem) as the distributional kernel which recreates the evaluation functional. In other words, it's defined solely by its ability to satisfy the equation shown at 4:20 for any function in place of sin. The fact that it works as an impulse is actually a consequence of this: one may define "impulse" as something which integrates to a "switch" or a step function, and the fact that delta does this follows directly from the definition.
Think you nailed it w.r.t to the idea of using it to pick out a value from another function - about to review some QM & Fourier stuff so this vid will probably demystify the mathspeak for me a little bit!
Beautiful Explanation! Another analogy for the Dirac delta is sort of like the derivative of the Heaviside Unit Function, if anyone here wants to know where the idea of the function originated
Hey parth can we please get more videos on quantum mechanics, previously you mentioned you would make one on the Darwin term and other correction terms added to the Hamiltonian for the H-atom. ALSO can we get videos of QFT maybe a video on the Dirac equation
I would say it is both a function and a distribution. If you integral delta multiply a function , you just get what the function exactly is at a given point for the delta function.
What is more fun is when you consider two vert close particules of opposite charges. The density of charge becomes the derivative of the dirac "function".
Im an electrical engineering student, and we do study signal processing, you cna almost find dirac function in almost every line of the equations beside the fourier and laplace transforms
Small nitpick, the width of the Dirac delta is not zero, it is infinitesimal (smaller than any real number, but larger than zero). So the area (width times height) is an infinity times an infinitesimal, which can be a real number, as opposed to an infinity times zero, which is always zero. On topic, I would say that the use of Dirac delta is most prevalent in probability theory. It is an essential tool for relating probabilities over different measures. For example, expressing discrete probabilities as densities over a continuous domain.
' how can the area under an infinitesimally thin, infinitely tall function be a finite value,?' But surely it's not 'infinitesimally thin' - its width is precisely zero! It's defined at a unique point - not on a tiny interval of greater than zero value.I thought you were going to say that the integral was zero. It was a mathematical shock to h ear yo say the integral is 1. Clearly I have a lot of work to do on the math of this!
Actually the integral of the function equating to 1 makes sense because no matter where you calculate the integral, the sum of the integral area has to be 1 from 0. Consider that as something moves through integer space, it has to gain some net value. This gain in net value can be less than 1 but in the integral of the function, we are looking at all possible point between 0 and 1 thus the sum must be 1.
Parth G just an advice can you make a discord server for us so that we can discuss with other enthusiasts and have a community!that would be really great for us enthusiasts
I noticed a couple Electrical Engineers in comments were having PTSD. The delta function is a derivative of the unit step function... that aside, it's magic is discovered in Fourier transform pairs... Fourier transform of time domain sine(x) func for example gives you a delta in the frequency domain at 1/x. Visa versa, a Fourier transform of a time domain Delta function gives you all possible frequencies...yes, all of them... Imagine a step function in the frequency domain. And on a circuit model transformed via Laplace transform, we can get a response for all possible inputs.
Can the Delta function me modelled ny (say) nonstandard gaussian distributions which have a rising slope near zero but only in infinitismally small neighbourhoods of zero? And with the value a zero defined, but defined to me an infinite nonstandard real?
Does an Event Horizon not have zero depth and an infinite plane. All energy must have a Schwarzschild radius ,so why can it not be interpreted as this ?
I'm curious at to why physicists would want to dumb down the subatomic when that's exactly what we cannot reconcile with the macroscopic world of 3 (or 4, whatever) dimensions. Isn't it easier to keep the subatomic non point like, to integrate it (assimilate it) better to large scale phenomena? Ps: it *is* cool to use functions that way though :)
It would have been real nice if you had pursued the Impulse and perhaps explained how an impulsive force caused an acceleration over time .... not as small as a dirac moment but perhaps as important.
Interesting, although I might have used golf as an example of impulse rather than football. I think golf balls can reach 50,000gs of acceletation, not sure a footballer can achieve that!
This (and creation operators) was the point in math where i really started to feel like math is a made up tool I mean i always knew it was but this is where I really started to internalize it like my professor was like "well we really need this function to converge so we're going to stick a delta in it".
Math is NOT a made up tool, and this guy does not even understand the Dirac delta, which is NOT a function but a distribution, and has a very easy definition which is not even mentioned in this video. The problem is just most engineers and physicists are spreading a wrong use of math and don't even bother in doing things correctly.
As a mathematician, this dirac delta function infuriates me. Not because it's not useful, it clearly is, but because I'd file under abuse of notation, as it's not a valid function from ℝ -> ℝ to behave the way described with integrals / areas, without redefining integrals / areas to specifically support it. It's an extremely useful lie.
Consider the space all 'valid' functions from R to R. This space is incomplete in much the same sense that the rationals are incomplete --- there are sequences of functions which are in a sense 'Cauchy convergent' (in an analogous sense to sequences of rationals being Cauchy convergent). We complete this space just as we complete the rationals -- we define the 'missing' points in the function space to be equivalence classes of 'valid' functions that converge to it. One such point is the dirac delta .... _generalized function?_ Since you seem to have a bugaboo about using the term 'function' (you speak of it as a 'lie') we will try to be pedantic in your presence and make sure that we avoid the term 'function' without qualifications. Most of us drop the qualification in practice when we all know what we are talking about, and we don't consider ourselves to be 'lying' to each other. I feel your pain, I truly do, and I hope this helps. Cheers!
@@zapazap Ok, actually, this really changed my mind on this. Considering the function as itself a limit of functions that narrows to this one is a really interesting one.
@@Quargos It is pretty cool. :) I was not sure if your initial issue was with the validity of the concept (a conceptual issue) or with the appropriateness of calling it a function (a terminological issue). There is interesting paper called "Number idea and number concept" by a guy names Staffleu in which he contrasts tbe pre-mathematical _idea_ of number with the various mathematical _concepts_ of num pre-mathematical, and various mathematical _concepts_ that embody the idea. The development and study of various concepts of number (e.g. whole, rational, p-adic, quaternion) is mathematical -- investigation of the idea of number is philosophical. Perhaps what I described as a 'generalized function' was a function concept, and these new concepts of function enrich one's idea of a function.
@@Quargos P.S. I was exactly in tbe same frame if mind as you were vis-a-vis the dirac delta until someone clued me in to the same idea. I suspect that it is bad pedagogy that leads students early on to conceive of ideas in an overly narrow way. E.g. being told by high school textbooks that certain quadratic polynomials 'cannot be factored' but eliding the fact that this depends entirely on underlying field in question. This shows in all sorts of areas. My sister studied _and teaches_ piano, and I once gave her a book of introductory piano pieces by Bartok, and complained to be that he wrote the key signatures 'wrong' because they did not conform to the standard form she was taught, a form she implicitly learned to be sacrosanct! Forgive be for riding my hobby-horse briefly before you. Bad pedagogy is a curse! :)
@@DrDeuteron Pal is slang for friend.. words themselves cannot be agressive. Only when they are spoken or specific punctuation added, such as an exclamation mark in this case, could you make a tonal interpretation. We read that sentence two very different ways 😉
Doing f(a) means assigning the function f as an input into a linear functional which returns the number f(a) as an answer. That is extremely close to the definition of the Delta.
If you substitute 'is' for 'contains' in 'the subject contains the predicate' for an undecideable Kantian synthetic apriori proposition*, for example, the matter numeral '0' subject contains the predicate 'is the idea number zero', then you could argue that a synthetic apriori proposition violates the law of non-contradiction : nothing is both x and not-x and since the complex number, 0 = 0 ± 0.i , of general form z = a ± b.i for real numbers a and b, the imaginary number "i" equal to the square root of negative one and "." meaning multiplied by, is the augmented form of 0, then so does every complex number by induction : any geometric point in the complex plane is really and imaginarily constituted, raisng an issue over how you hedge against god affecting a complex number valued physical system through the imagination. * In contrast to either an analytic proposition that can be evaluated as either true, or false, by virtue of it's definitions, like 'zero is a number', or actually corresponding synthetic proposition, like 'this background is white'.
Delta function y = 0 / x Consider the function y = 1/ x two curvey 'L's in the top right and bottom left quadrants Y = 1 / 1,000 x is the same shape, but closer to the axes. Y = 1 / 1,000,000 x is even closer to the axes At the limit Y = 1/ infinity * x ~ 0 / x , the graph is the axes exactly 0 / 0 is undefined because it is all numbers positive, negative and imaginary. tlc
Nothing in the real world is made out of an infinite number of zero length points. Points are a mathematical abstraction, they are not a real thing. Furthermore, even in the actual theories of physics, objects aren't constructed of points. Their position is smeared around in a probability distribution, they don't take up a single point.
@@hOREP245 What about the dividing line between 2 adjoining plank lengths ? Further more there are many theories out there some have been proven ie relativity some have not ie QFT , though i do agree it is very close to the truth . no one has probed smaller then an atom, needles to say one millionth a billionth the size of an atom ie the plank scale, oh wait if space itself was quantized then there is a theory called "Lazy light" saying different frequencies would travel at different speeds depending on the substructure of space, slowing down if space was not uniformly smooth . all frequencies of light travel at the same speed in vacuum, and this has been measured over a billion years across a billion light years, and all frequencies travel at same speed, indication space itself is not quantized, just energy and matter . but im no expert and dont claim to be .
Hi friends, thanks for watching! For more videos about classical physics, check out this playlist: ruclips.net/p/PLOlz9q28K2e7UlSbJIwYTtR77CLmV5_3z
And as always, let me know what other topics to cover in future videos :)
I disagree. It is no less impossible than any other function that has an infinite value when given a finite input. f(x) = 1/(x-a) is also "impossible" in this universe.
The space of functions is simply the power set of the space of real numbers, and in those terms, the Dirac delta function is perfectly reasonable. My guess is that it bothers people because it has 2 properties people don't feel comfortable with:
1. The aformentioned infinte value for a finite input.
2. It is not analytical. Neither is the Lambert W function.
A bit of set theory and number theory can go a long way in feeling more comfortable with some of the math used in physics.
Physics student Dating profile: corrects people by saying the delta function is a distribution. Loves correcting people. Also doesn’t know what a distribution is.
How do you deal with studying physics when it gets boring and frustrating and when math gets so stupid and ridiculous? Thanks and hope to hear from you.
Andrew Dotson????!!!!!!, seriously he's here, it's a blessing!!!!
@@leif1075 by doing ur mom lmao
It is actually generalized function
@@leif1075 I struggle with the same, but watching videos from channels like this (Up And Atom, veritasium, The Science Asylum etc) helps me with my motivation since the "zoom" a bit out and helps more with the intuition rather than diving deep into the math and pure physics as they do in the lectures. This brings the fun and curiosity back to the physic and helps me to see the bigger picture, at least for me
Another great video. I really enjoy the way you explain things!
Thank you so much! Right back at ya :D
This is like sitting in my 2nd year Electrical Engineering lectures again! 😅
We engineers use the dirac-delta function as an approximation of physically sampling a signal. Then we string a bunch of time-shifted dirac-deltas - we call this a *dirac comb* - to get a mathematical approximation of a discrete sampled analog signal.
My brain exploded in those lectures ... 🤪
A Dirac comb was used in physics first as an approximation of the electric potential set up by an array of 1D atoms. Basically a solid.
It's use in this context is limited but servers as a primer for learning Block states and Bloch's theorem.
The really cool thing is that the Fourier transform of the Dirac comb (in the time domain) is a Dirac comb (in the frequency domain).
As someone who has studied Signal Processing (Electrical and Computer Engineering), this video is a great explanation of a concept that wasn't so clear back then.
Thank You, Parth!
- Eduardo
hats off for re-explaining integrals in each related video
Never will forget learning the Dirac delta function in differential equations. The brilliance of it hit me right away, particularly when considering how an impulse affects some model. It has the mathematical effect of multiplying by one, but it sets things in motion.
Nice way to explain the Delta function. I thought that this function is used only in Quantum Mechanics- done a short (and not so easy) course on QM (by HC Verma ji). But you have shown it can be used to evalauate Impulse- which is more familiar.
Thanks Parth for this nice video. Started seeing your videos recently and liked them
Keep going!!
The dirac-delta function is not exclusive to Quantum Mechanics at all!
Electrical Engineers use the dirac-delta function all the time. My fundamentals courses, plus digital control systems and signal processing courses made heavy use of the dirac-delta function.
It’s also used in circuit analysis.
@@pauligrossinoz okay man I got your excitement. You are electrical engineer.
To sum up it's used when you have very large amount of force for very short interval of time. According to what I studied in my mathematical physics class.
@@i2keepitrealInreseach - the point of the Dirac delta function is that it's integral is 1, so we use it as an idealised sampler.
For discretised system inputs we treat the continuous input function as the result of multiplying it by a sequence of equally spaced Dirac delta functions called the Dirac comb.
@@pauligrossinoz I know man. It's trivial for us now. I'm too in my masters physics and read this in BS physics
Parth G, thanks. RUclips, in its infinite algorithmic wisdom, recommended this video for me. Your bluntness about its curious combination of utility and inaccessibility in experimental physics helped pull together some thoughts I've been mulling over for months about the interplay of absolute conservation and spacetime localization in quantum physics.
Great explanation Parth, I find myself coming back to your channel for more and more things. You explain things very clearly, without any baggage. Cheers!
Coming from a math background I remember the first time I got introduced to distributions and it seemd a quite natural extension of what was going on (even in number theory measures are defined as continuous maps in the dual space). A lot of friends with a physics background seemed quite surprised (and sometimes unhappy) with distribution definition. I don't really get what's the big deal with shaping intuition with abstract concepts rather than obtusely frame everything with old unsuited concepts. Modern algebraic geometry was literally born by a mind fuck interpretation of space using ring spectra, enstabilishing a parallelism between geometrical intuitiion and abstract algebra behaviour. Embrace abstractness, people, it's all good.
I always found the definition of distributions very intuitive. Especially in the context of physical measurement. You use test functions with finite support because you can't measure anything on one mathematical point. The notion of convergence of distribution sequence is somewhat less intuitive, but every property still has a good motivation.
Nice job describing this. It's certainly a challenging concept to explain and you did a great job. Too many applications in engineering to cover in a short video so the engineers in the house will likely continue to list and discuss them. For me the elegance and broad application are a thing of beauty.
Hello P. This is so apropo for me. I'm a radio amateur and I'm trying to understand SDR (software defined radio). I have realised I must understand DSP (digital signal processing) first. It's a rabbit hole indeed! The undersanding of impulses is fundamental to signal processing and it's comforting to know that the Dirac function has it's roots truly imbedded in physics! I like to think of the Dirac function as clapping my hands together (whack!) but the outcome isn't intuative because of the infinite number of sinusoids so created (thank Fourier). Keep 'em coming...
Thank you so much for your help, I've spent weeks struggling with the concept of this Dirac, but your video turns everything easy.
I hope you will make videos about Fourier Series
@04:10,Shouldn't sin(x) be OUTSIDE the integral? Because otherwise, integral of sin(x) is a cosine function with a different value at (x-a)!!!!! Also, should we not be integrating by parts here?
Understandably he didn’t go into the detail, but the delta acts like an identity in this context, so indeed the value of that integral is sin(a). But that comes from the fact that the delta is defined as a distribution and that integration in this context is not regular Riemann integration.
The deeper understanding of physics, the easier you can explain it. That is what you are doing now. All the topic you are choosing is fundamental and essential, and also of course, interesting.
Technically the direc delta function is a distribution
What
@@naman4067 it's a probability distribution. The probability of 0 is 1, and the probability of everything else is 0.
"Theory of distributions" is a subdiscipline of mathematics, unrelated to the homonymous probability distributions. In theory of distributions, a distribution is a bounded functional on some space of nice functions (called test functions in this context). Distributions are more general than measures: Dirac delta is both a measure and a distribution, the derivative of Dirac delta is still a distribution, but not a measure anymore. While the theory is quite technical, it helped me to have a much deeper understanding of some derivations done in electrodynamics and QM.
I would say bollux but that would be impolite and upset the other nut cases who have answered your post. A distribution is generally considered to be continuous. The dirac delta function is discontinuous by definition. Analysis 101.
@@alphalunamare That's not accurate. A distribution in the context which is relevant for the dirac delta (ie the functional analysis definition) is commonly defined to be a continuous linear functional on the space of test functions (smooth functions R to R with compact support + some topology). It is given by the „evaluation at 0“ map.
However, the „continuous“ here refers to the test function you put in, ie if you change your test function a little bit, your distribution should assign only a slightly different value.
So it's a little more complicated and certainly not just „Analysis 101“.
All these years, been using Dirac Delta function to shred complicated problems yet I never understood it's mathematical interpretation until I saw this video and the connection was made. ❤️
I can't with how great the content is I admire this❤
Another use of this function is in probability theory. If the probability density function is described as the Dirac delta, it means that the quantity that it is related to is deterministic, and takes the value that the delta is shifted. Cool video!
This is one of your best! Thank you for the presentation. Cheers!
The lighting in your video is absolutely amazing
That one is good for convolution, to calculate reverb, when you take a sample of the sound in a particular place. Lots of harmonics, just clap your hands. Get a nice impulse recording. Hello matrices, transforms, linear stuff you know. Best for computers. You can apply the reverb of the acoustics to your sound.
Useful when you have to redub and you need that same reverb and it's the real one, you need to simulate it. And this is too good for the task. Pure Math
xd d³x/dt³, so Mathematics are so useful. So feared. Why?
Best physics content creator for a reason ❤️
To understand the area being 1. Start with a square with height and width both 1. The area of this figure is 1*1=1.
Now divide the width by some factor t and multiply the height with the same factor. Now still (1/t)*(1*t) =1.
Take the limit of t to infinity and you have our situation with infinite height and zero width!
Question about δ-Dirac "function": *DETECTED*
Laurent Schwartz book: *PROVIDED*
Shoutout to the course I took on Signal Processing. The professor drilled into our heads that anyone who mistook the Dirac pulse with a unitary pulse would essentially receive selta of a nonzero value for our grades
Akin to judging the ripeness of a watermelon when tapping it and listening to the impulse response from the tap.
This is great for getting a physical intuition of what the delta distribution is doing. I would just add that delta is defined (uniquely, by Schwartz kernel theorem) as the distributional kernel which recreates the evaluation functional. In other words, it's defined solely by its ability to satisfy the equation shown at 4:20 for any function in place of sin. The fact that it works as an impulse is actually a consequence of this: one may define "impulse" as something which integrates to a "switch" or a step function, and the fact that delta does this follows directly from the definition.
As always, great explanation. Life saving for me. Thank you.
Thanks for you videos its gratifying see thats videos thanks
Think you nailed it w.r.t to the idea of using it to pick out a value from another function - about to review some QM & Fourier stuff so this vid will probably demystify the mathspeak for me a little bit!
Beautiful Explanation! Another analogy for the Dirac delta is sort of like the derivative of the Heaviside Unit Function, if anyone here wants to know where the idea of the function originated
Plz do a video on lattice QCD or color confinement or quark confinement,something related to QCD/QF plz
Hey parth can we please get more videos on quantum mechanics, previously you mentioned you would make one on the Darwin term and other correction terms added to the Hamiltonian for the H-atom. ALSO can we get videos of QFT maybe a video on the Dirac equation
I would say it is both a function and a distribution. If you integral delta multiply a function , you just get what the function exactly is at a given point for the delta function.
What is more fun is when you consider two vert close particules of opposite charges. The density of charge becomes the derivative of the dirac "function".
It becomes painfully prosaic if you look at the most ordinary definition of the derivative of a function.
You are valuable.
Im an electrical engineering student, and we do study signal processing, you cna almost find dirac function in almost every line of the equations beside the fourier and laplace transforms
Keep uploading sir....you are my hope in physics
Small nitpick, the width of the Dirac delta is not zero, it is infinitesimal (smaller than any real number, but larger than zero). So the area (width times height) is an infinity times an infinitesimal, which can be a real number, as opposed to an infinity times zero, which is always zero.
On topic, I would say that the use of Dirac delta is most prevalent in probability theory. It is an essential tool for relating probabilities over different measures. For example, expressing discrete probabilities as densities over a continuous domain.
AWESOME VIDEO ! Really helped me !
' how can the area under an infinitesimally thin, infinitely tall function be a finite value,?' But surely it's not 'infinitesimally thin' - its width is precisely zero! It's defined at a unique point - not on a tiny interval of greater than zero value.I thought you were going to say that the integral was zero. It was a mathematical shock to h ear yo say the integral is 1. Clearly I have a lot of work to do on the math of this!
Love to see a video on Fourier series/transform.
Actually the integral of the function equating to 1 makes sense because no matter where you calculate the integral, the sum of the integral area has to be 1 from 0. Consider that as something moves through integer space, it has to gain some net value.
This gain in net value can be less than 1 but in the integral of the function, we are looking at all possible point between 0 and 1 thus the sum must be 1.
Minor error: x is the independent variable, not the dependent variable.
0/x equals zero for all values of x, except when x=0, then it is undefined.
Parth G just an advice can you make a discord server for us so that we can discuss with other enthusiasts and have a community!that would be really great for us enthusiasts
Please make a video on Gyroscopic Precession
Thank you Parth G! I am so interested in physics though ...
Thank sir !!
I noticed a couple Electrical Engineers in comments were having PTSD. The delta function is a derivative of the unit step function... that aside, it's magic is discovered in Fourier transform pairs... Fourier transform of time domain sine(x) func for example gives you a delta in the frequency domain at 1/x. Visa versa, a Fourier transform of a time domain Delta function gives you all possible frequencies...yes, all of them... Imagine a step function in the frequency domain. And on a circuit model transformed via Laplace transform, we can get a response for all possible inputs.
Very complicated topic, i spend 2 days understanding this
Can be aproximated, puede aproximarse como lim a->inf a*sech^2(ax)
Very very good as always :)
3:01 Okay so that settles it: zero x infinity = 1 ...I think i can live with that
Can the Delta function me modelled ny (say) nonstandard gaussian distributions which have a rising slope near zero but only in infinitismally small neighbourhoods of zero? And with the value a zero defined, but defined to me an infinite nonstandard real?
This guy understood colour theory!
sounds like its like an operator from QM...
but then again, i didnt really understand those either
Now, that was sure an interesting video 😃
I haven't seen anyone trying to explain the Dirac equation. Is it to complicated to explain in a format like this?
Can you talking about adiition of angular momentum
In environmental engineering, we use this to model instantaneous releases of pollutants into a fluid, e.g. the atmosphere, ocean or a stream!
I don't know why. But make me more interested in this......
I always feel delta function as getting surprise electric shock from a socket. It's a real example of delta. What do you think?
I heard about a concert hall that was analyzed acoustically with the sound of a spark from a spark plug as the delta function impulse.
It is an approximation of the delta.
I really think people studying physics should study measure theory first. And if they do, they should study it better.
damn man that was smooth!!
Amazing videos
Does an Event Horizon not have zero depth and an infinite plane. All energy must have a Schwarzschild radius ,so why can it not be interpreted as this ?
So, the wavefunction would be a (realistic) quantum version of the delta function ???
MAKE A VIDEO ON THE DIRAC EQUATION PLEASE but thank u
What does second degree integral represent.
Just calculate it. The repeated integral of delta would be zero for negative x and x for positive x, I guess?
Sir if I'm not wrong but you didn't say: impossible function as mentioned on thumbnail 😅
It’s not a function, it’s a distribution, technically
Can it be integradet by the lebesgue integral?
Nope. The support of Delta is a zero set.
it's not a function, it's a distribution
I'm curious at to why physicists would want to dumb down the subatomic when that's exactly what we cannot reconcile with the macroscopic world of 3 (or 4, whatever) dimensions. Isn't it easier to keep the subatomic non point like, to integrate it (assimilate it) better to large scale phenomena?
Ps: it *is* cool to use functions that way though :)
If you integrate the Dirac delta function from 0 to infinity, is the value ½?
It is not a valid operation. Reason is obvious: The Heaviside function is not a test function in the sense of distributions.
Search for:
"WHO INVENTED DIRAC'S DELTA FUNCTION?" by MIKHAIL G. KATZ AND DAVID TALL
for a nice essay on the Delta Function.
Insaaaaane!!
0:33 you say 'almost everywhere' is that an 'analytical term' or just the way the words came out?
5:32 ... just having a laugh here, I am taken by the 'visual' derivatives being so simple :-)
9:00 ..the conclusion ... isn't it fair to say that the Dirac delta function is merely playing the analogue of 'wave collapse'?
It would have been real nice if you had pursued the Impulse and perhaps explained how an impulsive force caused an acceleration over time .... not as small as a dirac moment but perhaps as important.
tell me plz, how can we integrate this function when it has infinite values, and the limit of itegral sums are not the nuber.
i think you need the lebesque integral definition for it to really make sense
Interesting, although I might have used golf as an example of impulse rather than football. I think golf balls can reach 50,000gs of acceletation, not sure a footballer can achieve that!
This (and creation operators) was the point in math where i really started to feel like math is a made up tool I mean i always knew it was but this is where I really started to internalize it like my professor was like "well we really need this function to converge so we're going to stick a delta in it".
Math is NOT a made up tool, and this guy does not even understand the Dirac delta, which is NOT a function but a distribution, and has a very easy definition which is not even mentioned in this video. The problem is just most engineers and physicists are spreading a wrong use of math and don't even bother in doing things correctly.
@@diarandor math is not made up? then where did it come from
As a mathematician, this dirac delta function infuriates me.
Not because it's not useful, it clearly is, but because I'd file under abuse of notation, as it's not a valid function from ℝ -> ℝ to behave the way described with integrals / areas, without redefining integrals / areas to specifically support it.
It's an extremely useful lie.
Consider the space all 'valid' functions from R to R. This space is incomplete in much the same sense that the rationals are incomplete --- there are sequences of functions which are in a sense 'Cauchy convergent' (in an analogous sense to sequences of rationals being Cauchy convergent).
We complete this space just as we complete the rationals -- we define the 'missing' points in the function space to be equivalence classes of 'valid' functions that converge to it.
One such point is the dirac delta .... _generalized function?_ Since you seem to have a bugaboo about using the term 'function' (you speak of it as a 'lie') we will try to be pedantic in your presence and make sure that we avoid the term 'function' without qualifications. Most of us drop the qualification in practice when we all know what we are talking about, and we don't consider ourselves to be 'lying' to each other.
I feel your pain, I truly do, and I hope this helps.
Cheers!
@@zapazap Ok, actually, this really changed my mind on this. Considering the function as itself a limit of functions that narrows to this one is a really interesting one.
@@Quargos It is pretty cool. :)
I was not sure if your initial issue was with the validity of the concept (a conceptual issue) or with the appropriateness of calling it a function (a terminological issue).
There is interesting paper called "Number idea and number concept" by a guy names Staffleu in which he contrasts tbe pre-mathematical _idea_ of number with the various mathematical _concepts_ of num pre-mathematical, and various mathematical _concepts_ that embody the idea. The development and study of various concepts of number (e.g. whole, rational, p-adic, quaternion) is mathematical -- investigation of the idea of number is philosophical.
Perhaps what I described as a 'generalized function' was a function concept, and these new concepts of function enrich one's idea of a function.
@@Quargos P.S. I was exactly in tbe same frame if mind as you were vis-a-vis the dirac delta until someone clued me in to the same idea.
I suspect that it is bad pedagogy that leads students early on to conceive of ideas in an overly narrow way. E.g. being told by high school textbooks that certain quadratic polynomials 'cannot be factored' but eliding the fact that this depends entirely on underlying field in question.
This shows in all sorts of areas. My sister studied _and teaches_ piano, and I once gave her a book of introductory piano pieces by Bartok, and complained to be that he wrote the key signatures 'wrong' because they did not conform to the standard form she was taught, a form she implicitly learned to be sacrosanct!
Forgive be for riding my hobby-horse briefly before you. Bad pedagogy is a curse! :)
Isn't that quite similar to an impulse function (in electronics)?
Yes, it''s the same.
If I have a function f(x) and I want to know the value at 'a', I just do f(a)... but that is boring I guess.
@@DrDeuteron Pal is slang for friend.. words themselves cannot be agressive. Only when they are spoken or specific punctuation added, such as an exclamation mark in this case, could you make a tonal interpretation.
We read that sentence two very different ways 😉
Doing f(a) means assigning the function f as an input into a linear functional which returns the number f(a) as an answer. That is extremely close to the definition of the Delta.
If you substitute 'is' for 'contains' in 'the subject contains the predicate' for an undecideable Kantian synthetic apriori proposition*, for example, the matter numeral '0' subject contains the predicate 'is the idea number zero', then you could argue that a synthetic apriori proposition violates the law of non-contradiction : nothing is both x and not-x and since the complex number, 0 = 0 ± 0.i , of general form z = a ± b.i for real numbers a and b, the imaginary number "i" equal to the square root of negative one and "." meaning multiplied by, is the augmented form of 0, then so does every complex number by induction : any geometric point in the complex plane is really and imaginarily constituted, raisng an issue over how you hedge against god affecting a complex number valued physical system through the imagination.
* In contrast to either an analytic proposition that can be evaluated as either true, or false, by virtue of it's definitions, like 'zero is a number', or actually corresponding synthetic proposition, like 'this background is white'.
I’m using dark mode tho
💓
If the area under the Dirac delta function is 1, does that mean that the sum of all possible positive real numbers is also 1?
Infinity × 1/infinity = 1
Some physicist from the early 20th century
Second
Delta function y = 0 / x
Consider the function y = 1/ x two curvey 'L's in the top right and bottom left quadrants
Y = 1 / 1,000 x is the same shape, but closer to the axes.
Y = 1 / 1,000,000 x is even closer to the axes
At the limit Y = 1/ infinity * x ~ 0 / x , the graph is the axes exactly
0 / 0 is undefined because it is all numbers positive, negative and imaginary. tlc
This video has more dislikes count than likes one but it's not shown. 😆
The function, that is not even a function.
Fascinating video. Well done. TY. However, you may be a bit dismayed to hear that I now suspect this function is God
You guys make this up as you go. It's not even a realistic concept.
How is it imposable to exist in the real world when literally everything is made of an infinite number of zero length points ?
Nothing in the real world is made out of an infinite number of zero length points. Points are a mathematical abstraction, they are not a real thing.
Furthermore, even in the actual theories of physics, objects aren't constructed of points. Their position is smeared around in a probability distribution, they don't take up a single point.
@@hOREP245 What about the dividing line between 2 adjoining plank lengths ? Further more there are many theories out there some have been proven ie relativity some have not ie QFT , though i do agree it is very close to the truth . no one has probed smaller then an atom, needles to say one millionth a billionth the size of an atom ie the plank scale, oh wait if space itself was quantized then there is a theory called "Lazy light" saying different frequencies would travel at different speeds depending on the substructure of space, slowing down if space was not uniformly smooth . all frequencies of light travel at the same speed in vacuum, and this has been measured over a billion years across a billion light years, and all frequencies travel at same speed, indication space itself is not quantized, just energy and matter . but im no expert and dont claim to be .
0 is infinite first of all. Second, the line of x value can stretch at the evident horizon just as the y value can.
Rah