the convolution, here it comes pretty intuitive. clearly explained the initial condition acts at the Gaussian solution. After a interpretation to convolution integral, it would be benefit more from this video. Thank you for your excellent teaching Professor.
Thank you very much! You are the type of teachers who we really need in my country. I hope I will be one in the future and your helps are enormous. Thank you again and again.
Thank you so much for your lessons, I passed 2 of my exams using them !! It actually saved my semester and I will recommend you everywhere ! :) (I'm a master student in applied mathematic master in France)
what concerns me is that I'm taking this as a sophomore applied physics student, and it's wayyy beyond my capabilities. I wish you all the best in the rest of your career!
Alexey, see my second reply to Naresh Kumar below. Professor Brunton goes into greater detail explaining the specific convolution identity above and discusses convolution in general, although I don't recall that he actually derived any other identities.
What a professional video. The details were in depth, the speed was moderately low (perfect), no background music, equations + words + diagrams. I loved it. Could you do a video with the wave equation, please? (I’m an electrical engineer and we use it in electromagnetics and in lossless transmission lines.)
So essentially, when we analyze the Fourier series of a function and derive its coefficients, we gain insight into how each frequency contributes to the energy of the signal. In a sense, each frequency is linked to a quantum of energy. Consequently, the total energy of a signal can be represented as the sum of all its coefficients multiplied by their corresponding harmonics. In simpler terms, it's the sum of all frequencies in the Fourier space, leading to the emergence of this identity.
Interesting. It reminds me of the problem of heat distribution through the earth. If we take the surface temperature as a sinusoidal then we only have a single omega and hence this would be a special case of your method, and measuring at a depth x, we should see another sinusoidal but with diminished amplitude via the Gaussian function.
Hi Steve, one question at 4:49, how did you get d U_hat/ dt on the left hand side. do you for F(tranform) on both sides? or do you mean d / dt is independent linear operator on t, so you can take it out? Thank you so much!
Good question. Yes, we Fourier transform both sides with respect to the "x" variable, and so the "d/dt" can come out (so F(d/dt(u)) = d/dt(F(u)) since these operations can switch orders)
@@Eigensteve Thank you so much for clarifying Steve! And actually one more question, in DFT, you showed us FT matrix is a square matrix, does it have to be square? in another words, do we have to pick same number of points, n , in frequency space as in X space? And this is one of the greatest channel for lectures, we really appreciate it!
@@kevinshao9148 You are very welcome! In principle, there is no reason why we need to compute and use all of the Fourier coefficients, so we could consider a smaller "rectangular" matrix consisting of a subset of the rows of the DFT matrix. This is often done in random/sparse sampling. But for the computational benefit of the fast FT (FFT), the square structure is important.
@@Eigensteve Ah, I see!!! Now I kind of recall you might have mentioned this in your following FFT lecture. I am watching your series 2nd time, need multiple times to grip solid understanding. Again, really appreciate your great lectures and it's super pleasure to learn from you and your lectures!!! 👍👍👍
@@Eigensteve Hi Steve, but how do you guarantee the limited bases you selected from frequency space (the transform matrix) represent the major contribution in original f(x) ? I also see other python fft tutorial, they actually just call fftfreq(number of data points) and get the sample frequencies, but there is no theory backup that those are the major frequencies in your original signal. Am I missing something?
First of all, thank you for nice explanation. I want to tell you that your lecture helps student in Korea a lot. I began to take an interest in Machine Learning since watching your lecture. Actually, I have a question at 05:25. I still don't understand how you solved ode. If you are still reading comments, it will be nice if you explain it for me. Thank you again.
Often solving ODE's is about recognizing familiar patterns with known solutions. Here he doesn't derive the solution so much as he recognizes that the solution he gives is the "well-known" solution for an ODE of the given form.
See Deriving the Heat Equation: A Parabolic Partial Differential Equation for Heat Energy Conservation (th-cam.com/video/9d8PwnKVA-U/w-d-xo.html) to understand Uxx.
I still don't quite understand how the Fourier Transform of a derivative acts independently of the variable over which the derivative is taken. What if we transform u sub tt, not u sub xx, is it the same?
I belive it wouldn't be possible to transform u sub t because the fourier transform implies that the function goes from -infinity to +infinity and in that case the time starts at 0. Maybe we could assume that the time goes form -infinity to +infinity, idk.
Mr. Kumar, the book is in a reference in the description under the video. "Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control" by Brunton and Kutz I haven't bought the book yet, but it's in my wish list on Amazon. The preview of the book on Amazon shows a bibliography with source references also.
Professor Brunton also has a playlist on his TH-cam site listed as "Engineering Mathematics", which are videotaped lectures from that class at University of Washington. In that series of courses, he derives in greater detail some of the formulas you will see in the current series. I have watched those lectures and recommend them to you.
Big respect to Pr. Brunton, i think such kind of projects should be supported.
the convolution, here it comes pretty intuitive. clearly explained the initial condition acts at the Gaussian solution. After a interpretation to convolution integral, it would be benefit more from this video. Thank you for your excellent teaching Professor.
I was mind blown by how nicely you described this! Thank you for the amazing video :)
This is really cool. Connecting the diffusion phenomenon to the convolution kernel was a really nice touch. I never thought about it that way before.
The best description I've seen on the topic.
Thank you very much! You are the type of teachers who we really need in my country. I hope I will be one in the future and your helps are enormous. Thank you again and again.
Really amazing and simple explanation. I never understood Fourier transform as I did from your videos. Thank you!
Thank you so much for your lessons, I passed 2 of my exams using them !! It actually saved my semester and I will recommend you everywhere ! :) (I'm a master student in applied mathematic master in France)
what concerns me is that I'm taking this as a sophomore applied physics student, and it's wayyy beyond my capabilities. I wish you all the best in the rest of your career!
Why the hell my university is giving me this course on my second year😢
Thank you so much for your video! Your explanation using the convolution is much easier to understand than what I have ever studied so far!!
I can't tell how grateful I am. Thank you so much
Thank you! Would be nice to see this linked to a video explaining convolution identities in detail.
Alexey, see my second reply to Naresh Kumar below. Professor Brunton goes into greater detail explaining the specific convolution identity above and discusses convolution in general, although I don't recall that he actually derived any other identities.
Good call. Just added a pointer to the convolution video at 6:42.
@@Eigensteve It isn't visible professor
Thank you! This was a great explanation of the heat equation as a smoothing operator.
What a professional video. The details were in depth, the speed was moderately low (perfect), no background music, equations + words + diagrams. I loved it. Could you do a video with the wave equation, please? (I’m an electrical engineer and we use it in electromagnetics and in lossless transmission lines.)
So essentially, when we analyze the Fourier series of a function and derive its coefficients, we gain insight into how each frequency contributes to the energy of the signal. In a sense, each frequency is linked to a quantum of energy. Consequently, the total energy of a signal can be represented as the sum of all its coefficients multiplied by their corresponding harmonics. In simpler terms, it's the sum of all frequencies in the Fourier space, leading to the emergence of this identity.
Interesting. It reminds me of the problem of heat distribution through the earth. If we take the surface temperature as a sinusoidal then we only have a single omega and hence this would be a special case of your method, and measuring at a depth x, we should see another sinusoidal but with diminished amplitude via the Gaussian function.
Absolutely superb lecture, but how did you find the Gaussian in x,t-coordinates ?
How does this method work when you have two or more space variables?
Hi Steve, one question at 4:49, how did you get d U_hat/ dt on the left hand side. do you for F(tranform) on both sides? or do you mean d / dt is independent linear operator on t, so you can take it out? Thank you so much!
Good question. Yes, we Fourier transform both sides with respect to the "x" variable, and so the "d/dt" can come out (so F(d/dt(u)) = d/dt(F(u)) since these operations can switch orders)
@@Eigensteve Thank you so much for clarifying Steve! And actually one more question, in DFT, you showed us FT matrix is a square matrix, does it have to be square? in another words, do we have to pick same number of points, n , in frequency space as in X space?
And this is one of the greatest channel for lectures, we really appreciate it!
@@kevinshao9148 You are very welcome! In principle, there is no reason why we need to compute and use all of the Fourier coefficients, so we could consider a smaller "rectangular" matrix consisting of a subset of the rows of the DFT matrix. This is often done in random/sparse sampling. But for the computational benefit of the fast FT (FFT), the square structure is important.
@@Eigensteve Ah, I see!!! Now I kind of recall you might have mentioned this in your following FFT lecture. I am watching your series 2nd time, need multiple times to grip solid understanding. Again, really appreciate your great lectures and it's super pleasure to learn from you and your lectures!!! 👍👍👍
@@Eigensteve Hi Steve, but how do you guarantee the limited bases you selected from frequency space (the transform matrix) represent the major contribution in original f(x) ? I also see other python fft tutorial, they actually just call fftfreq(number of data points) and get the sample frequencies, but there is no theory backup that those are the major frequencies in your original signal. Am I missing something?
Excellent. Explained it well., Should have also identified the error function.
Can we solve 2D heat equation with the same technique? If yes, could you share some sources?
First of all, thank you for nice explanation. I want to tell you that your lecture helps student in Korea a lot. I began to take an interest in Machine Learning since watching your lecture.
Actually, I have a question at 05:25. I still don't understand how you solved ode. If you are still reading comments, it will be nice if you explain it for me.
Thank you again.
Try to solve it this way: Integral (du/u) = Integral (-w^2*α^2) dt
@@user-sv6jh1fv5s Thank you!!!
Thank you! But what happens if the heat equation has a source term f(x, t)? Can we still solve this by using Fourier transform?
Very precise and clear explanantion thank you sir
Thanks for watching!
Hi Steve, one question about the solving of ODE at 5:19. can you explain this in more details? Much thanks!
Often solving ODE's is about recognizing familiar patterns with known solutions. Here he doesn't derive the solution so much as he recognizes that the solution he gives is the "well-known" solution for an ODE of the given form.
This is quite basic as far as ode's go. I would look at some diff eq courses before doing any Fourier or pde stuff
See Deriving the Heat Equation: A Parabolic Partial Differential Equation for Heat Energy Conservation (th-cam.com/video/9d8PwnKVA-U/w-d-xo.html) to understand Uxx.
This is really good, good job Sir.
Always had this question, can Steve sir write reverse or is the video flipped during post process?
Thank you for amazing explanation.
Glad you liked it
Awesome explanation.
It's interesting though that we have non-stationary process in time and still are allowed to do Fourier transform and convolution in space..
Great video! Thank you!
Thank you. This is amazing.
Puzzle: Is Prof. Brunton right or left handed?
How to use that board for teaching?? Is it a software??
Shouldn't the space variable x be in a bounded interval. Allowing x to be infinity is not physical either
What happens when I apply FFT over a constant term?
FFT I don't know but FT would give your constant multiplied by a dirac
If this is how Washington university lectures look like, I am really regret not choosing this university as my undergraduate
Thank you so much
Awesome! Thanks
Thank you so much!
I still don't quite understand how the Fourier Transform of a derivative acts independently of the variable over which the derivative is taken. What if we transform u sub tt, not u sub xx, is it the same?
I belive it wouldn't be possible to transform u sub t because the fourier transform implies that the function goes from -infinity to +infinity and in that case the time starts at 0. Maybe we could assume that the time goes form -infinity to +infinity, idk.
You are left handed and your image is flipped but not the board's?
Which books that you using?
Mr. Kumar, the book is in a reference in the description under the video. "Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control" by Brunton and Kutz
I haven't bought the book yet, but it's in my wish list on Amazon. The preview of the book on Amazon shows a bibliography with source references also.
Professor Brunton also has a playlist on his TH-cam site listed as "Engineering Mathematics", which are videotaped lectures from that class at University of Washington. In that series of courses, he derives in greater detail some of the formulas you will see in the current series. I have watched those lectures and recommend them to you.
Who knew that this very equation the Gaussian spread was the key to solve the Poincaré conjecture.
I like the part where he said "Gaussian".
That is a picture of gravity