13:30 as soon as he writes down the Fourier series and begins to explain it, the audio gets really epic and it's like the microphone is trying to reject the deep revelation he is sharing lol
Professor MathTheBeautiful, thank you for Solving the Heat Equation in 1Dimension and the Need for the classical Fourier Series in Partial Differential Equations. I also encountered Fourier Series in Signal and Systems Theory. This is an error free video/lecture on TH-cam TV.
Yes, everything continues to work. The solution becomes smooth instantly. This is clear from Fourier series analysis since after any finite amount of time, the Fourier coefficients decay exponentially with n.
08:47 _You're_ my new senpai :) 14:21 Can you tell the title of that paper by Riemann? Hmm... So if ANY function can be expressed as the sum of sines and cosines, why do we need to solve for those sines and cosines in the first place? The (arbitrary) function itself satisfies the equation, since we can always decompose it into sines and cosines which satisfy it on themselves :q
AugustoDRA It's the sum of second partial derivatives of a function with respect to all of its variables. In this case it's just the second total derivative of f(x).
Right. The operator itself is completely independent of coordinates but to be honest I have no idea what it means for an arbitrary coordinate system. My guess would be that if f is a function in a space and r is a vector in that space then the Laplacian(f) is the sum of the second partials of f(r) with respect to the chosen orthonormal basis for that space. I have no clue what this means explicitly though.
Here's the coordinate system-independent definition of what Laplacian is: For each particular point in the domain of your function, it tells you how much the function differs from the average (i.e. flat) level around that point. You may think of it as an operator that tells you the _curvature_ of the function graph's surface at any particular point (how much does it bend up or down). The second-order derivative is a 1D example of that: it tells you how much the graph of the function is bent on both sides of a point on it. There's a little catch, though: the function can bend up in one direction, and down in another direction, so on average it has 0 curvature despite the fact that it is not a flat surface.
13:30 as soon as he writes down the Fourier series and begins to explain it, the audio gets really epic and it's like the microphone is trying to reject the deep revelation he is sharing lol
13:26 when the P D E really starts to hit.
T H E R E C I P E. I S C O M P L E T E L Y O B V I O U S.
Impressive explanation of the need for the Fourier series. Thank you, very much, Sir.
I recommend the Darth Pavel voice for all future videos
Professor MathTheBeautiful, thank you for Solving the Heat Equation in 1Dimension and the Need for the classical Fourier Series in Partial Differential Equations. I also encountered Fourier Series in Signal and Systems Theory. This is an error free video/lecture on TH-cam TV.
How can I appreciate you with this amazing enlightenment!!!
are you an artist? lol
your 'penmanship' is really good
the sound gets "fuzzy" around 13:06
The "fuzzy" sound starts at 13:26 and ends at14:34. Quite annoying but you can skip it with not much loss of the talk
Dirichlet boundary equations ear-rape 2020?
14:35. The question I have always wondered
Could our initial conditions be discontinuous and if so how would that work with Fourier series? Does it 'become' continuous after some time step?
Yes, everything continues to work. The solution becomes smooth instantly. This is clear from Fourier series analysis since after any finite amount of time, the Fourier coefficients decay exponentially with n.
08:47 _You're_ my new senpai :)
14:21 Can you tell the title of that paper by Riemann?
Hmm... So if ANY function can be expressed as the sum of sines and cosines, why do we need to solve for those sines and cosines in the first place? The (arbitrary) function itself satisfies the equation, since we can always decompose it into sines and cosines which satisfy it on themselves :q
I don't know what the laplace operator is, but I think I followed well enough. Great lesson!
AugustoDRA It's the sum of second partial derivatives of a function with respect to all of its variables. In this case it's just the second total derivative of f(x).
This is true only in Cartesian coordinate system :q
Right. The operator itself is completely independent of coordinates but to be honest I have no idea what it means for an arbitrary coordinate system. My guess would be that if f is a function in a space and r is a vector in that space then the Laplacian(f) is the sum of the second partials of f(r) with respect to the chosen orthonormal basis for that space. I have no clue what this means explicitly though.
Here's the coordinate system-independent definition of what Laplacian is:
For each particular point in the domain of your function, it tells you how much the function differs from the average (i.e. flat) level around that point. You may think of it as an operator that tells you the _curvature_ of the function graph's surface at any particular point (how much does it bend up or down).
The second-order derivative is a 1D example of that: it tells you how much the graph of the function is bent on both sides of a point on it.
There's a little catch, though: the function can bend up in one direction, and down in another direction, so on average it has 0 curvature despite the fact that it is not a flat surface.
13:26 Enter Darth Pavel
How to determine Cn of fourier series U(x) ??
Fourier coefficient of initial condition
Yes the lecturer sounds very diabolic just as the key point is made. Kind of cool error.