Professor MathTHeBeautiful, thank you for an outstanding explanation and analysis of Laplace Eigenvalues on a Unit Disk in Partial Differential Equations. The derivations/problems are the heart and soul of Partial Differential Equations.The introduction to Bessel Functions is a powerful contribution/solution for Partial Differential Equations.This is an error free video/lecture on TH-cam TV with MATHTheBeautiful.
I'm not sure of the details of the mistake, but check out the paper about one of my scientific heroes, Tulio Levi-Civita "Toward a scientific and personal biography of Tullio Levi-Civita" and see page 7.
Hello, I was wondering why the eigenvalues of \Delta are negative using linear algebra. I know that it is a Fourier multiplier with symbol -4\pi^2 |\xi|^2.
MathTheBeautiful you mean we must have Theta(0) = Theta(2pi). In this case, we obtain A*cos(n*0) = A*cos(2*n*pi) + B*sin(2*n*pi), not just cos(n*0) = cos(2*n*pi), concluding n is integer. We don't have sin(2*n*pi) = 0, because n is not an integer yet.
05:08 But how do we know that for sure? Is there any proof for it somewhere? And how can we prove such things? (that some equation cannot be solved in terms of known functions) I asked this question to many mathematicians already, and no one could answer it in any meaningful way without a lot of hand waving. 05:40 Indeed. It looks _almost_ like the typical Cauchy-Euler, except that it has that nagging `r²` coefficient for `R` :P I can't believe how such an innocent thing can mess up the equation so much :q There _has_ to be a better way to solve such equations. But one can come up with much simpler-looking equations that behave like that. For example this one: y" = x²·y I have a feeling that there's some magical coordinate substitution that would simplify this, and these "special" functions look so complicated just because they are nested in each other in some fancy way. 07:30 Well, there _is_ a difference: We know what sines and cosines mean geometrically (the coordinates of a point on a unit circle). Same goes with hyperbolic sines and cosines: they measure the coordinates of a point on a hyperbola for a given hyperbolic angle. We also know what exponentials are (self-proportional growth). But we don't have any geometric interpretation for the Bessel functions or other fancy "special functions" :P We can only evaluate their power series to get some approximations (which is kinda what we do for sines and cosines too, true), but this is pretty much like having a formula for the digits of some fancy number but not knowing what that number is and what it represents :P We cannot use its algebraic properties in formulas or equations. And this drives me crazy everytime I deal with those "special functions". Because, in a way, they are not more "special" than sines and cosines or exponentials and logarithms (in a way that they are all defined as power series). But the lack of geometry makes them elude comprehension of the mind. They're more "fuzzy".
Professor MathTHeBeautiful, thank you for an outstanding explanation and analysis of Laplace Eigenvalues on a Unit Disk in Partial Differential Equations. The derivations/problems are the heart and soul of Partial Differential Equations.The introduction to Bessel Functions is a powerful contribution/solution for Partial Differential Equations.This is an error free video/lecture on TH-cam TV with MATHTheBeautiful.
I love how the lecturer is half yelling
WHAT ARE YOU SAYING?
Hello, Professor. Do you know is there any special functions concerning the radial eigenfunctions of Laplacian on higher dimensional balls?
Hello,
Do you know of any particular examples of these 'famous variable changes mistakes'? That seems like something interesting.
all of them lol
forgetting the Jacobian is the biggest one speciallly when going from Cartesian to Polar
I'm not sure of the details of the mistake, but check out the paper about one of my scientific heroes, Tulio Levi-Civita "Toward a scientific and personal biography of Tullio Levi-Civita" and see page 7.
Hello,
I was wondering why the eigenvalues of \Delta are negative using linear algebra. I know that it is a Fourier multiplier with symbol -4\pi^2 |\xi|^2.
Great question! Here's the answer: th-cam.com/video/ErgEzdLbBjk/w-d-xo.html, although there's a bit of buildup to that.
Ah, yes. Thanks.
Why is the positive constant n in 3:40 an integer? I don't get why "we wouldn't have a continuous function"
So that cos n*0 = cos 2*n*pi. This identity doesn't hold unless n is an integer.
MathTheBeautiful Thank you so much. You have no idea how this is a help. It's part of my final work in college, so i can get my degree
Why should Theta be 2pi periodic?
Because θ=0 and θ=π are the same physical point.
MathTheBeautiful you mean we must have Theta(0) = Theta(2pi). In this case, we obtain A*cos(n*0) = A*cos(2*n*pi) + B*sin(2*n*pi), not just cos(n*0) = cos(2*n*pi), concluding n is integer. We don't have sin(2*n*pi) = 0, because n is not an integer yet.
05:08 But how do we know that for sure? Is there any proof for it somewhere? And how can we prove such things? (that some equation cannot be solved in terms of known functions) I asked this question to many mathematicians already, and no one could answer it in any meaningful way without a lot of hand waving.
05:40 Indeed. It looks _almost_ like the typical Cauchy-Euler, except that it has that nagging `r²` coefficient for `R` :P I can't believe how such an innocent thing can mess up the equation so much :q There _has_ to be a better way to solve such equations.
But one can come up with much simpler-looking equations that behave like that. For example this one: y" = x²·y
I have a feeling that there's some magical coordinate substitution that would simplify this, and these "special" functions look so complicated just because they are nested in each other in some fancy way.
07:30 Well, there _is_ a difference: We know what sines and cosines mean geometrically (the coordinates of a point on a unit circle). Same goes with hyperbolic sines and cosines: they measure the coordinates of a point on a hyperbola for a given hyperbolic angle. We also know what exponentials are (self-proportional growth). But we don't have any geometric interpretation for the Bessel functions or other fancy "special functions" :P We can only evaluate their power series to get some approximations (which is kinda what we do for sines and cosines too, true), but this is pretty much like having a formula for the digits of some fancy number but not knowing what that number is and what it represents :P We cannot use its algebraic properties in formulas or equations. And this drives me crazy everytime I deal with those "special functions". Because, in a way, they are not more "special" than sines and cosines or exponentials and logarithms (in a way that they are all defined as power series). But the lack of geometry makes them elude comprehension of the mind. They're more "fuzzy".