It's so awesome to see topics being discussed in class being discussed in an unexpected topic such as "Angles between functions" Such as the Fourier series and Legendre's polynomials! I've only ever heard about Legendre's polynomials from quantum mechanics and the Hydrogen Atom but didn't expect them to see them here as-well. Another amazing video! Keep up the good work! 😁
Expressing fourier series with inner product was definitely way more intuitive than just writing coefficient with integrals. Thanks for providing great videos!!!
Very cool to see Grahm Schmidt applies to functions! Never thought about that at all. Being able to define an inner product between two things really gives you a lot of tools to analyze them. Love linear algebra!
The way you are pacing the conversation and the explanation is excellent; you have a real talent for presenting mathematics! Bravo!
@betelgeusecardioid85753 ай бұрын
ルジャンドル多項式ここでも出てくるんや 電磁気学で学習した事と繋がって嬉しい
@afrolichesmain7773 ай бұрын
I remember seeing 13:02 during my partial differential equations class and completely disregarding it. Now looking at it, Im upset at myself for not seeing back then, like its so obvious! Great video as always!
@apppples3 ай бұрын
i learned this in real analysis. also something about derivatives being linear maps, which when dealing with functions with finite terms were matrices maybe? i don't remember... its been so long. great video though!!
@atrophysicist3 ай бұрын
Your videos never disappoint!! They always start with a seemingly innocent question but dive deeper and deeper!
very clear and concise introduction to this mathematical concept! i only minored in mathematics, so i didn't need to go far too into linear algebra, i never even had to do this in all of my physics classes!
Great video, orthogonal polynomials are some of my favourites objects in maths (as a numerical analyst). Some things i want to say are 1: Some additional remarkable properties of Legendre polynomials involve: -three term recursion relations which tell us that (in the case of legendre) (n+1)Pn+1(x) = (2n+1)xP_n(x) - nP_n-1(x) This extremely useful in computing such polynomials fast and accurately. -gaussian quadrature. The roots of legendre polynomials can be used to numerically integrate functions very quickly and accurately - expressing functions: using the orthogonality relation, it is easy to express functions on computers using orthogonal polynomials. Then, you can solve many differential equations easily (spectral method) 2: In fact, if you change the inner product to a general one of the form (f,g) = ∫f(x)g(x)w(x)dx where w is a weight function (that satisfies certain integrability conditions...) You can get other sequences of orthogonal polynomials Look up Chebyshev polynomials of 1st and 2nd kind, which are heavily tied with fourier series Laguerre polynomials, on half real line and Hermite polynomials, on whole real line, which turns up in probability theory due to having the normal distribution as the weight function. There is nothing special about Legendre polynomials. There are analogous properties for all other orthogonal polynomials :D
@weegee79243 ай бұрын
Regarding the weight function, lots of differential equations come equipped with a natural choice of weight function, so solutions to these equations can be constructed by constructing an orthonormal basis of functions and then superposing them! All part of a class of problems called Sturm-Liouville problems, which encompass many physical differential equations, such as Poisson's equation, the Diffusion equation, and the wave equation. Bessel functions and spherical harmonics fall out naturally when solving these kinds of problems in specific coordinate systems. They're quite tame and well-behaved once you realize they're just basis functions!
@acborgia13443 ай бұрын
Great video, I learned a lot!
@eggyolk67353 ай бұрын
I remember doing the Gram-Schmidt process when doing linear algebra/vector spaces and seeing it here made everything click so beautifully!
If it adds like a vector and it scales like a vector, it's a vector. Functions add like vectors and scale like vectors, so they're vectors. (They're also an inner product space, which technically isn't a requirement to be a vector, but it's a nice bonus and important for orthogonality.)
@anandasatria77343 ай бұрын
I learned the concept in linear algebra, but they never expanded it that it can be expanded to Fourier series. This is very interesting
@a52productions3 ай бұрын
Oh! That's where the Legendre polynomials are from... they pop up all the time in physics, but my lecturers always brushed over them and I was left mystified by their definition. When you try to construct them manually like this, they become so simple -- it's just what you get when you try to turn a Taylor series basis into an orthonormal one!
@user-plr3 ай бұрын
As expected, it is the inner product of the continuous function space C[a,b]. Functional analysis on KZbin is really surprising!
@weegee79243 ай бұрын
With the last video and this video covering techniques used in Quantum Mechanics, I hope we get to see Zundamon-sensei's introduction to Quantum Mechanics!
I thought calculus was hard until I found about linear algebra
@kappasphere3 ай бұрын
From the thumbnail, I thought the question was how to find the angle between two function graphs, so I solved that problem before watching the video 😅 So anyways, here is my solution for finding the angle between two function graphs, using complex numbers: if z=ae^iθ and w=be^iγ, then z/w = a/b e^i(θ-γ) gives the angle between z and w. To get a complex number that has an angle that represents the slope of a function f at (x, f(x)), you just use the point (1 + i f'(x)). Use this to get the angle between f and g: The angle between (1 + i f'(x)) and (1 + i g'(x)) is the angle of (1 + i f'(x))/(1 + i g'(x)) = (1 + i f'(x))/(1 + i g'(x)) (1 - i g'(x))/(1 - i g'(x)) = (1 + i f'(x))(1 - i g'(x)) / (1 + g'(x)²) = (1 + f'(x)g'(x) + i (f'(x) - g'(x))) / (1 + g'(x)²) the angle of this is arctan( (f'(x) - g'(x)) / (1 + f'(x)g'(x)) ) Let's try to find two functions that are always orthogonal. For these, 1+f'(x)g'(x)=0 for all x, so g'(x)=-1/f'(x). So g(x) = C - int 1/f'(x) dx. For example, with f(x)=ln(x), f'(x)=1/x, and g(x)=C - int x dx = C - x²/2. But after watching the video, orthogonality of the functions as vectors is a lot more interesting than the angle between the graphs.
That length operator for functions at 2:32 is super close to the definition of the Root Mean Square (RMS). The only difference is you divide by 1/(b-a) to attain the average of the squared function along the interval [a,b]. It has very important applications in AC power systems, especially in delivering a certain consistent voltage to homes (for America, this is usually 240/120V RMS at 60 Hz). It equates AC Power Delivery to a device to some DC Power Delivery in a sense. This is done because AC power has this weird phenomenon where certain devices (equivalent to either capacitors or inductors), will have slight periodic behavior (passing charge/energy back and forth between each other and-or holding extra energy for some period of time instead of immediately dissipating it like a resistive load would). So engineers and physicists wanted a way to ignore this effect as this “reactive” power wouldn’t actually provide any power towards the device, so they remove it out of the equation for AC power systems using this cool little RMS thing.
@holery92153 ай бұрын
At first, i consider the video just explain about inner product (dot product). Because the thumbnail is asking about angle between two vectors. So, i guess that is just rearranging the equation of dot product or cosine rule. Unfortunately, my guesses are wrong. You make me very pleasant for this explanation. I learned a new thing, like assuming function as exponential series and manipulated the equation where has a different expectations. Thank you🙌 Ps: Please apologize for my bad english
It should have been stated that by definition the norm of a vector and inner product of vectors have to satisfy certain properties. This is especially important since at 12:12 one of these properties, linearity, is used, despite not having checked its validity for our definition. That said, it follows immediately from the linearity of integrals, but Metan rarely glosses over these details!
@jackolantern62013 ай бұрын
The length of a function resembles the RMS value of a function