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!
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!!
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!
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
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!
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.
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!
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.)
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!
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.
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!
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!
![[Eng Sub] Calculations in Another World | Tropical Geometry](http://i.ytimg.com/vi/eHVchEZuU7Q/mqdefault.jpg)
![[Eng Sub] Calculations in Another World | Tropical Geometry](/img/tr.png)
![[Eng Sub] Basel Problem: What is Behind the Famous Proof](http://i.ytimg.com/vi/jmtiWGnj5os/mqdefault.jpg)
![[Eng Sub] What is e to the Power of d/dx? | Shift Operator](http://i.ytimg.com/vi/INTKzi0qXpY/mqdefault.jpg)
![[Eng Sub] Something Like Differentiation | Half Derivative](/img/n.gif)
「直交」という表現は本来の漢字の意味から「計量線形空間内で定義される内積の値が0」という意味にかなり拡張されていますね。
「互いに素」の記号として当然のように⊥が濫用されてたりしますよねー、初めて見たのが気持ちを理解する前だったのでビックリした記憶があります
大学の教科書だと、これがメッチャあっさり導入されるよな
痒い所に手が届いて面白いや
線型代数「内積が定義できるヤツ全員友達」
初めてフーリエ変換について触れたとき、三角関数の直交性について面白さは感じつつも深掘りはしなかったからこの解説を聞いてとても感動している……。この投稿者さんの動画は他の解説と違って質の高さを感じるし毎回取り上げるテーマが面白くてすごい素敵。
線形代数「またオレ何かやっちゃいました？」
こういうお話を量子力学の講義の前に受けておきたかった。
そう言ってもらえるとうれしいです。
いつもご支援ありがとうございます！
大学化学で「2つの波動関数の積を全区間で積分して0なら、2つの波動関数は直交している」って習って、よくわかってなかったけど、この動画のあかげで分かった。ありがとう。
I learned about this in Linear Algebra, but this pleasantly extends it with more theory. Nice video!
ある意味無限次元ベクトルの内積と考えられるよね
This might be one of the best math channels on youtube right now. Thank you so much for the english subtitles!
フーリエ変換よく使ってるので解説助かります！
画像圧縮とか、スペクトラムアナライザ(音楽とかで周波数を表示するアレ)とかに使われてて、線形代数の応用力の高さを感じさせられますよね〜
15:30 ～ 公式を暗記していたフーリエ展開が、関数の内積を知って「もとの関数をベクトルと考えて
基底関数（cos、sin）の成分を求めていたんだ」と絵でわかった時、感動したのをおぼえています
内積を公理で定義しておけば、普通のベクトルの場合も関数の積分の場合も、公理を満たしていることをチェックするだけで、内積に関する必要な性質（内積から定義した「長さ」がちゃんと長さの性質である三角不等式を満たすこととか、シュワルツの不等式とか）がすべて自動的に分かる。
なぜなら、それらの性質は、内積の具体的定義によらずに内積の公理だけから証明できるから。
日本語を勉強しているコンピューターサイエンス専攻でこういう動画めっちゃ便利ね
このチャンネルが1番しっかりしてて好き(by数学専攻1年)
俺物理学科だけどやっぱり数学ってあるものを別な視点から見てそれをどんどん発展させる時が一番楽しい。いつもこういう動画ありがとう。
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! 😁
ベクトルの内積を学んだ時、ベクトルの内積は有限項の数列同士の積和になってるけど、無限項のべき関数の積和（=べき関数の積の積分）でもイケそうだなぁと思ったけど、できるんですね。
フーリエ級数は、正弦関数と余弦関数の位相が90度ズレてて、関数が直交してるイメージありました。直交した波動成分に分解してる感覚
機械力学で運動方程式を正規化する際に直交の関係を使ってたけどこういうことだったんだ…
普通に「直交」って"orthogonal"の事だと思って生きてたら"Cartesian"なんてものが出てきて泣いた大学数学の思ひ出
どうやってこういう面白いトピックに辿り着けるのか、自分で学ぶときにどうやっているのか、知りたすぎる。明らかにクオリティが高いので，動画作成の手法も待っています．
Expressing fourier series with inner product was definitely way more intuitive than just writing coefficient with integrals. Thanks for providing great videos!!!
高校中退生です！後半意味わからん、ムズすぎ！だけど、自分から新しく数学を開拓していく感じはええよなぁ。ありがとう！
ルジャンドル多項式ってなんかやったなぁ。でも思わぬところから出てきて余計に混乱しとる。貴様何故…。
今までふわっと使っていたルジャンドル陪関数がシュミットの直交化から導出できるのはよく考えると当たり前だけど勉強になった
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!
ん〜〜〜何かが拡張していくのは気持ちいいね プールで浮かんでるような気持ち良さ
関数の直交性といえば、三角関数を利用したフーリエ変換かな。フーリエ変換は神ツールだと思う。
振動モードを理解するのに必須の知識ですな。
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!!
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
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!
幾何ベクトルの「なす角」も、基本これと同じ定義なんだがな。
図面を使って「角度」が定義できると信じ込んでいる輩が多すぎる。
The way you are pacing the conversation and the explanation is excellent; you have a real talent for presenting mathematics! Bravo!
量子力学の講義やら課題でこういう計算を大量にやらされた記憶がある。何がわからないのかわからないままだったが、数学的な意味が分かってなかったのかも。
Zernike多項式というもので2次元平面の直交を扱ってるが
理解が深まった
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
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!
Your videos never disappoint!! They always start with a seemingly innocent question but dive deeper and deeper!
直行の定義に合う関数の組を眺め続けると、もしかすると夢の中でそれらが直交しているのが見えるかもですね。
Great video, I learned a lot!
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.
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!
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.)
量子力学だと関数の直交化はとてもなじみが深い
I remember doing the Gram-Schmidt process when doing linear algebra/vector spaces and seeing it here made everything click so beautifully!
毎度毎度シチュエーションが謎すぎて笑う
壮大な世界背景があるものと妄想しています
たぶん、ホビー漫画の話の規模がエスカレートして地球の危機にホビーで対抗するやつの数学バージョン。
大学の教科書全部この人に書いてほしいんだが…
線形代数はこの人に任せたいな
中の人が教科書書いたことがある教授さんだったりする可能性が微粒子レベルで存在している……………？
@@バックミンスターフラーレン
本当にそうだとしたら、教授も好きで教科書あんな雑に書いてる訳じゃないんだな
@@user-kl7hd2vv3e大学教授が本を書くのは、「本くらい書かないと碌な収入を得られないから」だという噂が……。
産学連携とかメディア出演とかで稼げる分野じゃなければ、本書いて『教科書』という体で学生に売りつけるくらいしか専門性を活かした副業がありませんから。
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!
My favorite math channel
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.
I learned the concept in linear algebra, but they never expanded it that it can be expanded to Fourier series. This is very interesting
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
As expected, it is the inner product of the continuous function space C[a,b]. Functional analysis on TH-cam is really surprising!
関数をどこぞのサラダおろしで細切れにして、無限個(実数の濃度)の値を持つベクトルとして扱ってるって解釈で合ってる？
サラダおろしが何を指すか分からんが基本その認識で良いと思う
解析や量子力学やるんだったら避けては通れない道かな（解析は詳しくないなら予想でしかないが）
しかも、大体の教科書はこれを「知ってる前提」でいきなり使うからね
@@user-kl7hd2vv3e 区分求積法(にんじんしりしり)、元は区分求積法の動画だけど積分に対してだいたいそのイメージが湧くようになった。
2:17 RMSみたいだなあ。
(音いじりが趣味のド文系並感)
ルジャンドルの多項式は正規直交系だけど完全ではないのかな
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!
The length of a function resembles the RMS value of a function
great video!
おしい。関数の「内積」は、対象とする関数の都合に合わせて調整されるので、一意に議論できるものじゃないことに言及してほしい。多項式の例とフーリエ級数では責任範囲が違ってるしね。関数2つから数値1個が得られる計算全てが内積に使えるよ。
Oh yes bois! Another great one!~
sinとcosを掛けると0になるやつですね
💕💕👏👏👏Super intereante el tema🤘 me encanto como una simple analogia genera mas matematematicas lo cual les llevo fourier🙌💕👍
単に関数のグラフを見るとそんな感じなさそうだけど, それとはまた別?
この動画は計量ベクトル空間(R∞)での関数の内積の定義についての説明
で合ってますよね？(🤔ちょっと自信ないです。
R∞がなんなのか分からんけどたぶん違う
そこらへんは積分可能性とかの煩雑な説明が必要だから省いていると思われる
Omg this is awesome!!!!!!!!
either way orthogonality means 90 degree, right?
すごく丁寧な動画だけど、構成が観づらく感じてしまうのは私だけでしょうか。
ℝ^nのユークリッド内積におけるシュミットの直交化を示してから、一般の内積空間の話に展開した方がスッキリすると思う。
あと、何を目標にして話が進んでいるのかが見えづらく、展開を追うのに少し苦労する。
ただ、とても分かりやすく楽しい動画だから、これをきっかけに数学好きな人が増えると好いなと思います。
parabéns, voçê acaba de encontrar o comentário em português que tanto procurava! faz o L.
積分区画に意味付けはできますか？
内積を定義できる範囲で、考えたい関数空間にあわせて、いい感じに積分区画を定義するといった感じのものでしょうか？
（-∞, +∞）の範囲になるが、量子力学やれば分かるよ
@@user-kl7hd2vv3e
量子力学では、例えば周期境界条件を考えたいときは-L〜Lとしたり、開放教会条件にしたい場合はL→∞です。空間次元が上がったり、対称性を考慮することで、積分区画を変えたりします。そう言う話は、あくまで物理的に考えたい積分区画として定義できます。
ここで質問している内容は、純粋に数学的に考えた場合に、積分区画に要請がかかっているかと言うことです。
so peak
Slight error in translation at 5:03 , 0!=1 (this is supposed to be a joke)
2h
エアプで申し訳ないけどこういう内容って数学科で学ぶの？
そんな学科入らなくても、どの分野であれ工学で勉強できる
@@user-tg6xp1ef2m 丁寧にありがとうございます！
なんなら数学科の方が見る機会少ない
そうなんですか！？
丁寧に教えてくださりありがとうございます