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Every waveform in nature is just exponentials added together. The Laplace Transform tells you exactly how much of each one — and that single idea turns unsolvable differential equations into basic algebra. ChatGPT, Claude, every robot arm and circuit on the planet runs on systems analyzed with this tool....

52,520 просмотров • 1 месяц назад •via X (Twitter)

Комментарии: 7

Фото профиля Bilal
Bilal1 месяц назад

It would be nice to mention the creator of this video on YouTube @EugeneKhutoryansky

Фото профиля Dr. Mathhattan
Dr. Mathhattan1 месяц назад

worth adding: everything's discrete now so it's actually the Z-transform doing the daily work in digital control systems, laplace is the continuous-time ancestor

Фото профиля Bellache
Bellache1 месяц назад

This could be pushed even further since many compression algorithms also use FFT.

Фото профиля Adeptus Econicus
Adeptus Econicus1 месяц назад

... ... ... while my 5 year old could answer by looking at the word, he cant spell strawberry reliably yet.

Фото профиля gabriel
gabriel1 месяц назад

Thanks for this. Thanks Laplace

Фото профиля 菅野正人
菅野正人1 месяц назад

数学的にはたった1本の単位ベクトルが支点を原点にして回転して描く単位円が全てですよ。 e ^i ωt=cos ωt+i sin ωt フラクタル自然数1単位ベクトル→の定義なければ、単位ベクトルの位相計算すら始まりませんよ。 リーマン予想 証明完了!

Фото профиля Craig Colley
Craig Colley1 месяц назад

The cardiac cycle is a pattern that resonates and underpins consciousness

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The Fourier transform runs inside every MRI machine, every audio compressor, every signal processor on earth. JPMorgan pays $350K to engineers who can derive it from scratch. There is one professor alive who explains it the way nobody else can. His name is Gilbert Strang. MIT. The most brilliant mathematical mind of his generation. His textbook sits in over 10 million homes. No other person on earth holds this combination of depth and clarity in one head. He opens with one confession. The Fourier transform is unreasonably effective. It solves problems it was never designed for. The goal is not to compute it. The goal is to understand why it works at all. Then the core idea. A periodic signal breaks into pure sine waves. Each with a frequency, amplitude, and phase. The transform finds all three simultaneously. A complicated signal in time becomes a simple picture in frequency. Then 1807. Fourier invented the transform not for sound but for heat. How does warmth spread through a metal rod? In time the equation is a partial differential equation - hard. In frequency space it becomes ordinary - easy. Transform in, solve it, transform back. Then convolution. Convolution in time equals multiplication in frequency. Filtering a signal, removing noise, compressing audio - all multiplication in frequency space. Without the transform: thousands of computations. With it: a few multiplications. Then the delta function. Zero everywhere except one point where it is infinite. Integral equals one. Every mathematician in 1900 said it was not a function. Dirac used it anyway. It took 40 years to justify what engineers had been doing the whole time. Watch the moment Strang shows that the Fourier transform of a delta function is a constant - every frequency in equal measure. The more concentrated in time, the more spread in frequency. This is the uncertainty principle. Not quantum physics. A theorem about any signal at all. A signal processing engineer I know rewatched this before their first project at Apple. Said it was the first time the Fourier transform felt like a change of coordinates rather than a formula to memorize. Free on YouTube, MIT OpenCourseWare. bookmark this and watch later - after this lecture every sound, every image, and every signal will feel like a sum of sine waves waiting to be separated

Zyphor

123,452 просмотров • 1 месяц назад