Zyphor
@Zyphorrl • 1,422 subscribers
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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
Zyphor120,380 просмотров • 6 дней назад
MIT filmed a mathematician proving the theorem behind every equation Maxwell ever wrote. Most engineers use it daily without understanding why any surface gives the same answer - and the ones who can explain it earn $350K at aerospace firms. Stokes' theorem is the reason Maxwell could write four equations instead of forty. Every result in electromagnetism, fluid mechanics, and differential geometry sits on top of it. Understanding it changes how you see every circulation and flux problem in physics. This is Denis Auroux. MIT, 18.02, Multivariable Calculus, Fall 2007. Lecture 31. It covers Stokes' theorem - the result that unifies line integrals, surface integrals, and curl into one statement. He opens with one confession. Then the statement. The work done by a vector field along any closed curve equals the flux of its curl through any surface bounded by that curve. Not a specific surface. Any surface you want. You pick whichever one makes the calculation easier. Then Green's theorem disappears. The result from the plane that converts line integrals to double integrals turns out to be Stokes' theorem in disguise. Same formula, restricted to a flat surface in the xy-plane. What looked like two separate theorems is one. Then orientation. Walk along the curve with the surface to your left and the normal vector points up. This rule is not chosen for elegance. It is the price of having a consistent coordinate system in three dimensions. Then the proof. Cut your surface into tiny flat tiles. Apply Green's theorem to each one. Sum everything up. All interior boundary edges cancel. Only the outer edge survives. The sum of tiny Green's theorems gives you Stokes' theorem for any surface. Watch the moment he computes the same line integral two ways. First directly - answer is pi. Then through a paraboloid instead of the obvious flat disk. Longer calculation, stranger setup, same answer. The theorem is not an abstraction. You can watch it work. A fluid dynamics engineer I know kept this lecture open during their first week modeling atmospheric circulation. Said it was the first time Stokes felt like a tool rather than a formula to verify on exams. Free on YouTube, MIT OpenCourseWare, 18.02. bookmark this and watch later - after this lecture every circulation, every flux, and every curl you encounter will feel like three views of the same object
Zyphor10,999 просмотров • 5 дней назад
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