Zyphor

[Zyphor @Zyphorrl]

MIT FILMED A LECTURE WHERE A PROFESSOR EXPLAINS WHY EVERY WEATHER FORECAST ASTEROID PREDICTION AND ECONOMIC MODEL ON EARTH RUNS ON ONE TRICK MOST STUDENTS NEVER UNDERSTAND - AND ALMOST NOBODY WATCHES IT

This is David Jerison, MIT, lecture 10 of Single Variable Calculus. He opens with a GPS satellite orbiting Earth and shows that the difference in time between your watch and the satellite clock is proportional to the ratio of velocity squared to the speed of light squared. Special relativity reduced to one linear approximation that fits on half a line.

Then he makes the point that defines the whole lecture. Every engineering system - satellite design, weather forecasting, asteroid tracking, economic modeling - involves dozens of these relationships. If you don't reduce each one to a linear approximation first, you cannot think about what is happening. The math becomes impossible to untangle. Simplification is not laziness. It is the only path to understanding.

The quadratic approximation extends this. Six functions - sine, cosine, exponential, logarithm, powers - each approximated near zero by at most three terms. Every complicated function on earth is locally one of these six. The rest is algebra.

Watch the moment he builds the graph of 3x minus x cubed using only the sign of the first and second derivatives. No calculator. No plotting. Just logic about where the function rises, falls, bends up and bends down. The complete shape of the curve emerges in under 4 minutes.

A control systems engineer I know uses this lecture as the first thing he sends to new hires. Says if they understand why linearization works they can learn anything else on the job.

Free on YouTube, MIT OpenCourseWare, Creative Commons license.

bookmark this and watch later - after this lecture the phrase "good enough approximation" will feel like one of the most powerful ideas in engineering

[the lich @thelichhh]

https://t.co/aqsY18NvVz

[Zyphor @Zyphorrl]

The pattern that looks like powers of two until it isn't. This is the most famous trap in math - and LLMs still fall for it when asked to reason about sequences.

Moser's circle problem breaks exactly where Pascal's triangle stops aligning. The real answer involves Euler's formula, combinations, and a connection that Jane Street mathematicians use to test candidates.

20 minutes. Bookmark & watch today. The coincidence has a proof.

[fintex @_yusufknl]

https://t.co/JUtRUEnagJ

[Zyphor @Zyphorrl]

Everyone thinks attention is a new invention. An attention matrix is an adjacency matrix, an object from a field Euler started in 1736.

PageRank, AlphaFold and GPT all run on graphs. This 16-minute intro builds one from scratch.

Bookmark & watch today.

[fintex @_yusufknl]

https://t.co/JUtRUEnagJ

[Zyphor @Zyphorrl]

THE MOST LEGENDARY PROFESSOR IN MIT HISTORY EXPLAINED THE ONE FUNCTION THAT RUNS EVERY BANK AND EVERY AI TRAINING LOOP ON THE PLANET - SO SIMPLY THAT YOU WILL WONDER WHY NOBODY TAUGHT YOU THIS WAY BEFORE

This is Gilbert Strang, MIT, the professor whose textbooks sit in over 3,000 universities across 135 countries. He opens with one sentence - this is the only function whose slope equals itself - and spends 40 minutes showing why that single property explains almost everything.

He builds the function from scratch. No formula given upfront. Just a rule - slope must equal the function - and a starting point of 1. The series falls out term by term automatically: 1 + x + x²/2 + x³/6 and so on. Each term forced by the previous one. No choice, no guessing.

Then he multiplies two of these infinite series together and shows they collapse into one - proving that e to the x times e to the y always equals e to the x+y. The binomial theorem kills the algebra in one move.

Then the bank. Start with $1 at 100% interest compounded once a year - you get $2. Compound monthly - you get more. Compound every second - you approach exactly e. The number 2.718... isn't arbitrary. It's what happens when a bank compounds interest infinitely fast and the limit stabilizes.

Watch the moment he shows that e to the minus x is simply 1 divided by e to the x. Multiply them and you get e to the zero which is 1. The whole symmetric shape of the graph falls out in 30 seconds.

A finance quant I know showed this lecture to a junior analyst who kept confusing exponential and linear growth. Said he never confused them again after watching.

Free on YouTube, MIT OpenCourseWare, Gilbert Strang at a chalkboard.

bookmark this and watch later - after this lecture compound interest will feel less like a bank trick and more like a law of nature

[the lich @thelichhh]

https://t.co/aqsY18NvVz

[Zyphor @Zyphorrl]

THE MOST LEGENDARY MATH PROFESSOR IN MIT HISTORY EXPLAINED IN 40 MINUTES WHAT MOST STUDENTS SPEND A FULL SEMESTER FAILING TO UNDERSTAND - AND HIS STUDENTS GO ON TO EARN MORE THAN MOST PEOPLE SEE IN A LIFETIME

This is Herb Gross, MIT, the professor whose teaching style has been called the clearest explanation of calculus ever recorded. Oxford students, MIT graduates, self-taught engineers - all credit this exact course for passing exams they had no business passing.

He opens with one idea - if you know the solution to the easy version of an equation, you can find the solution to the hard version. Replace the constants with functions, let them vary, and the structure of the equation itself tells you what those functions have to be.

Two conditions fall out automatically. One you choose freely to keep things clean. The other the equation forces on you. Together they form a system of 2 equations in 2 unknowns - and the only thing that breaks it is if your two starting solutions are secretly the same solution in disguise.

Watch him solve an equation where the answer comes out as x·sin(x) + cos(x)·log|cos(x)|. Nobody guesses that. Nobody sees it coming. The method produces it mechanically in 6 steps.

Then he checks it by differentiating twice and substituting back. Everything cancels. The mess was correct the entire time.
A physics PhD I know used this to model a damped oscillator with a non-standard forcing term. Said it was the only method that didn't require guessing the answer first.
Free on YouTube under MIT Creative Commons license.

bookmark this and watch later - after this lecture you will never need to guess the answer to a differential equation again

[the lich @thelichhh]

https://t.co/aqsY18NvVz

[Zyphor @Zyphorrl]

THE HEAD OF OXFORD MATHEMATICS WHO WROTE OVER 200 PAPERS AND BUILT HIS CAREER ON THIS EXACT TOPIC - SAT DOWN AND EXPLAINED IT FOR FREE IN 45 MINUTES

The professor has published over 200 research papers and authored 12 mathematics textbooks used across 40 countries. He opened this lecture by writing a single symbol whose square equals minus one.

Two complex numbers multiply together and the cross terms cancel perfectly, leaving a real number every time. It's the structure that runs every electrical system, signal processor and wave equation in modern physics.

He then proves that any polynomial of degree n has exactly n roots - but only if you allow complex numbers. This result took mathematicians centuries to prove and is called the fundamental theorem of algebra.

Watch the moment a student interrupts to question his definition. The professor stops, listens, then says - you're being formulaic, that's not mathematics.

A physicist I know rewatched this after 3 years of university. Said it was the first time complex numbers felt like something discovered rather than invented.

Free on YouTube, filmed live in Oxford with 200 first-year students.

bookmark this and watch later - after this lecture imaginary numbers will feel more real than most things you were taught at school

[the lich @thelichhh]

https://t.co/aqsY18NvVz