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40,825 görüntüleme • 1 ay önce •via X (Twitter)

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When I first took ordinary differential equations, the pattern was always the same. Week 1 turns into a speedrun of methods: separation of variables, integrating factors, variation of parameters, Bernoulli, exact equations… and by Week 2 or 3 the course has quietly degenerated into hammer-picking. Spot the form, apply the recipe, move on. Mechanical! Fuuuuck!😫😫😫😫 The problem is what you don’t walk away with. You leave with a toolkit, but without a feel for what a differential equation even is, especially geometrically. And that’s a big deal, because in real modeling the equations you meet are rarely nice enough to reward memorized recipes. So you end up trained to solve toy forms, while the actual subject...the behavior, the flow, the shape of solutions stays blurry. This is why I’m biased toward the old-timers. Their old-school way of doing things always surprises me:...they’ll spend time on one idea until it sticks, instead of sprinting through a syllabus checklist. One lecture from them and you start noticing a contrast. A lot of modern teaching feels like "finish the content,". You get marched through techniques, but you’re not left with a single thought that keeps bothering you later...the kind of thought that actually pushes you toward research-level curiosity. MIT OpenCourseWare’s Professor Arthur Mattuck did that to me in his very first ODE lecture. One lecture, and your whole relationship with dy/dx = f(x,y) changes. In this segment, Prof. Mattuck is basically saying: A first-order ODE is a slope field, and a solution is a curve that moves everywhere tangent to that field. The math breakdown Write the ODE as dy/dx = f(x,y). At each point (x,y) you attach a tiny line segment with slope f(x,y). A function y = y₁(x) is a solution exactly when its graph follows those slopes:. At every x, the slope of the curve equals the slope prescribed by the field at the point on the curve. That’s the single line that unifies both viewpoints: y₁′(x) = f(x, y₁(x)). So solving the ODE and drawing an integral curve are the same statement in two languages!👌🏻 Once you see that, you can stop obsessing over whether you can write y(x) in closed form. You can start asking the questions that matter: where do solutions flow, where do they get trapped, where do they blow up, and where does existence/uniqueness fail just because the field isn’t even defined? That’s the perspective shift I wish every ODE course forces early and it’s exactly why I keep pairing math with animation. #DifferentialEquations #ODEs #VectorFields #MathAnimation #Mathematics

Mathelirium

53,338 görüntüleme • 7 ay önce

A woman who spent nine years gluing paper models in a print shop just told a room of physicists their whole field stands on a mistake, one line, no hedging: "Every theory we have takes space and time for granted, like a bagel that forgot it was once a flat sheet of paper." That's Bianca Dittrich. She has a free lecture course that asks one question: what is left of geometry once you quantize it? The answer is: far less than you can picture. Quantum gravity looks like one more field theory. Buried inside is something stranger. Every quantum theory we have puts fields on a fixed stage, flat or curved, and does the physics on top of it. Here the stage itself is the thing being quantized. Your gut reads an atom of spacetime as a tiny grain sitting somewhere. Wrong. It cannot sit anywhere, because it is the somewhere. The move is invisible to human intuition, which is exactly why the people who get anywhere stop asking where the pieces are and start asking what a measurement even means. In 3D the whole thing collapses in a way that should scare you. No matter, no cosmological constant, and gravity is locally flat everywhere. Six degrees of freedom per point, all of them eaten by diffeomorphism symmetry. A field theory that ends up with finitely many real degrees of freedom, sometimes zero. None of it is hidden. Cut a parallelogram out of paper, glue the edges, and you have a torus. Flat everywhere, and yet two numbers survive that no local measurement can see. The lecture is free. Here is the trap: you feel every equation you can solve, every geometry you can draw, as progress. What you cannot feel is the gap you have to cross. Ten to the forty-one, from the Planck scale back up to the world you live in. And crossing it is the only thing that pays. Almost every approach still cannot show it recovers ordinary physics at the far end, and almost everyone quits long before then. The math is free to learn. The nerve to stop trusting your picture of space, that part you still have to bring yourself.

Zyron

110,850 görüntüleme • 1 ay önce

Today we introduce Stochastic Differential Equations (SDEs). I find that the best way to introduce these complex concepts is to look at an application. This is part I of the lecture🙂 We look at the theory behind electromagnetic scattering/radar clutter which leads to anomaly detection on scattering statistics. When a narrowband wave scatters off a messy cloud of particles, the complex field at your receiver is a random phasor sum...at time t you can write the electric field as E_N(t) = Σⱼ₌₁ᴺ e^{iθⱼ(t)}, each term a unit arrow in the complex plane from scatterer j. This is exactly where the magic of Brownian motion appears naturally and in the most reasonable way. Think of all the microscopic chaos...tiny motions, index fluctuations, path jitters, Doppler shifts that shows up as small random kicks to the phases θⱼ(t) over very short times. If you just made θⱼ(t) random in an ad-hoc way (say, resampling independent angles at each time), the field would jump around unrealistically with no temporal structure. Brownian motion is what you get when you let each phase take the continuous-time limit of many tiny, independent kicks...it’s continuous in t, it has the right cumulative variance growth, and it remembers just enough of its past to look physical. So we model each phase as a Brownian walk, θⱼ(t) = θⱼ⁰ + σ_θ Bⱼ(t), with independent Brownian motions Bⱼ(t) and a phase-diffusion rate σ_θ. Brownian motion here isn’t window dressing...it’s the clean way to compress all the small random stuff into a single process that actually matches how the phases wander in time. #StochasticProcesses #BrownianMotion #ItoCalculus #RadarClutter #RayleighScattering #SignalProcessing

Mathelirium

55,319 görüntüleme • 8 ay önce

Ask anyone who’s taken a course in Ordinary Differential Equations (ODEs) what a solution to an ODE represents geometrically, and most of them won’t have a clean answer. When I first took ordinary differential equations, the pattern was always the same. Early on it turns into a speedrun of methods: separation of variables, integrating factors, variation of parameters, Bernoulli, exact equations. Then pretty quickly the course slides into hammer-picking. Spot the form, apply the recipe, move on. Too mechanical! And the real problem is what you don’t walk away with. You leave with a toolkit, but without a feel for what a differential equation even is, especially geometrically. That matters because in real modeling the equations you meet are rarely nice enough to reward memorised recipes. So you get trained to solve toy forms, while the actual subject stays blurry. The behavior. The flow. The shape of solutions. It wasn't until I watched the first lecture of Professor Arthur Mattuck that I realized I didn’t actually know what a solution to a differential equation represents geometrically. His point is almost embarrassingly simple. A first-order ODE is a slope field, and a solution is a curve that stays tangent to that field everywhere. The math breakdown: Write the ODE as dy/dx = f(x,y). At each point (x,y), attach a tiny line segment with slope f(x,y). A function y = y₁(x) is a solution exactly when its graph follows those slopes. At every x, the slope of the curve equals the slope prescribed by the field at the point on the curve. That’s the one line that ties both viewpoints together: y₁′(x) = f(x, y₁(x)). So solving the ODE and drawing an integral curve are the same statement in two languages. Once you see that, you stop obsessing over whether you can write y(x) in closed form. You start asking the questions that actually matter. Where do solutions flow. Where do they get trapped. Where do they blow up. Where does existence or uniqueness fail because the field isn’t even defined? That’s the perspective shift I wish every ODE course forces early. It’s also why I keep pairing math with animation. #DifferentialEquations #ODEs #VectorFields #AppliedMathematics #Mathematics #

Mathelirium

40,841 görüntüleme • 6 ay önce

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

Zyphor

10,999 görüntüleme • 6 gün önce