10 ms·
PDEs you should know
- ramshanker 5y agoNo navier-Stocks equation? or is it in there with a different name.
- frakt0x90 5y agoI know I could look it up but having an explanation of what any of the variables mean would make this not useless.
- killjoywashere 5y ago> an explanation of what any of the variables mean As I recall, that's pretty much Electricity & Magnetism 2 for physics undergrads. E.g. https://www.colorado.edu/sei/departments/physics/activities/courses/electricity-magnetism-ii-course-materials https://www.colorado.edu/sei/departments/physics/activities/...
- actusual 5y agoCool, why?
- valbaca 5y ago"You should know" because, you should. (sarcastic b/c I had the same question)
- The_rationalist 5y ago
- ccosm 5y agoNo Navier-Stokes? Elasticity?
- valbaca 5y agoPartial Differential Equations you should know but with no explanation...I mean...shouldn't you...just know? /s
- marginalia_nu 5y agoPDEs are really useful if you are in the rare domains where they are useful. But most PDEs don't have even have closed form solutions for non-trivial boundary conditions. So unless you are a physicist or something adjacent, no, no you really don't need to know these. Hard to say what's the intended audience for the page though. Could be a message aimed at physics undergrads or something. If so, then indeed, you should know these.
- salty_biscuits 5y agoNavier stokes are incredibly important pdes for science and engineering, yet proof of even the existence of solutions is not currently available. Numerical methods are far more important than analytical solutions.
- medo-bear 5y agoMost useful equations don't have a closed form solution. This is why things like machine learning exist. More specifically PDEs are a hot topic in DNNs at the moment. See physics informed neural networks: https://en.m.wikipedia.org/wiki/Physics-informed_neural_networks https://en.m.wikipedia.org/wiki/Physics-informed_neural_netw...
- marginalia_nu 5y agoThe point is that equations without closed form solutions are pretty useless if you aren't into pretty hairy maths, Green's functions and the like. An equation everyone should know is Hooke's law. That's useful at a high school level.
- travisporter 5y agoWhy do i need to enable javascript to see the equations tex-style
- Kwpolska 5y agoThis website is using MathJax [0] to render math. MathJax and its faster and leaner competitor KaTeX are the only ways to display beautiful, human-friendly math on the Web. They can be run server-side, but many sites do it client-side. The alternative, MathML [2] is a pain for humans to write [3] — it’s a late-90s XML format — and is only supported by Firefox and Safari [4]. [0] https://www.mathjax.org/ https://www.mathjax.org/ [1] https://katex.org/ https://katex.org/ [2] https://en.wikipedia.org/wiki/MathML https://en.wikipedia.org/wiki/MathML [3] https://fred-wang.github.io/MathFonts/mozilla_mathml_test/ https://fred-wang.github.io/MathFonts/mozilla_mathml_test/ [4] https://developer.mozilla.org/en-US/docs/Web/MathML https://developer.mozilla.org/en-US/docs/Web/MathML
- mindcrime 5y agoand is only supported by Firefox and Safari [4]. That's no longer (entirely) true. Chrome are re-adding MathML support (thanks to Igalia) and a significant degree of MathML support is already there. It's hidden behind a feature toggle, but if you turn it on, it does work. It's not complete as far as handling every detail of the spec, but what's there is not insignificant from what I've seen. https://chromestatus.com/feature/5240822173794304 https://chromestatus.com/feature/5240822173794304 https://mathml.igalia.com/news/2021/02/15/mathml-plans-for-2021/ https://mathml.igalia.com/news/2021/02/15/mathml-plans-for-2...
- mjfl 5y agowhat's the best black scholes tutorial?
- _se 5y agoThe "Natenberg Bible": https://www.amazon.com/Option-Volatility-Pricing-Strategies-Techniques/dp/155738486X https://www.amazon.com/Option-Volatility-Pricing-Strategies-...
- Orangeair 5y agoIt's been awhile since I've had a Diff EQ class, but isn't the harmonic motion one an ODE?
- lucaspauker 5y agoYeah good catch
- ChrisRackauckas 5y agoODEs are one-dimensional PDEs.
- fithisux 5y agoBoltzmann equation?
- rudiger 5y agoThe Black–Scholes equation is basically identical to the heat equation. Divide through by σ^2 and let n = σ^2 * (T - t) if you want to derive it.
- dls2016 5y agoThe Schrodinger equation is the heat equation with complex time. Although qualitatively it’s dispersive, not dissipative.
- jcla1 5y agoThe difference is that in the Schrödinger case you're effectively 'turning' the solution (in the complex plane) which leads to the uncomfortable question of whether the solution to the heat equation you'd start with is still defined. When going from heat to Black-Scholes you're just rescaling in 'existing' dimensions which doesn't change the character of the PDE.
- dan-robertson 5y agoBlack–Scholes is the heat equation backwards, which has pretty different behaviour to the heat equation as the latter smooths things out over time and the former makes them less smooth over time. But this does make some sense: when an option reaches expiry you know exactly how much it’s worth (as a function of strike price) but the further you are before, the less well you can predict the strike price and the smoother the price function should be. Indeed your substitution reverses the direction of time but intuitions about the heat equation aren’t so applicable to Black–Scholes because intuitions are often directional.
- pc86 5y agoWho is "you" in this scenario? Who actually needs to know these by heart day to day?
- docfort 5y agoI think it’s better to know that sometimes we only know how to describe something by relating rates of change to other states. And that’s ok. Maybe it has a closed form equation, or maybe can only be solved numerically. But if I see that a differential equation looks like a wave equation, then I get intuition that it’s describing waves. And why do the waves appear? Because the physical process the PDE describes has a speed limit on information passing from time into space! Don’t like traffic waves? Well, why is there some limit on spatial information connected to temporal information? It’s because I cannot see through the cars in front of me. The “fog of war” creates the waves. The denser the fog (e.g. I’m surrounded by semitrucks), the greater the likelihood of waves developing. This intuition is formed by being able to recognize the form of the PDE with general knowledge of the solutions, without needing to actually solve the PDE. Sure, additional insights are possible if you solve it, but knowing that traffic is like springs gives you leverage to use your ordinary intuition to understand unfamiliar things. Point of fact, James Maxwell of E&M fame saw the wave equation and the separate electric and magnetic field PDEs and came up with a detailed spring model to give himself a more familiar analog to play with.
- dan-robertson 5y agoTo give Maxwell a little more credit (not that you aren’t), the wave equations and PDEs of today are much nicer thanks to modern knowledge and computational techniques. Maxwell didn’t have div, grad or curl and so he had dozens of equations to look at instead of just a few, and I think the terms and patterns weren’t as well known as they are today.
- aidenn0 5y agoDo you know if the surface/line integral forms taught to those without vector-calculus under their belts an actual stepping stone to the modern ones? The number of equations are the same, they are just a lot more hairy.
- dan-robertson 5y agoI don’t know enough about the history and I don’t know what forms you are talking about. When I learned ‘vector calculus’ at university, we were introduced to surface/line integrals and div/grad/curl at around the same time.
- Extigy 5y agoI too would have liked to have seen Navier-Stokes included, or least an inviscid Euler equation for modelling fluid flow.
- prof-dr-ir 5y agoThe author is an undergraduate student, and judging from this list it appears that he has yet to encounter non-linear PDEs? Besides the Navier-Stokes equations, which are already frequently mentioned, I would have very much liked to see Einstein's equations added as well.
- deleted 5y ago[deleted]
- groos 5y agoOne of them is not like the rest.
- elil17 5y agoI don’t really get the point of this. Who should know these? Why should they know them? What should they know about them? As a mechanical engineer, for instance, it’s usually a bad idea for me to think about these equations - it’s to “in the weeds”, so to speak.
- madengr 5y ago
- deleted 5y ago[deleted]
- dls2016 5y agoThe Unfinished PDE Coffee Table Book https://people.maths.ox.ac.uk/trefethen/pdectb.html https://people.maths.ox.ac.uk/trefethen/pdectb.html
- Koshkin 5y agoThis is much, much more useful (and beautiful, too).
- dang 5y agoYou should definitely submit that someday—just leave enough time for the hivemind caches to clear (ideally a few months). If you do, email us at hn@ycombinator.com so we can put it in the second-chance pool (https://news.ycombinator.com/pool https://news.ycombinator.com/pool, explained at https://news.ycombinator.com/item?id=26998308 https://news.ycombinator.com/item?id=26998308), so it will get a random placement on HN's front page.
- gtsnexp 5y agogreat! Shame it is "unfinished"...
- HFguy 5y agoThat's great. You should definitely finish the swing on this.
- bally0241 5y agoHelmholtz equation?
- brummm 5y agoThis seems like a random collection of equations from a 1st/2nd year undergrad physics class + Black Scholes.
- paulpauper 5y agoNot to be dick but how does this get to the front page. it's just a Wikipedia copy of some equations.
- JabavuAdams 5y agoOne of these is not like the others.
- aaaaaaaaaaab 5y agoTo me the most baffling thing about differential equations is the fact that somehow the Universe is able to solve them in real time. I mean, of course there are PDEs like the Navier-Stokes equation that describe phenomena emerging from the simple interactions of an immense number of particles, so you could say that the Universe doesn’t “solve” them per se, rather, it runs the discretized simulation on an extremely fine scale, and the whole continuous PDE is our “simplification” of the problem. However, there are equations like the Einstein field equations that operate on a seemingly continuous domain, and whose solutions are impossibly complex in nontrivial cases… So how does the Universe do it? One can say that this question is beyond what science should be concerned with; the Universe evolves according to these equations, because this is what the Universe is. Yet, from a computational point of view it irks me…
- deleted 5y ago[deleted]
- Koshkin 5y ago> the Universe is able to solve them in real time Not just the Universe - analog computers can do that, too.
- aaaaaaaaaaab 5y agoAnalog computers are part of the Universe, so IMO they give no insight into how the Universe solves PDEs. In contrast, digital computing can be modeled on a purely logical basis.
- aidenn0 5y agoMaybe we are just simulations living in a computer that is designed to calculate solutions to PDEs?
- aaaaaaaaaaab 5y agoSure, but you’ve just moved the problem of solving PDEs one layer higher to the simulating universe.
- sampo 5y agoThe harmonic motion equation is an ordinary differential equation (ODE), not partial differential equation (PDE).
- civilized 5y agoTechnically, ordinary derivatives and partial derivatives are the same thing on a function of a single variable, so ODEs and PDEs are the same thing in this case. On the other hand, the author should have included many other ODEs if they wanted to go down this path.
- jvanderbot 5y agoPet peeve: Define your constants (at least units!). If I know the constants by heart, I probably remember the equation.
- dan-robertson 5y agoMost of the units can be inferred, e.g. for the wave equation, say the units of u are A (for amplitude, but you can guess whatever), x is L (for length) and t is T for time (both are extremely conventional dimensions). You convert the pde to units and get: A/T^2 = units(c)^2 A/L^2, and can therefore say: units(c) = L/T And guess that c is the speed of the wave or something proportional to it (it is, in fact, the speed). For the simple harmonic oscillator you get: units(m) L/T^2 = units(k) L which is insufficiently determined but gives units(k/m) = 1/T^2, and you might guess m is mass (in kg say) and then k is kgs^-2, or force per distance, a reasonable set of units for a spring constant (the ode is just Hook’s law: F=kl, but F=ma) For the other equations it becomes harder but the point of the website isn’t really to teach you what the pde is. It’s extremely easy to search for the equation on Wikipedia (as the site gives their names) and look up the units and a bit about the equations there.
- rq1 5y agoBlack Scholes and Heat Equation are the same, up to a change of variable.
- quantum_state 5y agoKlein Gordon equation and Dirac equation should be included.
- tgsovlerkhgsel 5y agoI've read the linked page and skimmed the comments and I still don't know what "PDE" is supposed to mean. From a quick search, my best guess is https://en.wikipedia.org/wiki/Partial_differential_equation https://en.wikipedia.org/wiki/Partial_differential_equation?
- exclipy 5y agoYes
- hoseja 5y agoI see these as very nice, very elegant, mostly useless tautologies. Each one needs a couple pages of explanation to be useful and if you know the explanation you don't need the ~four symbol equation.
- deleted 5y ago[deleted]
- divbzero 5y agoNavier–Stokes is a notable omission. Perhaps because it comes in so many forms [1]? [1]: https://en.wikipedia.org/wiki/Navier–Stokes_equations https://en.wikipedia.org/wiki/Navier–Stokes_equations
- user-the-name 5y agoThe only people who will understand this page are those who already know those equations, so what's the point?
- eternityforest 5y agoI love how the title makes it seem like they're just ordinary general knowledge things that average people will totally get.