PDEs 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.
Most 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_netw...
i disagree. equations are useful only if you have use for them. this is by definition of the term. on the other hand, high schoolers rarely feel like Hooke's law is useful!
Yes, it is. Hooke’s law tells basically that the force is proportional to the deformation, or in standard textbook physics, f = - k x, the classical harmonic oscillator.
It’s not surprising if you see a solid material that way:
- it is in stable equilibrium when no force is applied, i.e., it is at an energy minimum
- therefore either extending it or compressing it increases its energy
- therefore the second derivative of its energy is positive.
On a high level, this sounds very much like a simple parabola, because it is one.
For more technical details, we can always locally approximate a function using a Taylor’s series. In this case, the constant term and the linear term are zero if we place the frame of reference correctly (not necessary but it simplifies the equations). So the leading term is quadratic. If we are close enough to the energy minimum (i.e. if the deformations are small), we can ignore the other terms. Therefore, a solid is, to a very good degree of approximation for most of them, a harmonic oscillator.
Alternatively, to a physicist, almost everything looks like a parabola.
If the energy is a parabola, then the force is a linear function, or, as it was written in the linked document, -dE/dx = m d^2 E/dt^2 = - k x . (The first step being Newton’s second law).
The analogy can be pushed a bit: a solid deformed quickly enough will have periodic deformations, periodically contracting and expanding, i.e. vibrating. This is strictly equivalent to a harmonic oscillator oscillating in its energy well.
I suspect (please tell me if I am wrong) the sine(x) solution you mention refers to elastic waves, which appear if we take the next step and consider the solid as a bunch of coupled harmonic oscillators. Long story short, if we do that, we end up with the wave equation, from which you get sines (or more accurately, complex exponential settings).
I think we need a better mutual definition of useful. IMO an equation is "useful" if someone out there is doing practical things with it. You are taking a sort of definition where anyone can do math with it.
Depends on what you're doing, but you can numerically simulate solutions to PDEs. To me that makes them useful even if we struggle to conjure up closed form solutions.
Well, to be useful, wouldn't you need to be able to use it? Most people outside of physicists and physics-adjacent fields are very far away from the mathematical tools to deal with these equations.
> There's a whole field to simulate PDEs that you can't solve exactly.
There are whole fields to simulate one particular PDE: Computational fluid mechanics for the Navier-Stokes equation. Computational electromagnetics for Maxwell's equations. Computational chemistry for the Schrödinger equation. Mathematical finance ...probably does also other things than just simulates the Black-Scholes equation.
I think what the other commenters are getting at is that PDEs can be used without having a closed form solution (and mostly are used that way as closed form solutions usually only come up in special artificial cases). You start your system in a real known state and then propagate it forward in time using (for example) the finite difference method on the equations to figure out the state at a later time.
https://en.wikipedia.org/wiki/Numerical_methods_for_partial_...
NS is a case in point. No general solution, but thousands of special cases that are solved and many more that can be understood using numerical methods
As an engineer, I've used lots of FEA to solve problems, but if you can't put part of your solution space where it can be approximated by a closed form solution (and there are many more than those shown, which are amenable to solution to appropriate methods), you're going to have a hard time building a trusted model. There's a reason we still have wind tunnels.
The most interesting parts of good FEA (where you've shown your model and reality match on measurable components), is that you can see hidden and unmeasurable variables, which may be design limiting.
> PDEs are really useful if you are in the rare domains where they are useful.
Aside from ‘rare’, this seems at best vacuously true.
> 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.
As others have said, while your first sentence is surely true, the latter doesn't follow from it (and I would argue isn't true—but it depends on how you define adjacency). There are lots of things one can usefully do with an equation besides finding a closed-form solution. (For an ODE example, the classical predator–prey model does not have a nice closed-form solution, but is still plenty useful.)
Being familiar with PDEs is useful even without knowing how to solve them.
Just the formulas alone teach you how physical quantities interact with one another and give you great insights on how the universe operates on a fundamental level.
Most people may not be using them at their everyday job, or at all for that matter, but knowing the core ones is just as enlightening if not more than having read major works in Philosophy.
I was once asked a (programming) interview question and suggested solving it with differential equations (it was a system of odes but could be made into pdes too). But that wasn’t the solution they were looking for.
Navier 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.
> Could be a message aimed at physics undergrads or something.
The Black Scholes equation is present though, which is used in pricing securities in financial markets. Perhaps this page is aimed at physicists who intend on jumping ship to become a quant!
This 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].
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.
The 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.
Black–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.
I 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.
To 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.
Do 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.
I 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.
This equation was initially thought of as the appropriate continuous version of the discrete problem investigated by Fermi-Pasta-Ulam-Tsingou (on a very early digital computer shortly before Fermi's death), but then it was realized the KdV equation was better for that.
The fun thing is that a PDE’s solution is emergent by definition: it is an interplay of the dynamics (specified by the the PDE), the boundary conditions, initial conditions, and maybe a forcing function (energy or information pump). Change one of those things and you’ll get a different solution. You might be reminded of fractals, which are a special case of iterated function systems, roughly a discrete version of iterated function systems. Or maybe you might think of cellular automata, all based on local update rules and some initial conditions.
The answer to your actual question is literally called the wave equation (first in the list on the linked webpage). The left side talks about some variable u and how it changes over time. The right side is how u changes over space. And the two sides are linked via a constant c. By observing the solution or by working out the units, we can understand c to be the phase velocity, or roughly, the wave speed. So the way that u evolves over space is limited by how it evolves over time (and vice versa)! u cannot react to all things in space instantaneously. Therefore, a wave emerges, carrying updates from one part of space to another.
You 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, explained at https://news.ycombinator.com/item?id=26998308), so it will get a random placement on HN's front page.
To 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…
Technically, 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.
Most 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.
101 comments
[ 3.9 ms ] story [ 169 ms ] threadAs I recall, that's pretty much Electricity & Magnetism 2 for physics undergrads. E.g. https://www.colorado.edu/sei/departments/physics/activities/...
(sarcastic b/c I had the same question)
(TLA - Three Letter Abbreviation)
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.
An equation everyone should know is Hooke's law. That's useful at a high school level.
Hooks law is an approximation to a material (and spring) property that sets the PDE up. Sin(x) is the solution.
But I could be wrong :S
It’s not surprising if you see a solid material that way:
- it is in stable equilibrium when no force is applied, i.e., it is at an energy minimum
- therefore either extending it or compressing it increases its energy
- therefore the second derivative of its energy is positive.
On a high level, this sounds very much like a simple parabola, because it is one.
For more technical details, we can always locally approximate a function using a Taylor’s series. In this case, the constant term and the linear term are zero if we place the frame of reference correctly (not necessary but it simplifies the equations). So the leading term is quadratic. If we are close enough to the energy minimum (i.e. if the deformations are small), we can ignore the other terms. Therefore, a solid is, to a very good degree of approximation for most of them, a harmonic oscillator.
Alternatively, to a physicist, almost everything looks like a parabola.
If the energy is a parabola, then the force is a linear function, or, as it was written in the linked document, -dE/dx = m d^2 E/dt^2 = - k x . (The first step being Newton’s second law).
The analogy can be pushed a bit: a solid deformed quickly enough will have periodic deformations, periodically contracting and expanding, i.e. vibrating. This is strictly equivalent to a harmonic oscillator oscillating in its energy well.
I suspect (please tell me if I am wrong) the sine(x) solution you mention refers to elastic waves, which appear if we take the next step and consider the solid as a bunch of coupled harmonic oscillators. Long story short, if we do that, we end up with the wave equation, from which you get sines (or more accurately, complex exponential settings).
Are you arguing that e.g. Navier-Stokes isn't useful?
edit: Just noticed that Navier-Stokes isn't even on here. This is frankly a weird list.
There are whole fields to simulate one particular PDE: Computational fluid mechanics for the Navier-Stokes equation. Computational electromagnetics for Maxwell's equations. Computational chemistry for the Schrödinger equation. Mathematical finance ...probably does also other things than just simulates the Black-Scholes equation.
That's what we have computers for, numerical solutions to PDEs ;-)
NS is a case in point. No general solution, but thousands of special cases that are solved and many more that can be understood using numerical methods
The most interesting parts of good FEA (where you've shown your model and reality match on measurable components), is that you can see hidden and unmeasurable variables, which may be design limiting.
Aside from ‘rare’, this seems at best vacuously true.
> 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.
As others have said, while your first sentence is surely true, the latter doesn't follow from it (and I would argue isn't true—but it depends on how you define adjacency). There are lots of things one can usefully do with an equation besides finding a closed-form solution. (For an ODE example, the classical predator–prey model does not have a nice closed-form solution, but is still plenty useful.)
Just the formulas alone teach you how physical quantities interact with one another and give you great insights on how the universe operates on a fundamental level.
Most people may not be using them at their everyday job, or at all for that matter, but knowing the core ones is just as enlightening if not more than having read major works in Philosophy.
The Black Scholes equation is present though, which is used in pricing securities in financial markets. Perhaps this page is aimed at physicists who intend on jumping ship to become a quant!
[0] https://www.mathjax.org/
[1] https://katex.org/
[2] https://en.wikipedia.org/wiki/MathML
[3] https://fred-wang.github.io/MathFonts/mozilla_mathml_test/
[4] https://developer.mozilla.org/en-US/docs/Web/MathML
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://mathml.igalia.com/news/2021/02/15/mathml-plans-for-2...
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.
https://en.wikipedia.org/wiki/Burgers%27_equation
This equation was initially thought of as the appropriate continuous version of the discrete problem investigated by Fermi-Pasta-Ulam-Tsingou (on a very early digital computer shortly before Fermi's death), but then it was realized the KdV equation was better for that.
https://en.wikipedia.org/wiki/Fermi%E2%80%93Pasta%E2%80%93Ul...
https://en.wikipedia.org/wiki/Korteweg%E2%80%93De_Vries_equa...
Intro to Traffic Flow Modeling and Intelligent Transport Systems
(Note that thanks to aggressive monetisation of MOOCs they shut off access to the course a few weeks after you enrol in it.)
https://www.edx.org/course/intro-to-traffic-flow-modeling-an...
The answer to your actual question is literally called the wave equation (first in the list on the linked webpage). The left side talks about some variable u and how it changes over time. The right side is how u changes over space. And the two sides are linked via a constant c. By observing the solution or by working out the units, we can understand c to be the phase velocity, or roughly, the wave speed. So the way that u evolves over space is limited by how it evolves over time (and vice versa)! u cannot react to all things in space instantaneously. Therefore, a wave emerges, carrying updates from one part of space to another.
Besides the Navier-Stokes equations, which are already frequently mentioned, I would have very much liked to see Einstein's equations added as well.
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.
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…
Not just the Universe - analog computers can do that, too.
In contrast, digital computing can be modeled on a purely logical basis.
On the other hand, the author should have included many other ODEs if they wanted to go down this path.
For the simple harmonic oscillator you get:
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.