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.
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.
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
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!