Laplace Applications

Laplace's Equation Applications

In this video, I give some very neat applications of Laplace's equation. In particular, I explain why harmonic functions are called harmonic, and I give a really cool probability application. Enjoy!

[welcometothemadhouse]

There is a cool application of the Laplace Equation in physics again for finding an analytical solution for the motion of a vibrating cylinder following stokes law's, using Laplace transforms and viscosity measurements. Usually used to derive analytical solutions for the moment of elastic cylinders in Newtonian Fluid. I hope you're dealing well in lockdown, not going too crazy I hope ;)

[aliexpress.official]

solutions to Laplace\Poisson equation also tell you the electric fields produced by various charge distributions! (actually the electric potentials but the E field can be derived from that) probably the first PDE physics students come across :)

[matrixstuff]

You can use the diffusion equation to model concentration of a substance in a fluid; Stokes flow; finding the electric potential for a given charge density to name a few

[dominicellis]

How would you model amplitude and pitch changes? I used a decay function to model crescendos and descends. But the attack is harder to fit. (Staccato articulations are impulses with non zero epsilons). How would you define the shape of a melodic line and use it to predict future lines?

[soadvent]

Hello,
I've got a question I hope someone will be able to answer:
In electrodynamics, and especially when talking about guided waves, there's a sentence stating that TEM waves can not exist in a simply connected domain. Whether you're familiar or not with electrodynamics is not important.
The proof given to me goes as follows:
With some manipulations of Maxwell equations, assuming an harmonic regime and a TEM mode, one can find the following equation (u corresponds to the electric potential):
laplacian of u(x, y) = 0.
With boundaries conditions that u must be constant on the boundary.
Therefore, a sufficient condition to prove the non-existence of TEM modes would be that u is constant on the whole domain, as, therefore, the electric and magnetic fields would lead to a trivial solution of Maxwell equations.
Does anyone know why u must be constant?
The explanation given to me was that, if a non-constant solution existed, there should be an extremum inside the domain, and one can therefore find (with the Hessian matrix and the laplace equation) that second partial derivatives u_xx and u_yy must be at the same time equal and opposite in sign. This yields to a contradiction if one can prove that the Hessian matrix must be positive-defined.
I don't know how to prove it, as a necessary condition for extremum is that the Hessian matrix is semi-positive defined

[davidkwon]

7:50 pde is infinite dimensional linear algebra.

[davidkwon]

10:10 Can you hear the shape of a drum? 16 - dim. Counterexample!

[AqibKhan-thcw]

Could you give me the reference from where i can study the application of laplace equation in borwnian motion. Its urgent please

[nadavslotky]

Nitpicking: maybe next time you could use a different letter for the function and for the domain?
That was a little distracting.

Other than that, very nice video!