Poissons Equation

[Nusc]

1. The problem statement, all variables and given/known data
So poissons equation takes the for uxx + uyy = f(x,y)
Laplace is where f(x,y). What does the f(x,y) physically represent?


2. Relevant equations



3. The attempt at a solution

[buzzmath]

Laplace equation is when f(x,y)=0. f(x,y) can represent many things physically. the solution of this problem can represent many things for example u could be a steady state temperature of the cross section of a rod with an electrical current.

[quasar987]

What you wrote does not make sense to me, but the question got throught nonetheless.

In Maxwell's theory of electromagnetism, the electromagnetic field is governed by a set of 4 equations and one of them is Poisson's equation where u is the electric field in space-time (x,y,z,t) and f(x,y,z,t) is an expression taking into account the density of charge and the rate of change of the magnetic field at the point (x,y,z,t) in space-time.

[Nusc]

Laplace equation is when f(x,y)=0. f(x,y) can represent many things physically. the solution of this problem can represent many things for example u could be a steady state temperature of the cross section of a rod with an electrical current.

But what is f actually doing to this cross section?

[Nusc]

What you wrote does not make sense to me, but the question got throught nonetheless.

In Maxwell's theory of electromagnetism, the electromagnetic field is governed by a set of 4 equations and one of them is Poisson's equation where u is the electric field in space-time (x,y,z,t) and f(x,y,z,t) is an expression taking into account the density of charge and the rate of change of the magnetic field at the point (x,y,z,t) in space-time.

And when f = 0 ? What does it mean in this case?

[quasar987]

Well it means that this particular Maxwell's equation (\nabla^2\vec{E}=0) is describing the evolution of the electric field in a region where there are no electric charges and where the magnetic field is constant.

[Nusc]

So say one is concerned with the heat distribution among a metal plate, what would f mean and what would f = 0 mean?

[Nusc]

Maybe I should have written this in the undergraduate physics forum.

[quasar987]

You can ask a mentor to move it.

[HallsofIvy]

I hate to keep saying this but mathematics is not physics! Quantities in a mathematical equation do NOT have any "physical" meaning and do not "physically represent" anything until you apply them to a specific physics problem.

(I guess I am like an ex-smoker. I started majoring in physics, then switched my major to mathematics, though staying in applied math (My doctoral disertation was on computing Clebsch-Gordon Coefficients in general SU(n)) but have steadily moved to more abstract mathematics since.)

That said, if you have \nabla \phi= \kappa \partial \phi/\partial t+ f(x,y,t) specifically applied to heat distribution on a plate, then f(x,y,z) might represent an external heat source applied to every point of the plate.