What is the meaning of LaPlace and Poisson's equation in English?

[yungman]

$\nabla^2 V = \nabla \cdot \nabla V$.

Let me first break this down in English from my understanding:

$\nabla V$ is the gradient of a scalar function $V$. $\nabla V$ is a vector field at each point P where the vector points to the direction the maximum rate of increase and $|\nabla V|$ is the value of the slope.


$\nabla \cdot \vec{A}$ at a point P is the divergence of $\vec{A}$ at point P. If $\nabla \cdot \vec{A}$ at a point P is not zero, there must be a source or sink because the inflow to point P is not equal to the outflow from point P.


So what is the meaning of the divergence of a gradient ($\nabla^2 V = \nabla \cdot \nabla V$)?

What is the meaning of Laplace equation where $\nabla^2 V = 0$?

What is the meaning of Poisson's equation where $\nabla^2 V =$ some function?

Please explain to me in English. I know all the formulas already, I just want to put the formulas into context.

Thanks

Alan

 

[phyzguy]

Well, in 1D, if the Laplacian is zero over some region, that means that the slope is constant. This means that the function cannot have a local maximum anywhere in the region, and the maximum can only occur at the boundary of the region. Generalizing this to higher dimensions, the same thing holds. If the Laplacian is zero, the function cannot have a local maximum anywhere in the region, and the maxima or minima cann only occur on the boundaries of the region. Although it doesn't mean that the slope is constant, it does define in some sense that the function is "smooth" and has no "kinks" or "peaks". If we go now to Poisson's equation, the places where the Laplacian is not zero are the sources of the function, and in these regions the function can have maxima ("peaks" or "kinks"). I'm not sure if this helps or not. You might try this:

http://en.wikipedia.org/wiki/Harmonic_function

 

[yungman]

Well, in 1D, if the Laplacian is zero over some region, that means that the slope is constant. This means that the function cannot have a local maximum anywhere in the region, and the maximum can only occur at the boundary of the region. Generalizing this to higher dimensions, the same thing holds. If the Laplacian is zero, the function cannot have a local maximum anywhere in the region, and the maxima or minima cann only occur on the boundaries of the region. Although it doesn't mean that the slope is constant, it does define in some sense that the function is "smooth" and has no "kinks" or "peaks". If we go now to Poisson's equation, the places where the Laplacian is not zero are the sources of the function, and in these regions the function can have maxima ("peaks" or "kinks"). I'm not sure if this helps or not. You might try this:

http://en.wikipedia.org/wiki/Harmonic_function

Thanks for the reply and remind me about Harmonic function is Laplace equation and the value can be found by knowing the value of the function on the boundary. I had studied this in the Green's function but just not relate to Poisson's and Laplace equation.

 

[MasterX]

A similar definition to the one that phyzguy gave:

In a 2D (or 3D) domain if ΔV=0 then the integral of ΔV in the entire domain is also zero. Using the divergence theorem, you get that the surface integral of the normal vector dot the gradient of V is zero. This means that the net flux is zero => equilibrium

To better understand the above statement, replace V with T (temperature). Then the gradient of T is the heat flux. If you have a system where the net heat flux is zero, the system is in equilibrium.

 

[blue_raver22]

In simple terms, LaPlace and Poisson's equations are mathematical equations that describe the behavior of a scalar function in a given space. The \nabla^2 V represents the Laplacian operator, which is a measure of the curvature or change in a function at a specific point. The \nabla \cdot \nabla V represents the divergence of the gradient, which describes the flow of a vector field at a given point.

In the context of LaPlace equation where \nabla^2 V = 0, this means that the function V is harmonic, or has a constant value at every point. This can be applied to various physical phenomena such as heat distribution, fluid flow, and electrostatics.

On the other hand, in Poisson's equation where \nabla^2 V = some function, the value of the function V is influenced by an external source or sink, represented by the function on the right side of the equation. This can be used to model situations where there is a varying source of a physical quantity, such as electric charge or mass.

In summary, LaPlace and Poisson's equations are fundamental tools in mathematical physics that help us understand and predict the behavior of scalar functions in different physical systems.