_____
->| -->| ---> ---->
->| -->| ---> ---->
->| -->| ---> ---->
->| -->| ---> ---->
====
^^ Pretend that is a box around the arrows.
Lets look at how much is coming in vs going out of the box. Let -> indicate a unit vector. (So --> has magnitude 2, ---> has magnitude 3, etc.)...
The higher the magnitude of the divergence, the more they diverge. . .
Let's say that those arrows are created by the vector function -- since I don't know what function you actually used--: (and now you must forgive me for lack of knowing TeX!)
f = (ax, ay)
The divergence of f is
div f =...
Well that's a little trickier. Here's wikipedia's explanation of the laplacian: "The Laplacian Δƒ(p) of a function ƒ at a point p, up to a constant depending on the dimension, is the rate at which the average value of ƒ over spheres centered at p, deviates from ƒ(p) as the radius of the sphere...
The laplacian acts on a scalar function and returns a scalar function. It is the divergence of the gradient. The gradient of the divergence would act on a vector function and return a vector function.
If you have a scalar function that gives the elevation at different points on a mountain...
I wrote a program that uses the FEM to approximate a solution to the heat conduction equation. I was lazy and wanted to test it, so I only allowed Neumann boundary conditions (I will program in the Dirichlet conditions and the source terms later).
When I input low values for the heat flux, I...