What is your interpretation of Laplace operator? When evaluating Laplacian of some scalar field at a given point one can get a value. What does this value tell us about the field or it's behaviour in the given spot?
I can grasp the meaning of gradient and divergence. But viewing Laplace operator as divergence of gradient gives me interpretation "sources of gradient" which to be honest doesn't make sense to me.
It seems a bit easier to interpret Laplacian in certain physical situations or to interpret Laplace's equation, that might be a good place to start. Or misleading. I seek an interpretation that would be as universal as gradients interpretation seems to me – applicable, correct and understandable on any scalar field.
Best Answer
The Laplacian measures what you could call the « curvature » or stress of the field. It tells you how much the value of the field differs from its average value taken over the surrounding points. This is because it is the divergence of the gradient..it tells you how much the rate of changes of the field differ from the kind of steady variation you expect in a divergence-free flow.
Look at one dimension: the Laplacian simply is $\partial^2\over\partial x^2$, i.e., the curvature. When this is zero, the function is linear so its value at the centre of any interval is the average of the extremes. In three dimensions, if the Laplacian is zero, the function is harmonic and satisfies the averaging principle. See http://en.wikipedia.org/wiki/Harmonic_function#The_mean_value_property . If not, the Laplacian measures its deviation from this.