Question

what's a laplace operator?

Answer 1

A very useful tool in differential equations:
http://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2010/video-lectures/lecture-19-introduction-to-the-laplace-transform/

Answer 2

I know laplace transform, but is laplacian similar?

Answer 3

oh the Laplacian. Completely different.

Answer 4

does that have anything to do with gradient of vector field?

Answer 5

The Laplacian is a differential operator on scalar fields,
\[ \phi : \mathbb{R}^n \rightarrow \mathbb{R} \]
defined by
\[ \Delta f = \nabla^2 f = \sum_{i=1}^n \frac{\partial^2 \phi}{\partial x_i^2} \]

Answer 6

Both the \( \Delta \) and \( \nabla^2 \) are standard notation for it.

Answer 7

It's related to grad as
\[ \nabla^2 \phi = div(grad \ \phi) = \nabla . \nabla \phi \]

Answer 8

but what does the tell us physically?

Answer 9

It tells us something above the the density or diffusion of a quantity. It comes up all over the place.

I find it tricky to interpret by itself. But if for example
\[ \Delta f = f_t \]
then we have wave motion.

If
\[ \Delta f = 0 \]
then we something like gravitational fields.

The wikipedia article does a good job pointing out a few examples. You need to work with it a bit to get a feel for it. It's not as easy to interpret as grad, div or curl.

Answer 10

yes, I am reading this article on wiki, and laplacian keep coming up

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

thanks for the help