What does Laplace operator represent?
I was wondering what's the physical meaning of Laplace operator. $\vec\nabla$ actually represent a field. I had seen that Laplace operator is written as $\vec \nabla \cdot \vec\nabla=\nabla^2=\Delta$ So Laplace operator is scalar quantity. I had found usage of Laplace operator in a book and Wiki-page.
I think the laplace operator is also representing "field" but it works like Wave. In a book I had seen that $$\nabla^2\vec A=\frac{1}{v^2}\frac{\partial^2 \vec A}{\partial t^2}$$
The above equation is representing wave. So does that mean Laplace operator represents wave?
1 answer
Laplacian acts like Divergence but not completely. If you take a function (called $\vec{A}$) and write that laplacian of that function is $0$ than it will be flat space.
$$\nabla^2\vec{A}=0$$
But if you write Laplacian like this :
$$\Delta \vec{A}=\text{c}\frac{\partial^2 \vec A}{\partial t^2}$$
than it will represent a wave.
Divergence of gradient actually means "changes in field with respect to that field".
In short we can say that Laplacian represent curvature or stress of the field.
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