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What does Laplace operator represent?

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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?

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Interpretations of the Laplacian (1 comment)

Comments on What does Laplace operator represent?

Interpretations of the Laplacian
Derek Elkins‭ wrote 11 months ago:

It doesn't make sense to say the "Laplace operator represents waves" any more than it would make sense to say the second partial derivative of time does. You could say that the solutions to that differential equations] models waves, but this isn't a property of the operator. (Admittedly, there is a close connection. The elements in the kernel of the Laplacian in $n+1$ Minkowski space are waves. That is, the wave equation in Minkowski space looks like $\nabla^2 f = 0$.)

There are decent "interpretations" here (https://physics.stackexchange.com/questions/20714/laplace-operators-interpretation) and here (https://math.stackexchange.com/questions/50274/intuitive-interpretation-of-the-laplacian-operator) that give an idea of what the Laplacian "measures". Google returns others. That said, as my example above illustrates, these interpretations may not always be straightforwardly applicable.

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