Is the Laplace Equation a Visualization of Zero Potential Gradient?

[danong]

Sorry but i have a question regarding Laplace Equation,
say if a potential function P represents the inverse square propotional field,
then how am i going to visualize taking twice partial derivative of P is equal to zero?

Because since grad of P is pointing inward (which looks to me is a sink at the centre of the gravitation point (x0,y0,z0)),

so how am i going to say that divergence of grad(P) is equal to zero? (since the grad(P) the vector is pointing inward),

I mean i have seen some proofs of it which leads to the final conclusion of Laplace Equation,
but how am i going to visualize it in a way that it makes sense that grad(P) is pointing no-where? (since divergence of grad(P) should be zero, which means neither sink or source, but isn't gravitation a sink? ).

 

[BAnders1]

A generalization of Laplace's equation is called Poisson's equation, which shows that the Laplacian (div dot grad) of P (in this case the electric potential) is proportional to the charge density enclosed by the region under consideration. If there is no charge in the region, then the Laplacian is equal to zero.

Similarly, with gravitational potential, Laplace's equation for the gravitational potential field is just a special case of the more general Poisson equation, which says that the Laplacian of
the gravitational potential is proportional to the mass enclosed in the region.

div dot grad P is only zero in regions with no charge/ mass.

 

[danong]

Thanks BAnders1, much appreciated =)