What function in PHYSICS that has ZERO DIVERGENCE and ZERO CURL everywhere... besides CONSTANT function?

Question

Can you give any physical example of that kind of vector function?

Answer 1

F(x,y,z) = sqrt(x^2 + y^2 + z^2)r

That is supposed to be an r-hat. Where r-hat is the unit vector pointing towards the origin.

Any uniform vector field should have zero divergence and zero curl everywhere. By the way, a function that has zero divergence and zero curl is said to be a "Laplacian vector field."

Answer 2

selket, the divergence of your example:

F(x,y,z) = sqrt(x^2 + y^2 + z^2)r

(where r is the radial unit vector) doesn't appear to be zero.

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Any vector function with zero curl must be the gradient of some scalar field Phi(x) and the condition of zero divergence gives the additional condition (Laplace equation):

Del^2 Phi(x) = 0

So Div V = Curl V = 0, if and only if V is the gradient of a harmonic function (a scalar field satisfying Laplace's equation).

Any everywhere harmonic function which goes to 0 at infinity must be a constant, so *physical* vector fields are curl and divergence free *only* outside of sources, but never everywhere. An example is the electric field due to a point charge:

V(x,y,z) = (x/r^3,y/r^3,z/r^3)

It's divergence and curl free everywhere except the origin (i.e. the source point). All static electric and magnetic fields are like this.

A simple (non-physical) example of vector field with zero divergence and curl *everywhere* is:

V(x,y,z) = (x,y,-2z)

Answer 3

If the field is conservative the divergence is equal to zero.Its simple. for example water flow in a pipe when the divergence is zero it means what go into to the pipe comes out of the pipe.