# Just what does it mean when a vector field has 0 divergence?

## Homework Statement

Yeah I've been pondering over that, my book doesn't really do the justice of nailing it down for me.

Does having 0 divergence means having "absolute convergence", like maybe at every point (or at a certain point) all the vectors are pointing towards a point?

Like what does div E = 0 really mean? E is electric field.

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tiny-tim
Homework Helper
hi flyingpig! zero divergence means that the amount going into a region equals the amount coming out

in other words, nothing is lost

so for example the divergence of the density of a fluid is (usually) zero because you can't (unless there's a "source" or "sink") create (or destroy) mass

the electric field is like a fluid, and the field strength E is like its density …

the field can spread out, but the amount of field stays the same, except where there's a charge, which we can regard as a source or sink (depending on sign), continually creating or destroying field hail tiny tim :!!)

hi flyingpig! zero divergence means that the amount going into a region equals the amount coming out

in other words, nothing is lost

so for example the divergence of the density of a fluid is (usually) zero because you can't (unless there's a "source" or "sink") create (or destroy) mass
At a point or everywhere? Sorry I just learned those words from LA and I am not 100% familiar with them yet.

the electric field is like a fluid, and the field strength E is like its density …

the field can spread out, but the amount of field stays the same, except where there's a charge, which we can regard as a source or sink (depending on sign), continually creating or destroying field Does it even make sense to say the divergence of a point?

tiny-tim
Homework Helper
hi flyingpig! At a point or everywhere? Sorry I just learned those words from LA and I am not 100% familiar with them yet.
(what's LA ? )

zero divergence everywhere Linear Algebra.

So does that mean nonconservative fields have a nonzero divergence?

SammyS
Staff Emeritus
Homework Helper
Gold Member
As I recall, finding the divergence of the Electric field at a point is equivalent to finding the flux (of the Electric field) coming out of a closed surface in the limit as the size of the volume enclosed goes to zero. You can look at this as Gauss's Law again! I'm sure you're be happy about that.

As I recall, finding the divergence of the Electric field at a point is equivalent to finding the flux (of the Electric field) coming out of a closed surface in the limit as the size of the volume enclosed goes to zero.

Just started computing easy surface integrals maybe I will get to that

You can look at this as Gauss's Law again! I'm sure you're be happy about that.

## Homework Statement

Yeah I've been pondering over that, my book doesn't really do the justice of nailing it down for me.

Does having 0 divergence means having "absolute convergence", like maybe at every point (or at a certain point) all the vectors are pointing towards a point?

Like what does div E = 0 really mean? E is electric field.
An analogy with water.
You've got a river with water flow. Into a river there's a fish cage.
Water is flowing through the cage, and if a molecule of water enters the cage through the boundary, somewhere else a molecule must exit the cage, (since water is incompressible).
If from the outside you plug into the cage a pipe of water, and you pump water into the cage, then you create water into it, and from the surface of the cage must in addition come out an amount of water equal to what you pump in.

In other words:
$$\iint_{S}\mathbf{v} \cdot \mathbf{n} \ ds = 0$$

where
v is the speed of water through the surface
n is the normal surface vector
S is the surface

Last edited:
SammyS
Staff Emeritus