Understanding Divergence & Curl of Vector Fields

[UndeniablyRex]

I know how to calculate the divergence and curl of a vector field. What I am lacking is any intuition on what these values mean.
example:
V= {x, y, z}
∇.V = 3
∇xV = {0,0,0}

F={-y, x, 0}
∇.F = 0
∇xF = {0,0,2}

G={0, 3y, 0}
∇.G = 3

I understand that that the divergence is a measure of how much the vector field "spreads out" and the curl measures the circulation, but what does it mean to have a divergence of 3? or a curl vector of {0,0,2}

The last example is merely because it has the same divergence as the first; the graphs look very different(the second doesn't even seem to "diverge"), yet have the same divergence.

Thank you for any help.

 

[mohammed.omar]

Hi UndeniablyRex,

The divergence of a vector field gives the density of field flux flowing out of an infinitesimal volume dV. It is positive for outward flux and negative for inward flux. The curl on the other hand, gives the rotation of the field around the three main axes, taken in the positive sense using the right hand rule.

Lets take your examples for an illustration.

For field 1: V={x,y,z}Shown in attached figure (field_1)
it divergence = 3 this means there should be 3 flux lines per unit volume flowing out of the volume dV.
On the other hand, its curl being 0, means the field has no net rotation about any direction axis.


For field 2: G={-y,x,0}Shown in attached figure (field_2)
It has no net flux, as you see the field lines are all closed on themselves, and it would have a positive. On the other hand, it has a rotation about the positive Z axis, whose measure is 2.

Attached is also your third field F in figure (field_3). You take a look at it and tell me if it makes sense now to you.

 

[UndeniablyRex]

It definitely makes more sense now, thanks.
If I understand it correctly, the reason field_3 has a positive divergence is because if you have a volume, dV, then there would be more lines out of the volume than into it, right?

 

[mohammed.omar]

You are right UndeniablyRex. Also, you'ld find it has no curl because it has no rotation about whatever axis