Problem interpreting the divergence result

[Karol]

Homework Statement

I take the divergence of the function:
V=x^2 \boldsymbol{\hat {x}}+3xz^2\boldsymbol{\hat {y}}-2xz\boldsymbol{\hat {z}}
And get zero. the answer doesn't make sense, since i expect to get a zero divergence only for a function that looks like the one in the drawing attached.

The Attempt at a Solution

\nabla \cdot V=2x+0-2x=0
I test to see whether the function V behaves like in the drawing.
The function V at an arbitrary point, (1,1,1) is:
V_(1,1,1)=1\boldsymbol{\hat {x}}+3\boldsymbol{\hat {y}}-2\boldsymbol{\hat {z}}
And, on another arbitrary point, let's say (2,2,2):
V_(2,2,2)=4\boldsymbol{\hat {x}}+6\boldsymbol{\hat {y}}-6\boldsymbol{\hat {z}}
The vectors are different, not the same like i expected.

 

[tiny-tim]

Hi Karol!

Think of a vector field V as showing the velocity of a fluid at each point.

If the fluid has constant density ("incompressible"), then divV = 0.

divV = 0 is another way of saying that, for a fixed region of space, the flow in equals the flow out.

Alternatively, if we look at a fixed mass of fluid, divV = 0 says that that fixed mass may change shape, but it will always have the same volume.

 

[Karol]

Does Div V=0 also mean that the vector V at point B must have the same length as the vector at point A, but may change it's direction, like in the new drawing attached?
But the function V doesn't even support this state since the magnitude of the two vectors i have calculated, at points (1,1,1) and (2,2,2) isn't the same!
I see no regularity in V, but, of course, i cannot know from inspecting only 2 points

 

[tiny-tim]

Hi Karol!

(just got up :zzz:)

Does Div V=0 also mean that the vector V at point B must have the same length as the vector at point A …

No.

Imagine V is the velocity of water flowing through a pipe that's wider near the end …

the water will slow down, but it won't change density …

|V| is less, but divV = 0 …

A disc of water will get wider but thinner … same volume, different shape

If you drew arrows to show the flow, you would have to draw shorter arrows, but more of them.​

 

[Karol]

Thanks, Tim, but the divergence is in a point, and you are talking about volumes of water

 

[tiny-tim]

Thanks, Tim, but the divergence is in a point, and you are talking about volumes of water

Agreed, but div is a derivative, so …

i] I'm allowed to talk about an infinitesimal volume

ii] I'm allowed to integrate over a finite volume

(also, you are talking about divV = 0 everywhere: so divV = 0 over a volume)