Understanding Divergence & Curl of Vector Fields

In summary: All in all, you got it right. Keep practicing and you'll get better at it. Good luck!In summary, the divergence of a vector field measures the density of field flux flowing out of a volume, while the curl measures the rotation of the field around the three main axes. A divergence of 3 means there are 3 flux lines per unit volume flowing out, while a curl vector of {0,0,2} indicates a rotation about the positive Z axis with a measure of 2. It is possible for two vector fields with the same divergence to have different appearances, as seen in the example of fields 1 and 2. Field 3 also has a positive divergence due to having more
  • #1
UndeniablyRex
12
0
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.
 
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  • #2
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.
 

Attachments

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  • field_2.png
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  • field_3.png
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  • #3
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?
 
  • #4
You are right UndeniablyRex. Also, you'ld find it has no curl because it has no rotation about whatever axis
 

1. What is a vector field?

A vector field is a mathematical concept used to represent the direction and magnitude of a vector at every point in space. It can be visualized as a collection of arrows, each pointing in a specific direction and with a specific length.

2. What is divergence of a vector field?

Divergence of a vector field is a measure of how much the vectors in the field are spreading out or converging at a given point. It is represented by the symbol ∇ · F and can be thought of as the flux or flow of the vector field out of a small region around the point.

3. What is curl of a vector field?

Curl of a vector field is a measure of how much the vectors in the field are rotating or swirling at a given point. It is represented by the symbol ∇ × F and can be thought of as the circulation or rotation of the vector field around the point.

4. How are divergence and curl related?

Divergence and curl are two important properties of vector fields and are related through the fundamental theorem of vector calculus. The divergence of a vector field represents the sources and sinks of the field, while the curl represents the rotation of the field around these sources and sinks.

5. How are divergence and curl used in real-world applications?

Divergence and curl are used in many areas of science and engineering, including fluid dynamics, electromagnetism, and weather forecasting. They are used to model and analyze complex systems and can provide valuable insights into the behavior of these systems.

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