Divergence of vector field help

In summary, the concept of divergence of a vector field can be understood by looking at the flux and the direction of the field lines. In some cases, the divergence can be positive or negative depending on the behavior of the field.
  • #1
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Hey guys!

So I've been trying to get my head around Divergence of a vector field. I do get the general idea, however I thought of a hypothetical situation I can't get my head around. Look at the second vector field on this page, http://mathinsight.org/divergence_idea

it has a negative divergence. It makes sense because if you take a circular surface and measure the flux, there is more flux going into the sphere then going out. Also you can see as a whole the field is compressing towards the origin. However what happens if you take the same vector field, but the field lines are shorter on the outside and get longer towards the origin?

This means if I take a surface and measure the flux, more flux is going out than in because the field lines are longer towards the origin. This suggests a positive divergence, however looking at the field as a whole its clear the field is compressing towards the origin at an accelerating rate so the divergence should be negative?

Thanks!
 
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  • #2
In your case, the divergence would be positive at the surface and negative at the center. The divergence doesn't have to have the same value everywhere.
 

1. What is divergence of a vector field?

The divergence of a vector field is a measure of the net flow of the field at a given point. It represents how much the vector field is “spreading out” or “converging” at that point.

2. How is divergence calculated?

Divergence is calculated by taking the dot product of the vector field with the del operator (∇) and then taking the result's divergence at a specific point.

3. What does a positive divergence value indicate?

A positive divergence value indicates that the vector field is spreading out or diverging at that point. This means that there is a net outflow of the field from that point.

4. How is divergence useful in physics and engineering?

Divergence is useful in physics and engineering as it helps to describe the flow of a field, such as fluid flow or electric flux, and can be used to calculate important quantities like flow rate and charge density.

5. What is the relationship between divergence and the concept of a source or sink?

A source or sink in a vector field refers to a point where the field either originates from (source) or converges towards (sink). The magnitude of the divergence at a source or sink point is directly proportional to the strength of the source or sink.

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