I have difficulty in visualizing the divergence of vector fields.

In summary, the divergence of a vector field is a measure of the net flux of a point in fluid mechanics and the total charge within a surface in Gauss' Law. This is calculated using integral theorems and is proportional to the charge density in arbitrary volumes. More information can be found in the link provided.
  • #1
hanson
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0
Hi all. I have difficulty in visualizing the concept of divergence of a vector field. While I have some clue in undertanding, in fluid mechanics, that the divergence of velocity represent the net flux of a point, but I find no clue why the divergence of an electric field measures the charge denity?

Can anyone tell me how to interpret the divergence of a vector field? Please kindly help.
 
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  • #2
I'll try...

Gauss' Law says that the flux of an E field is proportional to the total charge within the surface that you compute the flux.

Using integral theorems, the flux is a volume integral of the divergence. But, the total charge is the volume integral of charge density. This works out for arbitrary volumes so the divergence must be proportional to the charge density.
 
  • #3

1. What is the divergence of a vector field?

The divergence of a vector field is a measure of how much the vector field behaves like a source or a sink at a given point. In other words, it measures the net flow of the vector field out of or into a small region surrounding the point.

2. Why is it difficult to visualize the divergence of a vector field?

The divergence of a vector field is difficult to visualize because it is a three-dimensional concept, and it can vary at different points in the vector field. It is also difficult to visualize because it is a measure of the flow of a vector field, rather than the vectors themselves.

3. How is the divergence of a vector field calculated?

The divergence of a vector field is calculated using a mathematical formula involving partial derivatives. Specifically, it is the sum of the partial derivatives of the vector field with respect to each coordinate direction.

4. What is the physical significance of the divergence of a vector field?

The physical significance of the divergence of a vector field is that it represents the net flow of a vector field out of or into a given point. This can have applications in fluid dynamics, electromagnetism, and other areas of physics and engineering.

5. Can the divergence of a vector field be negative?

Yes, the divergence of a vector field can be negative. This means that the vector field is behaving more like a sink than a source at that point. Conversely, a positive divergence indicates that the vector field is behaving more like a source than a sink at that point.

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