What does divergence tell us about vector fields?

[Isaac0427]

Hi guys! I am using many different sources to self teach myself about divergence. I understand it, however there is one thing that is confusing me. For example, a divergence of 0 could mean that i, j, and k don't change at all, or it could mean that i changes by 1, j by -1, and k by zero (or many other combinations). Just telling me that divF=0 is very vague. Now, could you tell me divX=1, divY=-1 and divZ=0? I'm not sure if you can have divergences of particular variables, so I may be completely off on this. Is there any other way to express a divergence to make it more clear? Thanks in advance.

 

[Dale]

Now, could you tell me divX=1, divY=-1 and divZ=0?

No. The divergence of a vector field is a scalar field. What you are describing would be another vector field.

In physics the divergence represents the "source" of a vector field. The field you describe turns at that spot rather than having a source there. That is why the divergence is 0.

 

[Isaac0427]

No. The divergence of a vector field is a scalar field. What you are describing would be another vector field.

In physics the divergence represents the "source" of a vector field. The field you describe turns at that spot rather than having a source there. That is why the divergence is 0.

Then how can divergence be explained to be less vague?

 

[Dale]

It isn't vague. It is clearly defined and useful.

 

[Isaac0427]

It isn't vague. It is clearly defined and useful.

As I have seen it, divF=0 can mean one of many things. Am I missing something?

 

[Isaac0427]

For example, F=xi-yj and F=i+j both have a divergence of zero, however they look very different.

 

[Dale]

As I have seen it, divF=0 can mean one of many things. Am I missing something?

It does not mean one of many things. It means only one thing. A zero divergence means that the vector field has no source. That is it.

For example, F=xi-yj and F=i+j both have a divergence of zero, however they look very different.

They may look very different, but they both share the property of being source-free.

A stop sign and an apple may be very different and yet share the fact that they are red. The fact that an apple is different from a stop sign does not mean that red is vague.

 

[boneh3ad]

Divergence doesn't tell you how a vector field looks as a whole. It tells you specific things about how it behaves. In the case of your two examples there, the two fields exhibit the same behavior in that neither exhibits any source/sink behavior, even though the fields as a whole look quite different.

 

[boneh3ad]

It does not mean one of many things. It means only one thing. A zero divergence means that the vector field has no source. That is it.

I suppose I'll expand on this slightly because I think I get what the OP is getting at here. Divergence finds utility in many fields concerning the study of vector fields, and can mean a variety of things in different fields. However, these are all related by the fact that the divergence measures the degree to which a field acts like a source, and each possible physical interpretation can be related back to that concept, so really it only means one specific thing that can have a number of different consequences in different fields.

 

[Isaac0427]

Ok, can someone explain a source to me.

 

[anorlunda]

Ok, can someone explain a source to me.

A sink and a source.

 

[Isaac0427]

A sink and a source.

I mean in terms of divergence and of a vector field.

 

[anorlunda]

I mean in terms of divergence and of a vector field.

Lift your eyes off the paper and look around you. You are surrounded by fields in real life. Think of what is needed to describe them mathematically. If you had to write the vector equations to describe those water current and wind fields, what properties must they have?

 

[Isaac0427]

Lift your eyes off the paper and look around you. You are surrounded by fields in real life. Think of what is needed to describe them mathematically. If you had to write the vector equations to describe those water current and wind fields, what properties must they have?

Ok, so I have one last question. What is the difference between a divergence of one and a divergence of 2.

 

[boneh3ad]

The magnitude of the divergence represents the strength of the source (positive) or sink (negative).

 

[Isaac0427]

Hold on, I think I'm getting this better. Is this correct: a source goes away from the origin and a sink goes towards the origin, and the higher the absolute value of the divergence, the more drastically the field's magnitudes change.

 

[anorlunda]

Hold on, I think I'm getting this better. Is this correct: a source goes away from the origin and a sink goes towards the origin, and the higher the absolute value of the divergence, the more drastically the field's magnitudes change.

Yes except replace origin with "any point"

 

[Isaac0427]

The magnitude of the divergence represents the strength of the source (positive) or sink (negative).

Is it the strength or how the strength changes, because I know that the del operator represents change in something.

 

[boneh3ad]

Divergence essentially represents the amount of something carried by the vector field that crosses an imaginary boundary. It is the strength of the source/sink, i.e. the rate at which that something crosses that boundary.

 

[gleem]

Take a look at this web site
http://mathinsight.org/divergence_idea

 

[jtbell]

Consider the electric field, for which $\vec \nabla \cdot \vec E = \rho / \epsilon_0$ (one of Maxwell's equations), where $\rho$ is the charge density.

For a point charge, $\rho = 0$ everywhere, except at the location of the charge. Therefore $\vec \nabla \cdot \vec E = 0$ everywhere, except at the location of the charge, which is the source of the field.

For a non-point charge, i.e. an object with charge distributed throughout it, each point inside the object acts as a source for the field, and $\vec \nabla \cdot \vec E \ne 0$ inside the object. Outside the object, where there is no charge, $\vec \nabla \cdot \vec E = 0$.

 

[Dale]

Ok, can someone explain a source to me.

This vector field has one point source and one point sink. Can you visually spot them. The divergence is zero everywhere except at two points. At the source point it is positive and at the sink point it is negative.