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How useful is divergence?

  1. Aug 20, 2015 #1
    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.
     
  2. jcsd
  3. Aug 20, 2015 #2

    Dale

    Staff: Mentor

    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.
     
  4. Aug 20, 2015 #3
    Then how can divergence be explained to be less vague?
     
  5. Aug 20, 2015 #4

    Dale

    Staff: Mentor

    It isn't vague. It is clearly defined and useful.
     
  6. Aug 20, 2015 #5
    As I have seen it, divF=0 can mean one of many things. Am I missing something?
     
  7. Aug 20, 2015 #6
    For example, F=xi-yj and F=i+j both have a divergence of zero, however they look very different.
     
  8. Aug 20, 2015 #7

    Dale

    Staff: Mentor

    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.

    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.
     
  9. Aug 20, 2015 #8

    boneh3ad

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    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.
     
  10. Aug 20, 2015 #9

    boneh3ad

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    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.
     
  11. Aug 21, 2015 #10
    Ok, can someone explain a source to me.
     
  12. Aug 21, 2015 #11

    anorlunda

    Staff: Mentor

    A sink and a source.

    the-whirlpool.jpg

    5467681388_a37fed0990_b.jpg
     
  13. Aug 21, 2015 #12
    I mean in terms of divergence and of a vector field.
     
  14. Aug 21, 2015 #13

    anorlunda

    Staff: Mentor

    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?
     
  15. Aug 21, 2015 #14
    Ok, so I have one last question. What is the difference between a divergence of one and a divergence of 2.
     
  16. Aug 21, 2015 #15

    boneh3ad

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    The magnitude of the divergence represents the strength of the source (positive) or sink (negative).
     
  17. Aug 21, 2015 #16
    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.
     
  18. Aug 21, 2015 #17

    anorlunda

    Staff: Mentor

    Yes except replace origin with "any point"
     
  19. Aug 21, 2015 #18
    Is it the strength or how the strength changes, because I know that the del operator represents change in something.
     
  20. Aug 21, 2015 #19

    boneh3ad

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    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.
     
  21. Aug 21, 2015 #20
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