vector concepts of divergence, curl and gradient

Discussion in 'Math' started by PG1995, Sep 25, 2012.

  1. PG1995

    Thread Starter Active Member

    Apr 15, 2011
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    Hi

    These days I'm learning about vector concepts of divergence, curl and gradient on a basic level.

    Q1:
    I was reading this Wikipedia article on the curl where it was written:

    As the article says:

    The curl gives us a direction of rotation and length/magnitude of that vector of rotation. What does the line in red mean in the quoted text above? Please help me with it. Thank you.


    Q2:
    Another article I was reading says:

    I was thinking that in case of electric field the divergence is equivalent to magnitude of flux density. The magnitude of flux density is given as:

    [​IMG]

    But I think I was wrong. Divergence = Flux/volume. The divergence is flux per unit volume but flux density is flux passing per unit area. What do you say? Please let me know. Thanks.

    Regards
    PG
     
    Last edited: Sep 27, 2012
  2. WBahn

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    Mar 31, 2012
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    Imagine a ball is suspended in the are in front of you. The motion you would see would be very different if the axis of rotation were vertical, versus horizontal left-to-right, versus horizontal going straight away from you. That's the 'direction' part. The magnitude is the (angular) speed, namely how fast the ball is spinning.

    There is a classic book called, "Div, Grad, Curl and All That" by Schey that many people rave about. I didn't find it overly helpful, but a lot of people have. What I found to be an absolutely superb discussion of these concepts was Chap 1 of "Electromagnetic Fields" by Wangsness.
     
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  3. nsaspook

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  4. PG1995

    Thread Starter Active Member

    Apr 15, 2011
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    Thank you, WBahn, nsaspook.

    Thanks a lot for letting me know about that book. I understand that direction part of the curl but I don't get it where the Wikipedia article says that the curl's magnitude is half of the angular speed? Why is the magnitude half of the angular speed? So, please help me if possible. Thank you.

    Regards
    PG
     
    Last edited: Sep 26, 2012
  5. amilton542

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  6. nsaspook

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    PG1995 likes this.
  7. PG1995

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    Apr 15, 2011
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    Thank you for the links, nsaspook.

    In this video the presenter calculates the divergence to be 1/2 or 0.5. To me, this means that the divergence is positive but the velocity of particles going to the right is decreasing. But check out the vector field drawn in the view, it shows that the velocity of the particles increases as they move to the right. In my view, if the divergence were, say 2, then this would mean that the velocity of the particles gets increased by factor of '2' as they move to the right. Do I have it right? Please let me know. Thank you.

    Regards
    PG
     
  8. nsaspook

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  9. PG1995

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    Apr 15, 2011
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    Thank you, nsaspook.

    So, you say that what is said in the video is correct. I will use the example in the video for my query below and divergence is "1/2" as was calculated in that video.

    A derivative is rate of change or simply a slope where slope=rise/run=(corresponding change in y's value)/(change in x's value). The divergence is also rate of change and in that video divergence was calculated to be 1/2. What does it mean in that context? Does it mean that the particles enter an infinitesimal volume with 1 unit/sec velocity and leave the same volume with double the velocity, or the particles enter the volume with 2 unit/sec velocity and leave the volume with half the velocity? Now please help me with it. Thanks.

    Regards
    PG
     
  10. nsaspook

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    I don't mind helping if you're stuck as it's brings back into memory things I studied a long time ago but you really need to put some effort into visualizing a mental picture of what's happening. I was never good at vector math as a means to express the structure of fields but I could create a mental 3D picture of something and animate it in my head because I subconsciously understood the rules from daily observations of objects in motion.

    So put down your pencil for a moment, close your eyes and think visually about the problem.
     
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