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:

Intuitive interpretation

Suppose the vector field describes the velocity field of a fluid flow (such as a large tank of liquid or gas) and a small ball is located within the fluid or gas (the centre of the ball being fixed at a certain point). If the ball has a rough surface, the fluid flowing past it will make it rotate. The rotation axis (oriented according to the right hand rule) points in the direction of the curl of the field at the centre of the ball, and the angular speed of the rotation is half the magnitude of the curl at this point.

As the article says:

In vector calculus, the curl is a vector operator that describes the infinitesimal rotation of a 3-dimensional vector field. At every point in the field, the curl of that field is represented by a vector. The attributes of this vector (length and direction) characterize the rotation at that point.

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:

In vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point.

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:

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

 

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.

 

I found this about the similarity between the equations that
govern fluid dynamics and the equations that govern
EM fields.

TORNADOGENESIS “EM-LIKE” TORQUING
https://ams.confex.com/ams/pdfpapers/164116.pdf

 

Thank you, WBahn, nsaspook.

magine 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.

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

 

The book - Inductance: Loop and Partial [ http://www.amazon.co.uk/Inductance-...1888/ref=sr_1_2?ie=UTF8&qid=1348762829&sr=8-2 ] gives some clean-cut examples for this.

Just a thought.

 

(dangerous analogy ahead)
The curl is similar to circulation density.


http://keep2.sjfc.edu/faculty/kgreen/vector/Block4/circ/node9.html

What is the curl? Chris Tisdell
http://www.youtube.com/watch?v=kOl9c2cU3UE

 

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

 

The divergence is a rate of change in density. A positive rate always has the "outgoingness" increasing.

http://betterexplained.com/articles/divergence/

 

The divergence is a rate of change in density. A positive rate always has the "outgoingness" increasing.

http://betterexplained.com/articles/divergence/

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

 

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.