Intuition on divergence and curl

[cgiustini]

Hi,

I'm looking at the following graph, but there are a few things I don't get. For instance:

• curl should always be zero in circles where the field lines are totally straight (right-most figure)
• curl should always be non-zero in circles where the field lines are rotating (center figure in 2nd from right figure)
• divergence should always be zero in circles where field lines around pointed in one general direction - ie there is no field "source" (two left-most figures)

The writing in the picture below seems to contradict them - is my understanding incorrect?

Thanks,
Carlo

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[andrewkirk]

In the first diagram the vector field everywhere except at the origin appears to be $(1,0)$ in polar coordinates, which is $\left(\frac x{\sqrt{x^2+y^2}},\frac y{\sqrt{x^2+y^2}}\right)$ in rectangular coords. What do you get when you calculate the divergence of those at an arbitrary point away from the origin? (I get a nonzero result, whether I do the calc in polar or rectangular coords, but I may have miscalculated)

Intuitively that makes sense because the field lines become farther apart as you move away from the origin, and that 'divergence' process continues no matter how far away you go (although it slows, so as to asymptotically approach zero from above).

 

[D H]

The writing in the picture below seems to contradict them - is my understanding incorrect?

You are correct: Your understanding is incorrect. I'll look at your three items point by point.

Curl should always be zero in circles where the field lines are totally straight (right-most figure)
The figures in the opening post use a fairly common technique where the density of lines represents the strength of the vector field. Note that the density of lines is not constant in the right-most figure. A better way to think of curl is "if one places a test object that is free to rotate about an axis at some point the field (e.g., a waterwheel), would the field make it rotate?" In this case, the increasing field strength to the right would make the test object rotate.

Mathematically, the 2D curl in cartesian coordinates of \vec F = M(x,y)\hat x + N(x,y)\hat y is \nabla\times \vec F = \frac{\partial N}{\partial x}-\frac{\partial M}{\partial y}. The rightmost figure displays something like \vec F = \frac{\hat y}x, where the x-axis points leftward and the origin is somewhere off the right edge of the figure. This has curl -\frac 1 {x^2}, which is non-zero.

Curl should always be non-zero in circles where the field lines are rotating (center figure in 2nd from right figure)
The key difference between the center figure and the second from the right figure is the density of lines. In the center of figure, the density of lines decreases by 1/r, while the density is constant in the the second from the right figure. The center figure portrays a rotating flow that is irrotational. The second from the right figure portrays a rotating flow that is not irrotational, and the rightmost figure portrays a flow that is not rotating but is nonetheless not irrotational.

Divergence should always be zero in circles where field lines around pointed in one general direction - ie there is no field "source" (two left-most figures)
There obviously is a field source in the two leftmost figures. The second to left figure depicts a central radial force field, while the leftmost figure depicts a central radial force field plus a radial force field emanating from the boundary of a circle centered around the central field. What those little circles are trying to depict is whether there is net flux into or out of a closed region. In the leftmost figure, the dashed circle crosses the boundary of the outer source, so the divergence there is nonzero. There are two circles in the second to left figure, one at the center where the net flow is obviously out of the circle (so a nonzero divergence) and another well removed from the center where the net flow is zero (so a zero divergence).

 

[chiro]

Hey cgiustini.

There is a book by the name of Div, Grad, Curl and all that.

https://www.amazon.com/dp/0393925161/?tag=pfamazon01-20

I'd recommend going through the table of contents and sample pages before deciding whether you think it is worth purchasing.