This question is a bit long, the next two paragraphs give some context, but you can skip it. Thank you.
I have seen many different explanations for the meaning of curl, or what exactly does it measures, but so far none of those gives a truly unambiguous, rigorous, definitive and clear explanation that clear all my doubts. Often it is mentioned that curl measures "the tendency of the vector field to rotate" or "curl measures local rotation", without mentioning what exactly does "local" mean, or what is "tendency".
There is no mention of what exactly is rotating about what. And what exactly is the measurement of the rotation, is it "angular speed"? In which case, is it measured in radians/unit time? But this seems very arbitrary, and does not feel right, as mathematical concept should not rely on a physical concept.
So, I found this to give the most sense. But I still have lots of doubts regarding the explanation.
Here is what I understood from reading it; along with some hypothesis I made (which may be wrong), which I believe is necessary to make the entire reasoning logically consistent. Please verify if my understanding is correct, help to correct any mistake in the assumptions/hypothesis I made. And most importantly, the following is written in natural language, which is not precise, but I lack all of the analysis machinery to spell all things out in precise mathematical terms. Please help to rephrase the story using fully rigorous analysis.
- Suppose we are dealing with 3 dimensional space. Given a vector field F, curl F is also a vector field, such that at each point (x, y, z), curl F measures this:
draw a sphere centered at (x, y, z) with infinitely small radius, the sphere is infinitely small but not a singularity, so there are other points in the interior other than (x, y, z). F will give the direction and magnitude of the vector at each of the interior points. Then curl F gives 2x the total average instantaneous "angular velocity" of all the interior points about (x, y, z). It's not a real angular velocity as it does not carry the same physical units or meaning. It would be more precise to say it's 2x the total average instantaneous " ratio between the length of tangential component of vector to the radius " of all the interior points about (x, y, z)
The rotation of the points as defined in 1 can be broken down into 3 orthogonal components, for example in 3-D euclidean coordinate system the x, y and z components. Hence it is valid to compute individual components of curl F separately in each of the three orthogonal planes.
When computing one of the components (say z component in the x-y plane). Because we are only concerned with a sphere with infinitely small radius, we can regard the sphere as just a plane disc. As each of the level plane of the small sphere will have the same average "angular velocity".
Similarly, when computing the average rotation of the disc, because we are letting the disc's radius tending to zero, so every ring on that disc will again have the same average "angular velocity". Thus, it is valid to just consider a single ring around the point (x, y, z). Hence the computation simply involves calculating the circulation of F around (x, y, z) with infinitely small radius. Hence the analogy of a small "paddle wheel" in the linked notes.
But there is a problem here, if we apply the same argument in 3 and 4, we can say that the ring will shrink to a point, so that the value of the vector field F is the same at all points. But then this means that there will be no rotation, and the curl of any field is the zero vector. This is obviously wrong. But why is it valid to apply 3 and 4, but not 5? I need an analytic proof.
Also, why should we be concerned with a small sphere, but not a small enclosed shape of any sort? A cube, a tetrahedron or some irregular shape? My hypothesis is that because for every closed shape one can come up with, there will be two spheres that bound the shape from inside and outside. If we can show that as the radii tend to zero the average "angular velocity" of all interior points in the two spheres are the same, then whatever shape one comes up with must also have the same average "angular velocity".
with regard to 3-6, there seems to be a sense of when boundary shapes are "topologically" equivalent when considering limits. When or when not can someone "simplify" the shape to another form. Is there a formal branch of mathematics that deal with this, and gives the precise analysis tool to reason about this?
the term "instantaneous" is important in "total average instantaneous angular velocity of all the interior points about (x, y, z)". Because, this way we will not be concerning ourselves with physics. Since, the outcome of filling the small sphere with a elastics/inelastic rigid-body/fluid will be different. If it was a rigid body, with perfect in-elasticity, then in general the object will disintegrate as F does not guarantee that all points will always have the same relative distance and orientation after some movement specified by the vector field. If it was a elastic rigid body, then some parts of the object could be compressed, and some parts will be stretched out. So, we are only concerned (on a more abstract level) about all the points enclosed by the sphere we draw, nothing to do with physics.
