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I understand that, by definition, a vector field is irrotational if the rotation is zero, but what does this intuitively mean?

I have an idea of what it could physically be, which I've concluded by reading various things online, but I'm not sure if it's completely correct:

A gravitational field is an irrotational vector field (and so the rotation will be zero). This also means that the field is conservative (no matter what path you follow, the net work will always be the same), this is approximately how it is defined in my coursebook, though in there it's pure mathematically.
Intuitively this would mean that all vectors in the field are in the same direction, just with different starting points and magnitudes.

Also: What would be a physical example of a non-irrotational (rotational?) vector field?

Thanks in advance!

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Edward Purcell's undergraduate book on electromagnetism does a good job building intuitions about vector fields.

An example of a non-irrotational vector field that you might think about is the current flowing in a wide river. The water flows faster in the middle of the river. Near the banks, it flows more slowly. As you move from the bank toward the center, the velocity increases. The velocity vectors in the flow are increasing in a direction perpendicular to their length. This is a non-irrotational vector field.

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The simplest, most obvious, and oldest example of a non-irrotational field (the technical term for a field with no irrotational component is a solenoidal field) is a magnetic field. A magnetic compass finds geomagnetic north because the Earth's magnetic field causes the metal needle to rotate until it is aligned.

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Another intuition is if you think of a small enough neighborhood of a point in $\mathbb{R}^3$. By the action of the force field $\vec{F}$, everyone in the neighborhood will rotate around an axis... which has it's direction given by the vector $\operatorname{curl \hspace{1pt}} \vec{F}$, calculated at the initial point.

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Simplest idea that I came across for imagining the rotational field is to put a wheel in it (a hydro-wheel). Put the hydro wheel in the field and see if it rotates. You should be careful about the direction though. In an example of river, the axis of your wheel should be in the direction of river flow. If the wheel starts rotating, then the river flow vector field is not non-rotational.

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  • $\begingroup$ I don't believe this uses mathematically supported reasoning to answer the question, which is about a vector field (not a river). $\endgroup$
    – hardmath
    Sep 6, 2015 at 19:02

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