I know that acceleration $\vec{a}$ is not a vector field but $\vec{F}$ is. Can you view the angular velocity $\vec{\omega}$ as a kind of vector field on a rigid body? Can you do the same for the position vector $\vec{r}$ as well?
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2$\begingroup$ What is $\vec F$? Is it the net force on a point particle, or the force per unit volume exerted at a point on an extended object, or something else? $\endgroup$– J. MurrayCommented Nov 22, 2022 at 17:37
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1$\begingroup$ Normally vector fields are mapped for postion-dependent vector quantities like an electrostatic or gravitational force. You could map angular momentum for $dm$ particle in a rigid body at some distance $r$ from the axis of rotation. However, it may not be useful, as finding net angular momentum is always handy in many cases. $\endgroup$– Kshitij KumarCommented Nov 22, 2022 at 19:08
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$\begingroup$ Why is acceleration not a vector field? It's a vector field by the definition I am used to using, which suggests you may be learning a definition that is different than mine. It may be useful to explain why acceleration is not a vector field but force is, for those of us who have a different definition of a vector field. (for reference, the definition I am thinking of is the one used in Wikipedia's page on Vector fields.) $\endgroup$– Cort AmmonCommented Nov 22, 2022 at 20:16
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2$\begingroup$ Acceleration of what? The acceleration of a point particle is not a vector field. The acceleration of a fluid is. $\endgroup$– GhosterCommented Nov 22, 2022 at 21:12
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1 Answer
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In rigid body motion, the angular velocity $\vec\omega$ is a single (pseudo-)vector characterizing the rotational state of the body. You could construct a "field" by assigning every point on the rigid body the same angular velocity vector. However, it would necessarily be a boring field given the assumption of rigid body motion, in the sense that it would be the same for every point on the body. Additionally, I don't see what information you are trying to capture by performing this construction, vs simply saying the body has an angular velocity.
There are scenarios where it would make sense to talk about a velocity field, such as when describing fluid flow (notably not a rigid body). Similarly, it is sometimes interesting to talk about the angular velocity field of the fluid -- although instead of angular velocity, one typically speaks of the vorticity field, which you can think of as being proportional to the mean angular velocity of particles in a fluid element relative to the center of mass of the fluid element.
