[To expand on Wojowu's [comment](https://mathoverflow.net/questions/416910/why-are-solenoidal-fields-called-solenoidal#comment1069786_416910).]

**Q:** *"Why the description of a divergence-free field as solenoidal? I expect that this name had historical origins but its unlikely that it was so named without some link to some aspect of what is generally meant by a solenoid."*

**A:** The name *solenoid* for a helical coil was invented by Ampère (1823), who is quoted as follows in <A HREF="https://en.wikipedia.org/wiki/Solenoid">Wikipedia:</A>

> l'assemblage de tous les circuits qui l'entourent,
> assemblage auquel j'ai donné le nom de solénoïde électro-dynamique, du
> mot grec σωληνοειδὴς, dont la signification exprime précisement ce qui
> a la forme d'un canal, c'est-à-dire la surface de cette forme sur
> laquelle se trouvent tous les circuits.

<sub>the assembly of all the circuits around it, to which I have given the name of electro-dynamic solenoid, from the Greek word σωληνοειδὴς, the meaning of which expresses precisely what has the shape of a channel, that is to say the surface of this form on which all the circuits are located.</sub>

The magnetic field lines created by a solenoid are divergence free, which motivates the general name "solenoidal" for a divergence free field; it might be possible to locate some early incidences of this use, but the link to Ampère's solenoid seems beyond debate.

<IMG SRC="https://upload.wikimedia.org/wikipedia/commons/f/f5/Finite_Length_Solenoid_field_radius_1_length_1.jpg" WIDTH="300"/>