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I am reposting a thread from "physics stack exchange" : I was wondering - Why do we use the word "field" to describe a vector field? i.e., a field is "an expanse of open or cleared ground, especially a piece of land suitable or used for pasture or tillage" (from dictionary.com). So, why historically this was the name chosen for the description of a function $F:\mathbb{R}^n \to \mathbb{R}^n$?

Thank you!

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  • $\begingroup$ "Historically" this is the ultimate generalization , so it has to have some prehistory: a field was a piece of land most likely belonging to someone, that is, some entity exercised a force - ownership - over it. Trying to avoid actions at distance fields were introduced in physics for this analogy; originally it was a "force field", later conceived as a vector field. $\endgroup$
    – sand1
    Jun 16, 2021 at 16:51
  • $\begingroup$ Isn't a Field in maths a collection f 'entities; which have certain behaviours and properties under some operation. Fields are associative and commutative, and have identity elements and inverses. Forex. R is a Field under + and x; Z is a field under + but not x. I always thought a vector field was a field in tat sense $\endgroup$ Jun 16, 2021 at 17:06
  • $\begingroup$ Thank you both for your suggestions! $\endgroup$
    – Dr. John
    Jun 18, 2021 at 6:49
  • $\begingroup$ I would be surprised if the English terminology was not copied from the German. $\endgroup$ Jun 18, 2021 at 22:40
  • $\begingroup$ Field (algebra) is an article in wikipedia with a historical section; See also jeff560.tripod.com/f.html "E. Moore (1862-1932) was apparently the first person to use the English word field". The algebraic sense appears to be totally disconnected from the physical one. $\endgroup$
    – sand1
    Jun 20, 2021 at 20:24

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From Mathword "Earliest Known Uses of Some of the Words of Mathematics"

VECTOR FIELD is found in 1905 in “The Present Problems of Geometry” by Dr. Edward Kasner in Congress of Arts and Science, Universal Exposition, St. Louis, 1904: “The vector field deserves to be introduced as a standard form into geometry.”

We could investigate whether this was before or after the corresponding terms in physics, like "magnetic field".

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  • $\begingroup$ Thank you for your input! $\endgroup$
    – Dr. John
    Jun 18, 2021 at 6:50
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Consider a magnet. By playing around with it for a short time, you'll observe that the magnet exerts an influence on other magnetic objects, which is stronger when they're closer. This influence is outside the physical body of the magnet itself - it extends into the space around it, whether that space is occupied or not. We can form a concept of the magnetic field, and assign a quantity of some kind to different points in space, representing the strength and direction of the attraction.

In a similar way, by looking closely at the sun and the planets, we can theorize that they exert a force on each other, which extends out into the empty space between. (We might, or might not, imagine that the force is carried by some substance called "the aether", but in any case we have a way to assign a quantity to each point in space.) This is a gravitational field.

Again, considering water flowing in a pipe, you can assign a speed and direction to each tiny bit of water, as Leonhard Euler did in his work on hydrodynamics (Mechanica sive motus scientia analytice exposita, 1736).

The mathematical formulation of a vector field is a way of giving formal structure to these notions. The term field was used before the formalization of vector field, scalar field, etc., to talk about the area of influence around a body. It took some time to develop the unified mathematical theory, and to sort out various competing notions of what was going on physically - so people like Kepler or Newton might not have had the same mental picture as we do in modern times. But there was certainly some understanding of bodies exerting a sort of influence on the surrounding space, even if a particular bit of that space is not occupied.

One of the critical papers for this development is Maxwell's paper On physical lines of force (1861). He spends some time talking about scattering iron filings near a magnet to see that they form lines, and trying to justify from similar observations that it is right to model magnetism by giving a magnitude and a direction (i.e., a vector) for each point surrounding the magnetic object. Later in the paper, he mentions "a field of magnetic force such as that of the earth", and generally "the magnetic field". What he's doing here is giving a new mathematical intepretation (in modern words, a vector field) of the intuitive concept that magnets have a field of influence around them.

The English word field is suitable for the intuitive concept, because of its sense as a marked-off physical area that's dedicated for some common reason - in agriculture, or as a battlefield, that kind of thing. Its abstract meanings, as in "a field of study", are also well-established. So it's a plausible term for the area of influence. In other European languages we use similar words; German Feld, French champ, Italian campo, Russian поле.

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