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I would like to ask if a vector field is mathematical field which is defined to be a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.

I can see that a vector field satisfies addition, subtraction, multiplication but I am not sure about division. Is this just terminology or is a vector field a field in the mathematical sense?

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    $\begingroup$ No, absolutely not. It is just a matter of giving the same name to different things. $\endgroup$ Jan 24, 2018 at 17:58
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    $\begingroup$ No. In other languages, e.g. in Polish, these object have different names: ciało (of numbers) and pole (of vectors). $\endgroup$ Jan 24, 2018 at 18:00
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    $\begingroup$ A vector field in general does not have multiplication in the sense of a field. Vector field has scalar multiplication (e.g. $F \times V \to V$), field has multiplication of elements (e.g. that would be $V \times V \to V$). $\endgroup$ Jan 24, 2018 at 18:01
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    $\begingroup$ In French as well: 'corps' vs 'champ'. $\endgroup$ Jan 24, 2018 at 18:03
  • $\begingroup$ Thanks all for the clarification and interesting linguistic information. $\endgroup$
    – SAMCRO
    Jan 24, 2018 at 18:19

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No, these are distinct concepts. A field (in Algebra) is what you think a field is. But a vector field is, roughly speaking, an assignment of a vector to each point in a space.

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No. Vector field and field (math) are quite different objects. And it quickly becomes a little trouble if we venture into physics, because in physics, a vector field is simply called a field, creating potential confusion as to what "field" means without further qualifier.

In general, field (physics) means a section of a vector bundle over a base space (typically the spacetime), of which one primary example is a vector field. Since physics heavily involves mathematics, and theoretic physics is almost entirely mathematics, it is quite unfortunate to have two unrelated fundamental concepts with the same name.

I still remember how shocked I was many many years ago when I realized I could not easily distinguish "场论" (field theory (physics)) and "域论" (field theory (math))—two clearly different things in Chinese—in English. Context becomes important, or specific qualifiers need to be added.

In many languages, the two "fields" correspond to distinct words. The following is compiled from wikipedia. (Italian and Russian also use the same word for two fields.)

English: field (math) / field (physics) / vector field

German: Körper / Feld / Vektorfeld

French: corps / champ / champ de vecteurs

Italian: campo / campo / campo vettoriale

Russian: поле / поле / векторное поле

Spanish: cuerpo / campo / campo vectorial

Chinese: 域 / 场 / 向量场

Japanese: 体 / 場 / ベクトル場

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I believe by field they are referring to space(manifold), think of a paddy field.

Vector field is assigning a vector to each element of this field.

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A vector field in differential geometry is a smooth section of the tangent bundle. A field in abstact algebra is a commutative ring in which every nonzero element is a unit. Two entirely different notions.

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