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Wikipedia says:

In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space.1 A vector field in the plane (for instance), can be visualised as: a collection of arrows with a given magnitude and direction, each attached to a point in the plane. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from one point to another point.

Is "assignment" a canonical term in math? What does "assignment" mean here?

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The wording assignment is probably chosen to avoid the technically correct term mapping (or function, morphism,...) because a mapping needs a source (domain) and a target (codomain) and, in the present situation, the latter is a bit awkward to describe: As the vector field on a manifold $M$ assigns to each $x\in M$ an element $F(x)$ of the tangent space $T_xM$ which varies with $x$, a possible target is the union $\bigcup_{x\in M} T_xM$ and in many cases it is even better to take a disjoint union (direct sum) $\bigcup_{x\in M} \{x\} \times T_xM$. This is the tangent bundle of the manifold and indeed an important object -- however this technicality disturbs somehow the simple intuition.

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  • $\begingroup$ Thanks for your answer. Does this kind of function have to be linear? $\endgroup$ – fu DL Jul 9 '19 at 21:41
  • $\begingroup$ @fu DL, the points of the suspect of space/manifold might not be able to be added together with the sum lying in the same thing, so it doesn't really make sense to ask whether the function assigning vectors is linear $\endgroup$ – Mark S. Jul 10 '19 at 1:50
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In such contexts, the terms "assignment/mapping/rule/association" are all synonyms for "function". If you wish to know the precise definition of "function", refer to any respectable book on set theory.

At a working level, a function can be thought of as a triple of information $(f,A,B)$, where we call $f$ the function, $A$ the domain of $f$, and $B$ the codomain/target space of $f$. We like to think of $f$ as a "rule" which tells us "where to send elements of $A$". More explicitly, given an element $x \in A$, the function $f$ "tells" us to send it to a certain element of $B$, which we denote by $f(x)$.

Now, for the definition of vector field. The simplest definition of a vector field (but geometrically not pleasing) is:

1.) Let $A \subset \Bbb{R}^n$ be a subset. Then, a vector field on $A$ is a function $F$ with domain $A$, and target space $\Bbb{R}^n$. (Or more concisely, we might say, a vector field on $A\subset \Bbb{R}^n$ is a function $F:A \to \Bbb{R}^n$).

The above "definition" is often suitable when working with Euclidean spaces, but it hides a lot of the underlying geometry, because it doesn't capture the information of "where the vector starts from". A slightly more geometrically pleasing definition may seem more "abstract", but it is worth understanding.

The simplest "definition" of a vector field is "a collection of vectors which are each attached to certain points in space". This is also how the wikipedia article "defines" it. To formalise this, we need to be precise about what the domain and target space of the "assignment" (i.e function) are.

First, you need to know what is meant by "tangent space". Given a point $p \in \Bbb{R}^n$, define the tangent space of $\Bbb{R}^n$ at $p$ to be \begin{equation} T_p\Bbb{R}^n = \{p\}\times \Bbb{R}^n \end{equation} So, the tangent space at $p$ is basically just regular $\Bbb{R}^n$, but with an extra label "$p$", to remind us that we imagine the elements of $T_p\Bbb{R}^n$ as "vectors emanating from $p$". Now, define the tangent bundle of $\Bbb{R}^n$ to be \begin{align} T\Bbb{R}^n = \bigcup_{p \in \Bbb{R}^n} T_p\Bbb{R}^n \end{align} Now, we can define what we mean by "a vector field on $\Bbb{R}^n$".

2.) A vector field on $\Bbb{R}^n$ is a function $F$ with domain $\Bbb{R}^n$, and target space $T\Bbb{R}^n$ (so $F:\Bbb{R}^n\to T\Bbb{R}^n$) such that for every $p \in \Bbb{R}^n$, it is true that $F(p) \in T_p\Bbb{R}^n$.

So, once again, a vector field is just a function with a certain property. The only "abstract" thing in this definition is being careful about what the domain and target space of $F$ are. Lastly, the condition "for every $p \in \Bbb{R}^n$, $F(p) \in T_p\Bbb{R}^n$" is there to impose our intuitive idea that we want $F(p)$ to be a vector "starting from $p$".


If you really want to, you can absract this even further (once you define manifolds and tangent spaces/bundles) appropriately.

3.) Let $M$ be a (Banach) manifold, and let $A \subset M$. A vector field on $A$ is a function $F:A \to TA$, (where $TA := \bigcup_{p \in A}T_pM$) such that for every $p \in A$, $F(p) \in T_pM$.

(In differential geometry jargon, we might say a vector field on $A$ is a section of the tangent bundle $TA$ of $A$.)

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As the other answers mention, it is a synonym for "function".

To understand how a vector field can be described by a function intuitively, think about it like this. One commonly used conceptual metaphor for a function is the input/output metaphor: it is a "black box" for which you can insert something in on one end, and it pops something else out the other, and it always pops the same thing out at the other end whenever you insert the same thing on the first: here, a vector field is a function which eats a point in space, and puts out a vector corresponding to that point.(*)

An apt visualization, or apt "box", for this situation, may be a sort of "meter", which can be moved around and held at different points in space, and which "samples" the vector field at each place at which you put it. When you emplace it at a certain point, it shows you an arrow pointing in some direction (imagine it has a three, or whatever, dimensional display). That is the direction of the vector at that point. It also reports a number - the vector's length, or magnitude. As you move this meter around, the arrows and number may vary. We can thus describe the vector field as the collection of all the meter readings taken at every point in the entire space.

A simple physical example of this, albeit of only partial accuracy (no magnitude, also a compass is 2D, and this field in in 3D space with any directions being possible) so should be taken more as just a way to get a feel, is a compass and magnetic field, the latter of which is usually described using a vector field. A compass, when emplaced at different points in space, gives you the direction of the magnetic field at that point. The full magnetic field can be described as the collection of all compass readings taken at each and every such point.

In symbolic notation, if a point in space is denoted by $P$, and the vector field by $\mathbf{F}$, the vector at $P$ is $\mathbf{F}(P)$ - the function $\mathbf{F}$ applied to the single input, or "argument", $P$. More often, we may describe $P$ by its coordinates, and imagine $\mathbf{F}$ as taking these as arguments instead, e.g. $\mathbf{F}(x, y, z)$ when in three dimensional Euclidean space - but note that this requires us to specify such a coordinate system on the space first before the notation makes sense, and with different coordinate systems, we will have different realizations of $\mathbf{F}$-as-a-function-of-a-point in terms of $\mathbf{F}$-as-a-function-of-multiple-coordinates.


(*) This gets into the interesting distinction between a point and a vector, which is often abused. A vector is an element of a vector space, while a point is an element of a suitable "purely geometric" space. Vector spaces are distinguished by the fact that they have notions of algebraic operations (also functions, but we write them with different notation and use somewhat different conceptual metaphors) - addition and scaling - defined on them, while geometric points do not, and such also serve to distinguish a special point therein, called the origin. Given that functions are typically described as operating between sets only, it is hard to make this distinction rigorous in a particularly elegant and clean fashion in "standard" mathematics - what you really would like is a kind of "data type" annotation, as often used in computer programming, and while you can build that into your mathematical foundations, it is not commonly done. "Standard mathematics" is "not type safe", to use computer programming-ese, and points and vectors could end up as being the same as sets, yet are conceptually different and should not be mixed simplistically.

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  • $\begingroup$ Where does the compass point at the poles? $\endgroup$ – Jochen Jul 9 '19 at 8:34
  • $\begingroup$ @Jochen : Yeah. You'll probably want a three-dimensional compass :g: Qualified. $\endgroup$ – The_Sympathizer Jul 9 '19 at 8:36

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