Vector space or vector field?

Question

I seem to be having a problem distinguishing between a vector space (which I know to be a set of vectors over some scalar set) and a vector field. I know that in Multivariable Calculus a vector field is a vector-valued function (being a derivative for conserved fields.) I feel they are different but how does one explain in layman terms the difference between the two?

Answer 1

A vector field takes a point and glues a vector ("arrow", if you prefer) to it.

A vector space is a set, with an underlying field (algebraic structure, usually $\Bbb R$ or $\Bbb C$), and two operations. The elements of the set are called "vectors". Just a name.

They're entirely different: one is a function, the other is not.

Answer 2

Informally, you can imagine a vector field as a collection of little floating "arrows" attached to points in space. For example, a vector field might represent the velocity of the air in a room: at each point in space, you can ask the question "How fast and in what direction is the wind moving at this point?", and represent that with a vector that is "pinned" (so to speak) to that point in space. The room is then "filled" with little arrows, one at every possible location. The wind velocity at one location is not necessarily the same at another location, so these vectors are not all the same. Nor is it meaningful to "add" vectors that are attached to different points in the room, so the individual vectors don't live in a single vector space.

On the other hand, if you pick a single point in the room, and ask the question "What are all the possible wind velocities at this location?" then you have a vector space. At that one point, there are possible vectors pointing in every direction and with every length. Adding those vectors together is a meaningful operation. At each individual point, there is a vector space associated with that point.

So informally, a vector field can be thought of as choosing, at each point in the underlying space, a single vector from the vector space at that point.

Answer 3

Well, to give you the clearest idea of the difference between a vector space and a vector field, let me first define the a certain structure called the tangent space.

Intuitively, the tangent space is a vector space defined at each point $p$ of euclidean space in the following way:

1) the underlying set is elements of the form $(p;v)$ where $p$ is the point, and $v$ is a vector in the same euclidean space containing $p$.

2) addition and scalar multiplication: $(p;v)+(p;w)=(p;v+w)$ and $c(p;v)=(p;cv)$.

As you can see, the tangent space at $p$ is just a copy of euclidean space, but think of it as vectors starting from $p$ rather than zero. Finally, a vector field is a map $F$ which assigns to every point $p$ an element $(p;v)$ in the tangent space.