Is a vector field a subset of a vector space? This is not actually a duplicate although it seems very similar to the another question I found on stackexchange titled "vector space or vector field?"  If I have a vector field isn't each element of that field in n dimensions also an element of a particular vector space of similar dimension. Even if the subset does not satisfy all the properties of a vector space would it not still be a subset?  Just curious if the "object" that resides in a vector field the same object that appears in some vector space of similar dimension.  Thank you . Not really sure where this question belongs since one is a topic of linear algebra and the other is more in line with vector fields in calculus, which is what prompted my question to begin with. 
 A: Yes, any element of a vector field is a vector (with a dimension $n$), so ''all'' the vectors of a vector field are a subset of some vector space $W$. But a vector field is not simply a set of vectors, it is a function that assign a vector to any point in a  space, that is a function $f:V \to W$ where $W$ is a vector space and $V$ can be a set, but usually has some structure (as a manifold). Also if $V$ is a vector space,  this function $f$  may be not linear. So a vector field is  a lot more that a subset of a vector space.
A: The name "vector field" is commonly used to a particular association of points to vectors. In the case of Euclidean space, this becomes blurred, since we associate to each point of the plane a vector (that is, we make a function $V:\mathbb{R}^n \to \mathbb{R}^n$ which is better visualized as taking $x$ to a vector $V_x$ in each point $x$).
But consider a sphere (the sphere $S^2$, the "shell" of the unit ball in $\mathbb{R}^3$). You can imagine what a  vector field should be. For example, the wind flow on Earth should be an example of a vector field. We are associating to each point of the sphere a tangent vector to it. However, there is a problem. When we change every point, we are changing the space where our tangent vector lives. What we are actually doing is associating to each point $x \in S^2$ a vector in $T_xS^2$, the space of tangent vectors to $S^2$ based in $x$. This is what a vector field is: an association of a tangent vector to every point of a surface  (more generally, to every point of a manifold a tangent vector). We can give a structure to the collection of tangent spaces in order to visualize a vector field on the sphere as a particular kind of map $V: S^2 \to TS^2$, where this $TS^2$ is the "collection of tangent spaces". But note now that this $TS^2$ won't be a vector space. It will be what is called a vector bundle.
One could argue that associating to every point of the sphere a vector in $\mathbb{R}^3$ would also be a vector field. But I've never seen the name being used like this in the literature (if it were, it would be as reasonable as calling any section in any vector bundle a vector field - not that it isn't reasonable).
