# 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.

• Unless you give more details about what the term "vector field" means in the context of this question, it is not clear what this question is about. For what its worth, one meaning of vector field could be a map from some geometric set $S$ to a vector space $V$, and adding/scalar multiplying such maps as usual, the set of vector fields $S\to V$ forms a (rather large) vector space in itself. – Marc van Leeuwen Sep 18 '16 at 10:52
• Functions that assign a vector to a point in the plane or a point in space are called vector fields. – Sedumjoy Sep 18 '16 at 16:14
• So you answered your own question. Functions that assign a vector to a point in the plane are not subsets of a vector space, they are functions. Functions can themselves be vectors in a (different, large) vector space of functions (and they are here) but that makes them elements, not subsets, of a vector space. – Marc van Leeuwen Sep 18 '16 at 17:43
• No I did not answer my own question. You asked "what the term "vector field" means". This is not what I asked. I answered what you asked which is not what I asked. What I asked was answered by answer two and I acknowledged. I hope this clears any confusion. – Sedumjoy Sep 20 '16 at 0:29
• Of course you answered my question, not your own, but what I meant to say is that your answer to my question implies an answer to your own question (of the title) "is a vector field a subset of a vector space". I a vector field is a function (no matter of what type) then it cannot at the same time be a subset of the vector space. Functions are not subsets. From the answer that you accepted I gather that you confused "function" and "image of a function" (which indeed is a subset). To me your question as formulated simply has as answer "no that can never be the case". – Marc van Leeuwen Sep 20 '16 at 6:51

## 2 Answers

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.

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).