What is the difference between a vector valued function and a vector field?

Question

I think that I understand that if I parameterize $y=x\,$ I can write it as $f(t)=(t,t)\,$. So I'm assigning position vectors (coming from the origin) to the point $(t_0,t_0)$ for $t=t_0$. So I can say that that a vector valued function takes a scalar parameter and assign to a vector in $R^n$. And if a $(x,y)$ is going from $R^2$ to $R^n$, it is a vector field, because for each position vector I'm associating a vector ? (instead of each scalar $t$)

I'm confused because if vector valued functions can be going from $R^2$ to $R^2$ isn't it just like a vector field? (for each vector position I receive another vector in that point).

Answer 1

Sure. If you are in a region $U \subset \mathbb{R}^n$ a vector field on $U$ is a vector valued function on $U$ (taking values in $\mathbb{R}^n$).

In my mind (although maybe there are different conventions) a "vector valued function" is just any function from some space to a vector space. (I'm intentionally being a bit vague by what I mean by a "space", but most reasonable subsets of $\mathbb{R}^n$ you could write down should be fine) In particular you could define a vector valued function from some region in $\mathbb{R}^3$ to $\mathbb{R}^2$ if you wanted to.

A vector field is something more specific, it is an assignment to each point a vector inside the tangent space of your space at that point. As you suggested, for some open set $U \in \mathbb{R}^2$ this is just the same as a vector valued function $U \to \mathbb{R}^2$. However if instead our space is the unit circle $S^1 = \{(x,y) |x^2+y^2 = 1 \}$ in $\mathbb{R}^2$, then a vector field on $S^1$ is a map from $S^1$ to $\mathbb{R}^2$ with the condition that the vector assigned to a given point is tangent to the circle at that point (a harder condition to meet than just being an arbitrary vector valued function).