TITLE: Vector fields and tangent vector fields?
QUESTION [7 upvotes]: I am wondering if there are times when people would call a tangent vector field simply by a vector field?
Are not these two concepts different? 
For example,
a vector field assigns (say) to each point of $\mathbb{R}^{n}$ exactly one point of $\mathbb{R}^{n}$; whereas a tangent vector field assigns (say) to each point $x$ of $\mathbb{R}^{n}$ exactly one point of the tangent space $\{ x\} \times \mathbb{R}^{n}$. 
Clearly, the vector field has range in $\mathbb{R}^{n}$, but the tangent vector field has range in the tangent bundle $\mathbb{R}^{n} \times \mathbb{R}^{n}$ of $\mathbb{R}^{n}$.

REPLY [5 votes]: In full generality, you can talk about a section of any vector bundle $p : E \to M$: it's a map $s : M \to E$ such that $p(s(x)) = x$ for all $x \in M$. By definition, a vector field is a section of the tangent bundle $TM \to M$, and so there's no need to specify that it's a "tangent vector field" because it's already implied.
In your case, $M = \mathbb{R}^n$, and the tangent space $T_x\mathbb{R}^n$ at any point $x \in \mathbb{R}^n$ is canonically identified with $\mathbb{R}^n$ itself, in other words, $T\mathbb{R}^n \cong \mathbb{R}^n \times \mathbb{R}^n$. This is why you were able to say "a vector field assigns (say) to each point of $\mathbb{R}^n$ exactly one point of $\mathbb{R}^n$", because you know each tangent space is canonically identified with $\mathbb{R}^n$ and so you can forget the first coordinate for simplicity. But if you want to be completely rigorous, then yes, actually if you have a vector field $\xi$ then $\xi(x) = (x, \text{a vector})$.