Different notation for position vectors? Domain/Range? What is the difference between this notation for position vectors? Are there any differences in domain and range?
$$
\mathbf{r}=x{\mathbf{\hat{e}}_x}+y{\mathbf{\hat{e}}_y}+z{\mathbf{\hat{e}}_z} \qquad \mathbb{R} \rightarrow \mathbb{R}^3\text{?} \tag{1} 
$$
$$
\mathbf{r}(x,y,z)=x{\mathbf{\hat{e}}_x}+y{\mathbf{\hat{e}}_y}+z{\mathbf{\hat{e}}_z} \qquad \mathbb{R}^3 \rightarrow \mathbb{R}^3\text{?} \tag{2}
$$
$$
\mathbf{r}(t)=x(t){\mathbf{\hat{e}}_x}+y(t){\mathbf{\hat{e}}_y}+z(t){\mathbf{\hat{e}}_z} \qquad \mathbb{R} \rightarrow \mathbb{R}^3\text{?} \tag{3}
$$
$$
\mathbf{r}(u,v,w)=x(u,v,w){\mathbf{\hat{e}}_x}+y(u,v,w){\mathbf{\hat{e}}_y}+z(u,v,w){\mathbf{\hat{e}}_z} \quad \mathbb{R}^3\rightarrow \mathbb{R}^3\text{?}\tag{4}
$$
Are there any conventions for notation or it depend of the context?
 A: This is how I would intrepret that notation:


*

*The first is ambiguous.  It might mean that $\mathbf r$ is constant, or it might mean any of the others where the variable is suppressed on the LHS (and possibly RHS) to avoid extra writing when the variable is clear.  Note: Now that OP has added the domain and range to the question, I'd say this is probably supposed to just be the map (3) with the variable suppressed.

*This is a mapping $\mathbf r:D\subseteq \Bbb R^3\to\Bbb R^3$ given by $(x,y,z)\mapsto \mathbf r(x,y,z) = x\hat{\mathbf e_x} + y\hat{\mathbf e_y} + z\hat{\mathbf e_z}$.  This could, for instance, model heat flow.  At each point in the domain there's a vector attached which points in the direction the heat at that point is moving (i.e. in the coldest direction).  This is actually a very simple map though -- it maps to outward-pointing (from the origin) vectors at all points in $\Bbb R^3$ -- i.e. there's a heat source at the origin and it radiates outward symmetrically.

*This is a mapping $\mathbf r:D\subseteq \Bbb R\to\Bbb R^3$ given by $t\mapsto \mathbf r(t) = x(t)\hat{\mathbf e_x} + y(t)\hat{\mathbf e_y} + z(t)\hat{\mathbf e_z}$.  Geometrically, this represents some path through $\Bbb R^3$.  Each "time" $t$ maps to a specific point in the space.

*This is a composition of the mapping $(2)$ with a mapping $\mathbf \Gamma: D\subseteq \Bbb R^3\to \Bbb R^3$ given by $(u,v,w) \mapsto \mathbf \Gamma(u,v,w) = (x,y,z)$.  Explicitly, it is the mapping $\mathbf r\circ \mathbf \Gamma:D \subseteq \Bbb R^3\to \Bbb R^3$ given by $(u,v,w) \mapsto (\mathbf{r}\circ \mathbf \Gamma)(u,v,w)=x(u,v,w){\mathbf{\hat{e}}_x}+y(u,v,w){\mathbf{\hat{e}}_y}+z(u,v,w){\mathbf{\hat{e}}_z}$.  In your notation $\mathbf r\circ \mathbf \Gamma$ is just given the simpler name $\mathbf r$.  Geometrically (or physically) this is the same type of map as (2) only we can't say which direction the vectors point in at each point without an explicit rule for $x(u,v,w)$, etc.

A: This is how I would interpret that notation. In $(1)$ you are thinking of $r$ as a constant. In $(2)$ you are thinking of $r$ as a function of its coordinates which are variable. In $(3)$ the coordinates are functions of some parameter $t$ (e.g. time), so that the vector is a function of $t$. In $(4)$ each coordinate is a function of $u$, $v$, and $w$, so that the vector is also. Which one is appropriate will depend on the context.
