Different notation for position vectors? Domain/Range?

Question

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?

Answer 1

This is how I would intrepret that notation:

1. 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.
2. 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.
3. 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.
4. 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.

Answer 2

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.