I often see the electric field denoted \begin{align}\mathbf{E}(\mathbf{r})=E_x(\mathbf{r})\mathbf{\hat e}_x+E_y(\mathbf{r})\mathbf{\hat e}_y+E_z(\mathbf{r})\mathbf{\hat{e}}_z=(E_x(\mathbf{r}),E_y(\mathbf{r}),E_z(\mathbf{r}))
\end{align}
The position vector $\mathbf{r}$ confuses me. 

Is $\mathbf{r}$ a *constant vector*:
$$
\mathbf{r}=x\mathbf{\hat e}_x+y\mathbf{\hat e}_y+z\mathbf{\hat{e}}_z=(x, y, z)
$$

Or is $\mathbf{r}$ just a abbreviation for a *vector function* of a variable, say $u$, so:
$$
\mathbf{r}=\mathbf{r}(u)=x(u)\mathbf{\hat{e}}_x+y(u)\mathbf{\hat{e}}_y+z(u)\mathbf{\hat{e}}_z=(x(u), y(u), z(u))
$$
Which one is correct?