The position vector $\mathbf{r}$ in the electric field?

Question

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?

Answer 1

$\mathbf{r}$ represents an arbitrary position in space; it is not a constant vector, but it can be represented by $$ \mathbf{r}=x\mathbf{\hat e}_x+y\mathbf{\hat e}_y+z\mathbf{\hat{e}}_z=(x, y, z) $$

Think of it as a simplified notation.