Vector field and its components If I have a cartesian vector field just denoted $\mathbf{E}=\mathbf{E}(x,y,z)$ (e.g. a electric field), does it mean:
$$
\mathbf{E}(x,y,z)=x\mathbf{\hat{x}}+y\mathbf{\hat{y}}+z\mathbf{\hat{z}} \tag{1}
$$ 
or
$$\mathbf{E}(x,y,z)=x(x,y,z)\mathbf{\hat{x}}+y(x,y,z)\mathbf{\hat{y}}+
z(x,y,z)\mathbf{\hat{z}}\tag{2}
$$
or
$$\mathbf{E}(x,y,z)=E_x(x,y,z)\mathbf{\hat{x}}+E_y(x,y,z)\mathbf{\hat{y}}+
E_z(x,y,z)\mathbf{\hat{z}}\tag{3}
$$
Are there any differences? Any of them wrong?
 A: Remark: As notation is quite subjective, that's how I see it. (Note that my point of view already disagrees with Don Antonio's one)
If we write them using the standard notation they appear as:
\begin{equation}E(x,y,z)=\begin{pmatrix}x\\y\\z\end{pmatrix} \tag{1}\end{equation}
\begin{equation}E(x,y,z)=\begin{pmatrix}x(x,y,z)\\y(x,y,z)\\z(x,y,z)\end{pmatrix} \tag{2}\end{equation}
\begin{equation}E(x,y,z)=\begin{pmatrix}E_x(x,y,z)\\E_y(x,y,z)\\E_z(x,y,z)\end{pmatrix} \tag{3}\end{equation}
The general one is the third one, as $E_x,E_y,E_z$ are given functions of $(x,y,z)$ mapping into $\mathbb{R}$, so called components of the vector field.
The first one is just the vector field which associates to every point $P$, the vector $\vec{OP}$ ($O$ denoting the origin).
The second one is not really well defined: either $x(x,y,z)$ is the projection to the first coordinate $x$ (and so on), in which case it corresponds to the first, or $x(x,y,z)$ is meant to be a generic function, as in the third case, but the notation is very bad, so I would definitely avoid that one.
A: E = E(x,y,z) means that E is a generic vector field whose components depend on $x, y, z$. Namely, being a vector field in $\mathbb{R}^3$, E has $3$ components which depend on $x, y, z$. You can denote these three components however you like. For example, if you denote them as $E_x, E_y, E_z$, then the third notation would be the correct one.
