Why write $\mathbb R\times \mathbb R^3\to\mathbb R^3$ instead of $\mathbb R^4\to\mathbb R^3$? I often see this notation for a vector field
$$\mathbf A:\mathbb R\times \mathbb R^3\to\mathbb R^3$$
But isn't $\mathbb R\times \mathbb R^3=\mathbb R^4$, right?
So, is there any advantages using $\mathbb R\times \mathbb R^3\rightarrow\mathbb R^3$ instead of $\mathbb{R}^4\rightarrow \mathbb{R}^3$?
 A: The two sets are not equal, no. The first consists of pairs of the form 
$$
(t, (x, y, z))
$$
while the second contains elements of the form
$$
(t, x, y, z).
$$
With that notation, there's a pretty natural isomorphism between the sets, as you've guessed, and mathematicians generally get pretty glib about saying they're "the same". 
Why write the first way? Frequently, a vector field (the wind, for instance) is time-varying, and at each time $t$ and location $(x,y, z)$ there's a wind velocity. You'll frequently see that velocity denoted $F(t, v)$, where $v$ denotes a point in 3-space. You might want to talk about the field at one particular point $v_0$ changes over time, and if you like Leibniz notation, this becomes
$$
\frac{\partial F}{\partial t}(t, v_0).
$$
You might also want to talk about how the field changes with respect to position, at a fixed moment, and this would be written 
$$
\frac{\partial F}{\partial v}(t_0, v).
$$
It's true that you could write, in the $\Bbb R^4$ version, something like 
$$
\frac{\partial F}{\partial xyz}(t_0, x,y,z),
$$
but that's often cumbersome. 
Anyhow, the reason it's written that way is often just to make the naming of variables and their groupings easier to understand. Imagine if you did it the other way, but wrote $F(x, y, z, w)$ to indicate the wind speed at time $x$ and position $(y, z, w)$ ... that'd seem really awkward, because $x$ is not a natural name for a time coordinate, etc. 
