Question about notation used for planar vector fields and differential equations I have learned in my calculus courses that a planar vector field is just a continuous (or smooth) map $\mathbb{R}^2 \to \mathbb{R}^2$, say $V(x,y) = (f(x,y),g(x,y))$ where $f$ and $g$ are continuous (smooth) maps from $\mathbb{R}^2$ to $\mathbb{R}$.
But I've seen an example, in this wiki page on Hilbert's $16^{\text{th}}$ problem, where a planar vector field is written just as a system of $2$ ODEs, in this case
$$ \frac{dx}{dt}=P(x,y), \frac{dy}{dt} = Q(x,y) $$
for polynomials $P,Q$ in two variables.
Could one equivalently write the above vector field as $V(x,y)=(P(x,y),Q(x,y))$? Why is it written with ODEs instead of the other way? What differences are there between these two ways of constructing vector fields?

Yes, they are equivalent. It is just two different ways to look at the same thing.
Using your notation $V=(P,Q)$, you have an ODE in $\mathbb{R}^2$:
$$
\frac{d\vec x}{dt}=V(\vec x)
$$
where $\vec x:=(x,y)$.
One can write an (evolutionary) equation in higher dimensional space component-wise and have a system of equations.
There are lots of such examples. For instance, on this page the system of Navier-Stokes equations is written in the vector form while the official problem description in the Clay Mathematics Institute writes it component-wise.
This is similar to how you write a linear equation. For example,
$$
\begin{pmatrix}
1&2\\
3&4
\end{pmatrix}
\begin{pmatrix}
x\\
y
\end{pmatrix}
=
\begin{pmatrix}
3\\
7
\end{pmatrix}
$$
is equivalent to
$$
x+2y=3,\quad 3x+4y=7.
$$
