Given $F(x,y)$, what does it mean to compute $dF(X)$ for $X=x \frac{d}{dx}+y\frac{d}{dy}$? Given $F(x,y)$, what does it mean to compute $dF(X)$ for $X=x \frac{d}{dx}+y\frac{d}{dy}$?
My idea:
$dF(x,y)$ is the same as the Jacobian of $F(x,y)$. But in order to plug in $X$, then what should I do?

By definition, 
$$
dF(X) = \frac{\partial F}{\partial x} dx(X) + \frac{\partial F}{\partial y} dy(X).
$$
Here,
$$
dx(X) 
= dx(x \partial_x + y \partial_y) 
= \{ \text{ linearity } \} 
= x \, dx(\partial_x) + y \, dx(\partial_y)
= x \cdot 1 + y \cdot 0 = x
$$
Likewise, $dy(X) = y.$
Thus,
$$
dF(X) = \frac{\partial F}{\partial x} x + \frac{\partial F}{\partial y} y.
$$
