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?
 A: Let $F: M\to N$. Here, $M = \mathbb{R}^2$ and $N$ is yet to be defined. Let $(x,y) = p\in M$. Then, $X_p: M \to T_pM$, the tangent space of $M$ at $p$, and thus is a tangent vector of $M$ for a particular $p$. Intuitively, the tangent vector is the "direction" chosen, starting at the point $p$. The vectors $\frac{\partial}{\partial x}$ and $\frac{\partial}{\partial y}$ are the basis vectors of $T_p M$, so we can write $X_p = \begin{bmatrix}X_p^1 \\ X_p^2\end{bmatrix} = \begin{bmatrix}x \\ y\end{bmatrix}$ in this particular case. To generalize this concept, we can pair up the points $p$ with their associated tangent vector $X_p$ to get $(p,X_p)$, and we say the set of possible such pairs is called the tangent bundle of $M$, or $TM$. This lets us consider a function $X: M\to TM$, called a vector field, which will be a useful way of associating a tangent vector to any choice of $p$.
The differential of $F$, called $\text dF$, is a function which (roughly) takes in a starting point $p$ from $M$ and a "direction" from $T_p M$ and returns the starting point $F(p)$ in $N$ and the consequent "direction" in $T_{F(p)} N$. Formally, we say $\text dF: TM\to TN$, which should make some sense considering $p$ and it's direction as a pair in $TM$ and similarly for the output. Then, for some $p$ and direction $u$, $\text dF(p,u) = (F(p),D_{F(p)}u)$, where $D_{F(p)}$ is the Jacobian of $F$ at $p$.
Finally, in order to "use" $\text dF$, we need to have both a starting point and a direction. However, if we already have a particular direction we'd like to consider for each point, we can plug that into $\text dF$ instead. The vector field $X$ "primes" our input points by turning them into a pair with the associated direction, which is an element of $TM$, and thus exactly what kind of object $\text dF$ accepts. So, $\text dF \circ X : M \to T N$ is a function which takes in a point and returns the consequent "direction" of moving each point plugged into $F$ in the "direction" determined by the vector field $X$.
When it comes time to calculate this function, we simply need to do exactly what it seems: plug in the equation for the tangent vectors in for $u$. Then, the columns of $D_{F(p)}$ are $\frac{\partial F}{\partial x}\big|_{(x,y)}$ and $\frac{\partial F}{\partial y}\big|_{(x,y)}$, while $u = X_p$, so $$\text dF(X) =\left( F(p), \begin{bmatrix} \frac{\partial F}{\partial x}\big|_{(x,y)} & \frac{\partial F}{\partial y}\big|_{(x,y)}\end{bmatrix} \cdot \begin{bmatrix} x \\ y \end{bmatrix} \right) =\left( F(p), x\frac{\partial F}{\partial x}\biggr|_{(x,y)} + y\frac{\partial F}{\partial y}\biggr|_{(x,y)}\right)$$
To take it down one level of abstraction by "choosing" a point $p$, we can write everything with subscript $p$, in the same way that $TM$ and $T_p M$ are related. Thus, we can say $\text dF_p : T_pM\to T_{F(p)}N$, and $X_p: M\to T_pM$, so $\text dF_p \circ X_p: M\to T_{F(p)} N$ and $$\text dF_p(X_p) = D_{F(p)} X_p = x\frac{\partial F}{\partial x}\biggr|_{(x,y)} + y\frac{\partial F}{\partial y}\biggr|_{(x,y)}$$
A: 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,
$$
dF(X) = \frac{\partial F}{\partial x} x + \frac{\partial F}{\partial y} y.
$$
A: By definition the differential $\mathrm{d}F_p$ of the function $F$ at a point $p$ is a linear operator taking in input a tangent vector $X_p$ and returning the directional derivative of $F$ along $X_p$ at the point $p$, i.e. 
$$\mathrm{d}F_p(X_p):=X_pF = \frac{\mathrm{d}F}{\mathrm{d}t}(\gamma(t))\Big\rvert_{t=0},$$
where $\gamma\colon (-\varepsilon,\varepsilon) \to \mathbb{R}^2$ is a smooth curve such that $\gamma(0)=p$ and $\gamma'(0)=X_p$. 
From this definition you can see, for example, that $$\mathrm{d}x\left(\frac{\partial}{\partial x}\right) = \frac{\partial x}{\partial x}=1, \quad \mathrm{d}x\left(\frac{\partial}{\partial y}\right)=\frac{\partial x}{\partial y} =0.$$
Therefore, if $X=x\frac{\partial}{\partial x}+y\frac{\partial}{\partial y}$ and $p=(x,y)$ you get
$$dF_p(X_p)= X_pF=x\frac{\partial F}{\partial x}(x,y) +y\frac{\partial F}{\partial y}(x,y).$$
