Find $F$-related vector fields on $M\times N$, where $F(x)=(x, f(x))$ I am reading Lee's book on Differential geometry. In Chapter 4, Lee has this exercise.

Let $M$, $N$ be smooth manifolds, and let $f:M\to N$ be a smooth map.
Define $F : M\to M \times N$ by $F(x) = (x, f(x) )$. Show that for every smooth vector field
$V$ of $M$ , there is a smooth vector field on $W$ on $M \times N$ that is $F$-related to $V$.

I can understand that we must have $W_{(x,f(x))}=V_x\oplus Df(x)V_p$ for all $x\in M.$ The set $\{(x,f(x)):x\in M\}$ is a closed set in $M\times N.$ Therefore, if we can show that for all $(p,f(p))\in M\times N$ there is a neighbourhood $U_p$ and a smooth vector field on $U_p$ extending $V_x\oplus Df(x)V_x$ we are done by partition of unity. But I cannot show that. Can someone help me out?
 A: Allow me answer with notations that is more closely related to the book, which make it more pedantic than some answers here on MSE. Forgive me for that.
You have noticed that if we have a smooth vector field $Y \in \mathfrak{X}(M \times N)$ that is $F$-related to $X$, then for each point $p \in M$,
$$
dF_p(X_p) = \alpha^{-1} \circ \alpha \circ dF_p(X_p) = \alpha^{-1}\big( X_p, df_p(X_p) \big) = Y_{(p,f(p))}, 
$$
with $\alpha : T_{(p,f(p))}(M \times N) \to T_pM \oplus T_{f(p)}N$ is the isomorphism $\alpha(v) = (d\pi_M(v),d\pi_N(v))$. So we need to find $Y \in \mathfrak{X}(M \times N)$ such that its values on the graph $$\Gamma_f = \{(p,q) \in M \times N \mid p \in M , q=f(p)\}$$ satisfy the relation above. Define a continuous vector field $Y : \Gamma_f \to T(M \times N)$ as
$$
Y_{(p,f(p))} = dF_p(X_p).
$$
Now we need to extend $Y$ to $M \times N$, by extending $Y$ locally around each point first. That is for each $(p,f(p)) \in \Gamma_f$ there is a neighbourhood $W$ of $(p,f(p))$ and a smooth vector field $\widetilde{Y}$ on $W$ such that $\widetilde{Y}|_{W \cap \Gamma_f} = Y|_{W \cap \Gamma_f}$. The hint is that write down the expression for $Y$ in a local coordinates first. Then decide what is the form of $\widetilde{Y}$. Here is the detail :

Let $(p,f(p)) \in \Gamma_f$ be arbitrary, and choose smooth (boundary) chart $(U,x^i)$ contain $p$ and $(V,y^i)$ contain $f(p)$ with $f(U) \subseteq V$. Since $(U \times V, (x^i,y^j))$ is a smooth chart for $M \times N$ that contain $(p,f(p))$, then
\begin{align*}
    Y_{(p,f(p))} &= dF_p(X_p) = \alpha^{-1} \big( X_p,df_p(X_p) \big) = d\iota_M(X_p) + d\iota_N(df_p(X_p)) \\ &= X^i(p) \frac{\partial}{\partial x^i}\Big|_{(p,f(p))} + X^i(p) \frac{\partial f^j}{\partial x^i}(p) \frac{\partial}{\partial y^j}\Big|_{(p,f(p))},
\end{align*}
with $\iota_M : M \hookrightarrow M \times N$ and $\iota_N : N \hookrightarrow M \times N$ are inclusions $x \mapsto (x,f(p))$ and $x \mapsto (p,x)$ respectively. Since we know the local form of $Y$ as above, then clearly vector field  $\widetilde{Y} : U \times V \to T(M \times N)$ that we wanted is
$$
\widetilde{Y}_{(x,y)}:= X^i(x) \frac{\partial}{\partial x^i}\Big|_{(x,y)} + X^i(x) \frac{\partial f^j}{\partial x^i}(x) \frac{\partial}{\partial y^j}\Big|_{(x,y)}, \quad \forall (x,y)\in U \times V.
$$

