I'm having some trouble with this one:
"Given a smooth map $F:M\rightarrow N$ , prove that the map $F_*:TM\rightarrow TN$ defined as $(p,X)\mapsto (F(p),F_{*p}(X))$ for all $X\in T_p M$ is smooth. Which is its expression in local coordinates?"
My idea of the solution is: let $(x_1,\ldots,x_m)$ local coordinates on $M$ defined in a neighborhood of $p\in M$ and let $(y_1,\ldots,y_n)$ local coordinates on $N$ defined in a neighborhood of $F(p)\in N$. I get the expression of the vector $F_{*p}(X)\in T_{F(p)}N$ in local coordinates this way: \begin{eqnarray} F_{*p}(X)&=&\sum_{i=1}^n F_{*p}(X)(y^i)\left.\frac{\partial}{\partial y^i}\right|_{F(p)}=\sum_{i=1}^n X(y^i\circ F)\left.\frac{\partial}{\partial y^i}\right|_{F(p)}\\ &=&\sum_{i=1}^n \sum_{j=1}^m X^j\frac{\partial (y^i\circ F)}{\partial x^j}\left.\frac{\partial}{\partial y^i}\right|_{F(p)}=\sum_{i=1}^n Y^i \left.\frac{\partial}{\partial y^i}\right|_{F(p)} \end{eqnarray} where $Y^i=\sum_{j=1}^m X^j\frac{\partial (y^i\circ F)}{\partial x^j}$.
With this notation I can write $(F(p),F_{*p}(X))=(y^1,\ldots,y^n,Y^1,\ldots,Y^n)$ and I notice that $F_*$ is smooth iff $y^1,\ldots,y^n$ and $Y^1,\ldots,Y^n$ are smooth.
Is everything all right?
Thank you so much