Not understanding the definition of a differential of a map. I'am reading Loring.W.Tu's book on Manifolds and I'am stuck at the point where he defines differential of a smooth map between smooth manifolds $N$ and $M$.
If $F:N\rightarrow M$ is a $C^{\infty}$ map between two manifolds (smooth) then at each point $p\in N$ it induces a linear map of tangent spaces
$$F_{*}:T_{p}N\rightarrow T_{F(p)}M$$as follows.If $X_{p}\in T_{p}N$ then $F_{*}(X_{p})$ is the tangent vector in $T_{F(p)}M$ defined by 
$$(F{*}(X_{p}))f=X_{p}(f\circ F) \in \mathbb{R}$$ for $f \in C^{\infty}_{F(p)}(M)$.
Can anyone help me out in decoding this and explain why this is a good generalisation of Jacobians.
 A: The other answer covers it, but I think it is worthwhile to remark that the very definition of differential encodes how $(F_*)_p$ transforms tangent basis vectors, and so how it acts on $T_pN:$
if $p\in U\subseteq N$ and $(U,\phi)$ is a chart, then $\phi_*:T_pU\cong T_pM\to T_p \mathbb R^n$ is an isomorphism (because $\phi$ is a diffeomorphism) and so it makes sense to $\textit{define}$ for each $\ 1\le i\le n,\  \frac{\partial}{\partial x^i}:=\phi^{-1}_*(\frac{\partial}{\partial r^i})$, where $r^i$ are the standard coordinates on $\mathbb R^n$. The same analysis applied to $F(p)\in V\subseteq M$ using the chart $(V,\psi)$ gives tangent vectors $\frac{\partial}{\partial y^j}=\psi^{-1}_*(\frac{\partial}{\partial r^j})$ for $1\le j\le m$.
Then, a direct calculation shows that $(F_*)_p\left (\frac{\partial}{\partial x^i}\right )=\sum ^m_{j=1}\frac{\partial (\psi^j\circ F\circ \phi^{-1})}{\partial r^i}\cdot \frac{\partial}{\partial y^j}$.
The upshot of this is that the matrix of $F_*$ as a map from $T_pN$ to $T_{F(p)}M$ is precisely the Jacobian matrix of the function $\psi\circ F\circ \phi^{-1}$, which is a map between $\textit{Euclidean}$ spaces. And this result was ensured by the definitions. 
A: They talk about this in example 8.4. Let $f:\mathbb{R}^n\rightarrow\mathbb{R}^m$ be smooth and $p\in \mathbb{R}^n$. Take standard coordinates of $\mathbb{R}^n$ and $\mathbb{R}^m$ via $(x^1,\cdots, x^n)$ and $(y^1\cdots, y^m)$, respectively. Then, the map $F_*:T_p\mathbb{R}^n\rightarrow T_{F(p)}\mathbb{R}^m$ is a linear map. The entries of the matrix representation $[a_j^i]$ relative to the standard bases  (for the tangent spaces) is given via the formula $$F_*\left(\frac{\partial}{\partial x^j}\Big|_p\right)=\sum\limits_k a_j^k\frac{\partial}{\partial x^j}\Big|_{F(p)}.$$ To determine the $a_j^i,$ simply evaluate both sides on $y^i$. The right will give you $a_j^i,$ and the left, by the definition of the pushforward, will  give you $\frac{\partial F^i}{\partial x^j}(p),$ where $F^i$ denote the coordinates of $F$ in the $y$ coordinate system. Hence, the matrix representation with respect to the bases $\{{\partial}/{\partial x^j}|_p\}$ and $\{{\partial}/{\partial y^j}|_{F(p)}\}$ is $[{\partial F^i}/{\partial x^j}(p)],$ which is the Jacobian that we all know and love. 
So, this is a good generalization, since it matches what we should get in the Euclidean case.
