I understand the definition, but in the book, they tell about something more. Specifically:
A smooth map $f : M \rightarrow N$ naturally induces a map $f_∗$ called the differential map : \begin{equation} f_∗: T_{p}M \rightarrow T_{f (p)}N\end{equation}
The explicit form of $f_∗$ is obtained by the definition of a tangent vector as a directional derivative along a curve. If $g \in \cal{F}(N)$, then $g\cdot f \in \cal{F}(M)$. A vector $V \in T_{p}M$ acts on $g \cdot f$ to give a number $V[g ◦ f ]$. Now we define $f_{∗}V \in T_{f (p)}N$ by \begin{equation} ( f∗V )[g] ≡ V[g ◦ f ]\end{equation} or, in terms of charts $(U, \phi)$ on M and $(V,\psi)$ on $N$,
\begin{equation} ( f∗V )[g ◦ ψ^{−1}(y)] ≡ V [g ◦ f ◦ \varphi^{−1}(x)]\end{equation} where $x = \varphi(p)$ and $y = \psi( f (p))$. Let $V =V^{\mu}\frac{\partial}{\partial x^{\mu}}$ and $f_{∗}V = W^{\alpha}\frac{\partial}{\partial y^{\alpha}}$. Then
\begin{equation} W^{\alpha}\frac{\partial}{\partial y^{\alpha}} [g \cdot \psi^{−1}(y)] = V^{\mu}\frac{\partial}{\partial x^{\mu}} [g \cdot f \cdot \varphi^{−1}(x)]. \end{equation} If we take $g = y^{α}$, we obtain the relation between $W^{α}$ and $V^{μ}$,
\begin{equation} W^{\alpha} = V^{\mu}\frac{\partial}{\partial x^{\mu}}y^{\alpha}(x). \end{equation}
Note that the matrix $\frac{\partial y^{\alpha}}{\partial x^{\mu}}$ is nothing but the Jacobian of the map $f : M \rightarrow N$. The differential map $f_∗$ is naturally extended to tensors of type $(q, 0)$, $f_{∗} : \cal{T} ^{q}_{0,p}(M) →\cal{T} ^{q}_{0, f (p)}(N).$
For someone do not familiar with notations, $T_{p}M$ is tangent space at point $p$ on manifold $M$; $\cal{F}(M)$ is set of $C^{\infty}$ function $h:M\rightarrow \mathbb{R}$.
I do not know why they choose $g=y^{\alpha}$. As I think, if $g=y^{\alpha}$, $g\equiv\psi$, or another idea, just substitute $=y^{\alpha}$ into given equation, but I could not find the final result by using rules of derivative in both procedures. By doing that, what could we build? (I mean, when we do something, we all have purpose, aim to something else). Just to find the relation between components of tangent vectors and $W^{\alpha}$ (what is this called?)? I also read in "Introduction to manifold" by Tu, but there are nothing like above stuff.
Thank you!