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Let $F:N\rightarrow M$ be a $C^{\infty}$ map, At each point $p\in N$, the map $F$ induces a linear map of tangent spaces, called its differential at $p$,

$F_{*}:T_p N\rightarrow T_{F(p)}M$ as follows. $X_p\in T_pN$ then $F_{*}(X_p)$ is the tangent vector in $T_{F(p)}M$ defined by $$F_{*}((X_p))f=X_p(f\circ F)\in\mathbb{R}$$, I understand that a tangent vector $F_{*}(X_p)$ acts on a $C^{\infty}$ map on $M$, what is the role of $X_p=\sum_{i=1}^{n}a^{i}\partial/dx_i$ here in the definition?, where $\{\partial/dx_{i}\}_{i=1}^{n}$ are basis of tangent space at $p$ of $N$

well, let the basis for tangent space at $F(p)$ of $M$ be $\{\partial/dy_i\}_{i=1}^{m}$, so then $F_{*}(X_p)=\sum_{i=1}^{m}b^i\partial/dy_i$

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Exactly as below. The key point is that $f\circ F$ is a smooth function $N\to \mathbb{R}$. Since the derivative is local, you can choose a chart and represent the function $f\circ F$ as a map $\mathbb{R}^n\to \mathbb{R}$. In this form, what you wrote is just the standard directional derivative from calculus for a function $\mathbb{R}^n\to \mathbb{R}$ in the direction of $X_p$. –  Matt Aug 20 '12 at 18:15

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$X_p$ is a tangent vector in $T_pN$. It should read $$X_p = \sum_{i=1}^n \lambda_i(p) \frac{\partial}{\partial x^i}|_p$$ You somehow missed the coefficients. Applying this to $f$ is another way of stating that you take the derivative of $f$ wrt to that direction.

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You should be carefull not to mix up$\partial$ with $d$. Usually one denotes a basis of $T_p M$ by $\{ \frac{\partial}{\partial x_i}|_p \}_i$ and with $\{ dx_i |_p \}_i$ the dual basis of $\{ \frac{\partial}{\partial x_i}|_p \}_i$ of $(T_p M)^*$.

Now if you have $X_p = \sum_{i=1}^n a_i \frac{\partial}{\partial x_i}|_p$ where the $a_i$'s are constants you can write

$$F_{*}(X_p)=F_{*}(\sum_{i=1}^n a_i \frac{\partial}{\partial x_i}|_p)=\sum_{i=1}^n a_iF_{*}( \frac{\partial}{\partial x_i}|_p)$$

by the linearity of the differential. Furthermore:

$$F_{*}( \frac{\partial}{\partial x_i}|_p)=\sum_{j=1}^m \frac{\partial(y_j \circ F)}{\partial x_i}|_p)\frac{\partial}{\partial y_j}|_{F(p)}$$

$\{ \frac{\partial(y_j \circ F)}{\partial x_i} \}$ is called the jacobian of $F$.

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