Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 $F_{*}(X_p)=\sum_{i=1}^{m}b^i\partial/dy_i$

share|cite|improve this question
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

$X_p$ 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.

share|cite|improve this answer

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 $F$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.