Differential of a smooth map

Remarks

In charts $(x^1, \ldots, x^m)$ near $p$ and $(y^1, \ldots, y^n)$ near $F(p)$, with $F^j = y^j \circ F$, $$dF_p\left(\left.\frac{\partial}{\partial x^i}\right|_p\right) = \sum_{j=1}^{n} \frac{\partial F^j}{\partial x^i}(p) \left.\frac{\partial}{\partial y^j}\right|_{F(p)},$$ so the matrix of $dF_p$ is the Jacobian matrix of $F$ in these coordinates.

Example: for $F \colon \R \to \R^2$, $F(t) = (\cos t, \sin t)$, one gets $dF_t(d/dt) = -\sin t \, \partial_x + \cos t \, \partial_y$. Check: for $f \in C^\infty(\R^2)$, $(d/dt)(f(\cos t, \sin t)) = -\sin t \, f_x + \cos t \, f_y$ by the chain rule.

The formula $v \mapsto v(f \circ F)$ needs $F$ to be smooth: if $F$ is only continuous, $f \circ F$ need not be smooth and $v(f \circ F)$ is not defined.

The chain rule reads $d(G \circ F)_p = dG_{F(p)} \circ dF_p$. Other notations are $F_*$, $DF$, $TF$ and $F'$.