0x66a9ee19

Let \(M\), \(N\) be smooth manifolds and \((U,\varphi)\), \((V,\psi)\) respective local charts and \((x^i)\), \((y^i)\) the corresponding coordinate maps . For \(F\colon M\to N\) smooth , we have for \(p\in U\)

\begin{equation*} dF\biggl(\biggl(\frac{\partial }{\partial x^i}\biggr)_p\biggr)=\frac{\partial \hat{F}^j}{\partial x^i}(\hat{p}) \biggl(\frac{\partial }{\partial y^j}\biggr)_{F(p)}, \end{equation*}

where \(\hat{p}=\varphi(p)\) and

\begin{equation*} \hat{F}=\psi\circ F\circ \varphi^{-1} \end{equation*}

is the local representation of \(F\).

The \(d\hat{F}\) can be considered as the Jacobian in local coordinates of \(F\).