Let $F:\mathbb R^2 \to \mathbb R^2$ be the rotation of angle $\theta$. Find $F_*{\frac{\partial}{\partial x}}$.
Let $(u,v)\in \mathbb R^2$, $T_{F^{-1}(u,v)}F=F$, so \begin{align*} F_*{\frac{\partial}{\partial x}}(u,v)&=(T_{F^{-1}(u,v)}F) \left. \frac{\partial}{\partial x}\right |_{(F^{-1} {(u,v)}}\\& = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta &\cos\theta\end{pmatrix} \left. \frac{\partial}{\partial x}\right |_{(F^{-1} {(u,v)}} \\&=-\sin(\theta) \left.\frac{\partial}{\partial u} \right |_{(u,v)} +\cos(\theta) \left. \frac{\partial}{\partial v} \right|_{(u,v)} \end{align*} Is my answer correct? Any help is appreciated !