Let $x=r*cos(\theta)$ and $y=r*sin(\theta)$ represent the polar coordinates function $\mathbf f(r,\theta):\mathbf R^2\rightarrow\mathbf R^2$. Compute $\mathbf f_*(r\frac{\partial}{\partial r})$ and $f_*(\frac{\partial}{\partial\theta})$

My instinct was to calculate the directional derivative of $\mathbf f$ and I got: $$ D\mathbf f=\begin{bmatrix}cos\theta &-rsin\theta \\sin\theta & rcos\theta \end{bmatrix} $$

I then tried to find $\mathbf f_*(r\frac{\partial}{\partial r})$ as: $$\mathbf f_*(r\frac{\partial}{\partial r})=r*\begin{bmatrix}cos\theta \\sin\theta\end{bmatrix}=\begin{bmatrix}x \\y\end{bmatrix}$$ Is this correct? Should I be taking the transpose of the directional derivative?


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