# Push-Forwards and Pull-Backs: Polar to Rectangular Coordinates

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?