Suppose that f is a function of two variables (y and z) only. Show that the gradient $\nabla f = \frac{\partial f} {\partial y} \hat{y} + \frac{\partial f} {\partial z} \hat{z}$ transforms as a vector under rotations.
$$\overline{y} = y \cos{\phi} + z \sin{\phi}$$ $$\overline{z} = -y \sin{\phi} + z \cos{\phi}$$
$$ \begin{equation} \begin{cases} \overline{y} = y \cos{\phi} + z \sin{\phi} \\ \overline{z} = -y \sin{\phi} + z \cos{\phi} \\ \end{cases} \end{equation} $$
$$ \begin{equation} \begin{cases} \cfrac{\overline{y}- z \sin{\phi}} {\cos{\phi}} = y \\ \cfrac{\overline{z} + y \sin{\phi}} {\cos{\phi}} = z \\ \end{cases} \end{equation} $$
$$ \begin{equation} \begin{cases} \cfrac{\partial y} {\partial \overline{y}} =\cfrac{1} {\cos{\phi}} \\ \cfrac{\partial z} {\partial \overline{z}} =\cfrac{1} {\cos{\phi}} \\ \end{cases} \end{equation} $$
Using chain rule. $$ \cfrac{\partial f} {\partial \overline{y}} = \cfrac{\partial f} {\partial y} \cfrac{\partial y} {\partial \overline{y}} + \cfrac{\partial f} {\partial z} \cfrac{\partial z} {\partial \overline{y}} $$
$$ \cfrac{\partial f} {\partial \overline{z}} = \cfrac{\partial f} {\partial z} \cfrac{\partial z} {\partial \overline{z}} + \cfrac{\partial f} {\partial y} \cfrac{\partial y} {\partial \overline{z}} $$