Chapter 13.6, Problem 85ES

Let ${\text{u}}_r$ be a unit vector whose counterclockwise angle from the positive $x\text{-axis}$ is $\theta ,$ and let ${\text{u}}_{\theta }$ be a unit vector $90°$ counterclockwise from ${\text{u}}_r.$ Show that if $z=f\left(x,y\right),x=r\cos \theta ,\text{and}y=r\sin \theta ,\text{then}$

$\nabla z=\frac{\partial z}{\partial r}{\text{u}}_r+\frac{1}{r}\frac{\partial z}{\partial \theta }{\text{u}}_{\theta }$