Prove the formula of gradient in cylindrical coordinates. 
Let $U = (0, +\infty)\times (0,2\pi)\times \mathbb{R}\space$ and $\space V = \mathbb{R}^{3} - \lbrace (x,0,0)\in \mathbb{R}^{3}\mid x\geq0 \rbrace\space.$ Show that $\varphi: U \to V$ defined by $\varphi(r,\theta,z) = (r\cos \theta, r\sin \theta, z)\space$ is a diffeomorphism $C^{\infty}$. Given $f: V \to \mathbb{R}$ differentiable, explain the meaning and prove the formula
  $$\nabla f = \frac{\partial f}{\partial r}u_{r} + \frac{1}{r}\frac{\partial f}{\partial \theta}u_{\theta} + \frac{\partial f}{\partial z}u_{z},$$
  where $u_{r}, u_{\theta}, u_{z}$ are unit vectors tangent to curves $r, \theta$ and $z$ in $V$.

I already showed that $\varphi$ is a difeomorphism $C^{\infty}$ (basically, it comes down to coordinates). But, I have no idea about the meaning and proof of the formula. Any hint?
 A: The gradient of $V$ is characterized by the relation $Df(p)(v) = \langle v, \nabla f(p)\rangle$, for all $v$. So if you want to describe $\nabla f(p)$ using cylindrical coordinates, you must also describe the inner product in these coordinates. All the inner products can be described in a single matrix: $$\begin{pmatrix} \langle \varphi_r,\varphi_r\rangle & \langle \varphi_r, \varphi_\theta\rangle & \langle \varphi_r,\varphi_z\rangle \\  \langle \varphi_\theta,\varphi_r\rangle & \langle \varphi_\theta, \varphi_\theta\rangle & \langle \varphi_\theta,\varphi_z\rangle \\  \langle \varphi_z,\varphi_r\rangle & \langle \varphi_z, \varphi_\theta\rangle & \langle \varphi_z,\varphi_z\rangle \end{pmatrix} = \begin{pmatrix} 1 & 0 & 0 \\0 & r^2 & 0 \\ 0 
& 0 & 1\end{pmatrix},$$whose inverse is $$\begin{pmatrix} 1 & 0 & 0 \\0 & r^2 & 0 \\ 0 
& 0 & 1\end{pmatrix}^{-1} = \begin{pmatrix} 1 & 0 & 0 \\0 & r^{-2} & 0 \\ 0 
& 0 & 1\end{pmatrix}.$$We in particular see from the above that $u_r = \varphi_r$, $u_\theta = \varphi_\theta/r$ and $u_z = \varphi_z$. Hence $$\begin{align}\nabla f &= \begin{pmatrix} \frac{\partial f}{\partial r} & \frac{\partial f}{\partial \theta} & \frac{\partial f}{\partial z}\end{pmatrix}\begin{pmatrix} 1 & 0 & 0 \\0 & r^{-2} & 0 \\ 0 
& 0 & 1\end{pmatrix} \begin{pmatrix} \varphi_r \\ \varphi_\theta \\ \varphi_z\end{pmatrix} \\ &=\begin{pmatrix} \frac{\partial f}{\partial r} & \frac{\partial f}{\partial \theta} & \frac{\partial f}{\partial z}\end{pmatrix} \begin{pmatrix} u_r \\ u_\theta/r \\ u_z \end{pmatrix} \\ &= \frac{\partial f}{\partial r}u_r + \frac{1}{r} \frac{\partial f}{\partial \theta}u_\theta + \frac{\partial f}{\partial z} u_z,\end{align}$$as wanted.
