= Gradient in cylindrical coordinates
{title2=$\nabla\phi=\phi_r\mathbf e_r+r^{-1}\phi_\theta\mathbf e_\theta+\phi_z\mathbf e_z$}
For a differentiable <scalar field> and <cylindrical coordinates> with $r>0$, the <gradient> has the displayed components in the local <orthonormal basis>. The <chain rule> applied to $r=\sqrt{x^2+y^2}$ and a local angle $\theta$ gives $r_x=\cos\theta$, $r_y=\sin\theta$, $\theta_x=-\sin\theta/r$, and $\theta_y=\cos\theta/r$. Substitution in $\nabla\phi=\phi_x\mathbf e_x+\phi_y\mathbf e_y+\phi_z\mathbf e_z$ proves the formula. The angular factor $1/r$ appears because an angular change $d\theta$ sweeps <arc length> $r\,d\theta$.
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