Based on this pdf

Coordinate	$h_{1}$	$h_{2}$	$h_{3}$	$u_{1}$	$u_{2}$	$u_{3}$	$n_{1}$	$n_{2}$	$n_{3}$
Cartesian	$1$	$1$	$1$	$x$	$y$	$z$	$\overset{x}{^}$	$\overset{y}{^}$	$\overset{z}{^}$
Cylindrical	$1$	$s$	$1$	$s$	$ϕ$	$z$	$\overset{s}{^}$	$\hat{ϕ}$	$\overset{z}{^}$
Spherical	$1$	$ρ$	$ρ sin θ$	$ρ$	$θ$	$ϕ$	$\overset{ρ}{^}$	$\hat{θ}$	$\hat{ϕ}$

$\nabla f = \frac{1}{h _{i}} \partial_{u_{i}} f n_{i}$

$\nabla \cdot f = \frac{1}{h _{1} h _{2} h _{3}} \partial_{u_{i}} (h_{j} h_{k} f_{i})$

$\nabla \times f = \frac{1}{h _{j} h _{k}} ε_{ijk} \partial_{u_{j}} (h_{k} f_{k}) n_{i}$

Cartesian

$\nabla f = ⟨ \partial_{x} f, \partial_{y} f, \partial_{z} f ⟩$

Cylindrical

Spherical

$\nabla f = \partial_{ρ} f \overset{ρ}{^} + \frac{1}{ρ} \partial_{θ} f \hat{θ} + \frac{1}{ρ sin θ} \partial_{ϕ} f \hat{ϕ}$

$θ$ is the azimuthal angle

$\nabla \times f = \frac{1}{ρ sin θ} [\partial_{θ} (f_{ϕ} sin θ) - \partial_{ϕ} f_{θ}] \overset{ρ}{^} + \frac{1}{ρ} [\frac{1}{sin θ} \partial_{ϕ} f_{ρ} - \partial_{ρ} (ρ f_{ϕ})] \hat{θ} + \frac{1}{ρ} [\partial_{ρ} (ρ f_{θ}) - \partial_{θ} f_{ρ}] \hat{ϕ}$