For three commonly used coordinate systems we note

Cartesian coordinates: r = xi+yj+zk, dr = dxi + dyj + dzk.

Cylindrical polar coordinates: r = (r cos θ‎, r sin θ‎,z) dr = dre_r + rdθ‎e_θ‎ + dze_z where e_r = ( cos θ‎, sin θ‎, 0), e_θ‎ = (− sin θ‎, cos θ‎, 0), e_z = (0, 0, 1).

Spherical polar coordinates: r = (r sin θ‎ cos φ‎, r sin θ‎ sin φ‎, r cos θ‎), dr = dre_r + rdθ‎e_θ‎ + r sin θ‎dφ‎e_φ‎ where e_r = ( sin θ‎ cos φ‎, sin θ‎ sin φ‎, cos θ‎), e_θ‎ = ( sin θ‎ cos φ‎, sin θ‎ sin φ‎, cos θ‎), e_φ‎ = (– sin φ‎, cos φ‎,0).

Denoting the coordinates u₁,u₂,u₃, we see in each case that

dr = h₁du₁e₁ + h₂du₂e₂ + h₃du₃e_3,

where each h_i > 0 and e₁,e₂,e₃ is a right-handed orthonormal basis. Such coordinates are called orthogonal curvilinear coordinates, and general expressions for div, grad, and curl exist for such coordinates. (See appendix 16.) In particular the Jacobian equals h₁h₂h₃.