In cartesian coordinates (x,y,z) holds: →∇=∂∂x→ex+∂∂y→ey+∂∂z→ez , gradf=→∇f=∂f∂x→ex+∂f∂y→ey+∂f∂z→ez div →a=→∇⋅→a=∂ax∂x+∂ay∂y+∂az∂z , ∇2f=∂2f∂x2+∂2f∂y2+∂2f∂z2 rot →a=→∇×→a=(∂az∂y−∂ay∂z)→ex+(∂ax∂z−∂az∂x)→ey+(∂ay∂x−∂ax∂y)→ez In cylinder coordinates (r,φ,z) holds: →∇=∂∂r→er+1r∂∂φ→eφ+∂∂z→ez , gradf=∂f∂r→er+1r∂f∂φ→eφ+∂f∂z→ez div →a=∂ar∂r+arr+1r∂aφ∂φ+∂az∂z , ∇2f=∂2f∂r2+1r∂f∂r+1r2∂2f∂φ2+∂2f∂z2 rot →a=(1r∂az∂φ−∂aφ∂z)→er+(∂ar∂z−∂az∂r)→eφ+(∂aφ∂r+aφr−1r∂ar∂φ)→ez
In spherical coordinates (r,θ,φ) holds: →∇=∂∂r→er+1r∂∂θ→eθ+1rsinθ∂∂φ→eφgradf=∂f∂r→er+1r∂f∂θ→eθ+1rsinθ∂f∂φ→eφdiv →a=∂ar∂r+2arr+1r∂aθ∂θ+aθrtanθ+1rsinθ∂aφ∂φrot →a=(1r∂aφ∂θ+aθrtanθ−1rsinθ∂aθ∂φ)→er+(1rsinθ∂ar∂φ−∂aφ∂r−aφr)→eθ+(∂aθ∂r+aθr−1r∂ar∂θ)→eφ∇2f=∂2f∂r2+2r∂f∂r+1r2∂2f∂θ2+1r2tanθ∂f∂θ+1r2sin2θ∂2f∂φ2
General orthonormal curvelinear coordinates (u,v,w) can be obtained from cartesian coordinates by the transformation →x=→x(u,v,w). The unit vectors are then given by: →eu=1h1∂→x∂u , →ev=1h2∂→x∂v , →ew=1h3∂→x∂w where the factors hi set the norm to 1. Then holds: gradf=1h1∂f∂u→eu+1h2∂f∂v→ev+1h3∂f∂w→ewdiv →a=1h1h2h3(∂∂u(h2h3au)+∂∂v(h3h1av)+∂∂w(h1h2aw))rot →a=1h2h3(∂(h3aw)∂v−∂(h2av)∂w)→eu+1h3h1(∂(h1au)∂w−∂(h3aw)∂u)→ev+1h1h2(∂(h2av)∂u−∂(h1au)∂v)→ew∇2f=1h1h2h3[∂∂u(h2h3h1∂f∂u)+∂∂v(h3h1h2∂f∂v)+∂∂w(h1h2h3∂f∂w)]