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The operator

In cartesian coordinates (x,y,z) holds: =xex+yey+zez  ,  gradf=f=fxex+fyey+fzez div a=a=axx+ayy+azz  ,  2f=2fx2+2fy2+2fz2 rot a=×a=(azyayz)ex+(axzazx)ey+(ayxaxy)ez In cylinder coordinates (r,φ,z) holds: =rer+1rφeφ+zez  ,  gradf=frer+1rfφeφ+fzez div a=arr+arr+1raφφ+azz  ,  2f=2fr2+1rfr+1r22fφ2+2fz2 rot a=(1razφaφz)er+(arzazr)eφ+(aφr+aφr1rarφ)ez

In spherical coordinates (r,θ,φ) holds: =rer+1rθeθ+1rsinθφeφgradf=frer+1rfθeθ+1rsinθfφeφdiv a=arr+2arr+1raθθ+aθrtanθ+1rsinθaφφrot a=(1raφθ+aθrtanθ1rsinθaθφ)er+(1rsinθarφaφraφr)eθ+(aθr+aθr1rarθ)eφ2f=2fr2+2rfr+1r22fθ2+1r2tanθfθ+1r2sin2θ2fφ2

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=1h1xu ,  ev=1h2xv ,  ew=1h3xw where the factors hi set the norm to 1. Then holds: gradf=1h1fueu+1h2fvev+1h3fwewdiv 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)ew2f=1h1h2h3[u(h2h3h1fu)+v(h3h1h2fv)+w(h1h2h3fw)]