Consider the elliptical-cylindrical coordinate system (eta, psi, z)

Consider the elliptical-cylindrical coordinate system (eta, psi, z), defined by $x=a\cos h\eta \cos \psi ,y=a\sin h\eta \sin \psi ;z=z,\eta GE0,0LE\psi LE2\pi ,zR.InPS6$
it was shown that this is an orthogonal coordinate system with scale factors ${\displaystyle h_1=h_2=a{({\text{cosh}}^2\eta -{\cos }^2\psi )}^{\frac{1}{2}}.}$
Determine the dual bases ${\displaystyle (E1,E2,E3),(\eta ,\eta \psi ,z).Showt\widehat{:}f=a\frac{1}{a}\frac{{({\text{cosh}}^{2}eat-{\cos }^s\psi )}^1}{2}[\frac{f}{\eta }e1+\frac{f}{\psi }e2+\frac{f}{z}e3,\frac{f}{w}here(e1,e2,e3)}$ denotes the unit coordinate basis.