Question

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

Alternate coordinate systems

Consider the elliptical-cylindrical coordinate system (eta, psi, z), defined by $$x = a \ \cos h \ \eta \cos \psi, y = a \sin h\ \eta \sin \psi; z = z,\ \eta \ GE \ 0, 0 \ LE \ \psi LE \ 2 \pi, \ z R. In \ PS6$$
it was shown that this is an orthogonal coordinate system with scale factors $$\displaystyle{h}_{{1}}={h}_{{2}}={a}{\left({{\text{cosh}}^{{2}}\ }\eta-{{\cos}^{{2}}\psi}\right)}^{{{\frac{{{1}}}{{{2}}}}}}.$$
Determine the dual bases $$\displaystyle{\left({E}{1},{E}{2},{E}{3}\right)},{\left(\eta,\eta\psi,{z}\right)}.{S}{h}{o}{w}{t}\hat{:}{f}={a}\frac{{1}}{{a}}\frac{{\left({{\text{cosh}}^{{2}}{e}}{a}{t}-{{\cos}^{{s}}\psi}\right)}^{{1}}}{{2}}{\left[\frac{{f}}{\eta}{e}{1}+\frac{{f}}{\psi}{e}{2}+\frac{{f}}{{z}}{e}{3},\frac{{f}}{{w}}{h}{e}{r}{e}{\left({e}{1},{e}{2},{e}{3}\right)}\right.}$$ denotes the unit coordinate basis.

Given that magnitude of gradient along x and y direction is $$\displaystyle{h}_{{1}}={h}_{{2}}={a}{\left({{\text{cosh}}^{{2}}\ }\eta-{{\cos}^{{2}}\psi}\right)}^{{{\frac{{{1}}}{{{2}}}}}}.$$ and along z direction = h3 =1 Hence delf = vectors in direction of del/magnitude $$\displaystyle={\frac{{{1}}}{{{\left({\text{cosh}{\eta}}-{{\cos}^{{2}}\psi}\right)}^{{{\frac{{{1}}}{{{2}}}}}}}}}\rbrace{\left[{\frac{{{\frac{{\partial{f}}}{{\partial\eta}}}}}{{{e}{1}}}}+{\frac{{{\frac{{\partial{f}}}{{\partial\backslash\psi}}}}}{{{e}{2}}}}\right]}+{e}{3}{\frac{{\partial{f}}}{{\partial{z}}}}$$