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

Consider the elliptical-cylindrical coordinate system (eta, psi, z), defined by
it was shown that this is an orthogonal coordinate system with scale factors
Determine the dual bases $\left(E1,E2,E3\right),\left(\eta ,\eta \psi ,z\right).Showt\stackrel{^}{:}f=a\frac{1}{a}\frac{{\left({\text{cosh}}^{2}eat-{\mathrm{cos}}^{s}\psi \right)}^{1}}{2}\left[\frac{f}{\eta }e1+\frac{f}{\psi }e2+\frac{f}{z}e3,\frac{f}{w}here\left(e1,e2,e3\right)$ denotes the unit coordinate basis.

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Pohanginah
Given that magnitude of gradient along x and y direction is and along z direction = h3 =1 Hence delf = vectors in direction of del/magnitude $=\frac{1}{{\left(\text{cosh}\eta -{\mathrm{cos}}^{2}\psi \right)}^{\frac{1}{2}}}\right\}\left[\frac{\frac{\partial f}{\partial \eta }}{e1}+\frac{\frac{\partial f}{\partial \mathrm{\setminus }\psi }}{e2}\right]+e3\frac{\partial f}{\partial z}$