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I am calculating something related to the spheroidal membrane and want to ask you a question.

I consider a oblate spheroid (Oblate spheroidal coordinates can also be considered as a limiting case of ellipsoidal coordinates in which the two largest semi-axes are equal in length.)

In spheroidal coordinate, the relationship to Cartesian coordinates is

[tex]x=a\sqrt((1+u^2) (1-v^2))\cos(\phi)[/tex]

[tex]y=a\sqrt((1+u^2) (1-v^2))\sin(\phi)[/tex]

[tex]z=a u v[/tex]

Now, I want to know how to achieve the normal derivative to the surface of a spheroid, in terms of the derivatives of u, v and [tex]\phi[/tex].

Thank you very much.