Normal derivative in spheroidal coordinate

[dexturelab]

Hi PhysicsForums,
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
$$x=a\sqrt((1+u^2) (1-v^2))\cos(\phi)$$
$$y=a\sqrt((1+u^2) (1-v^2))\sin(\phi)$$
$$z=a u v$$

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 $$\phi$$.

Thank you very much.

 

[fresh_42]

Just calculate a basis of the tangent space and take the cross product.