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I am interested in the Frenet-Serret Formulas (theory of curves?) relationship to theory of surfaces.

1) Can one arrive to the Frenet-Serret Formulas starting from the theory of surfaces? Any advice on where to begin?

2) For a surface that contain a space curve: if the unit tangent vector to the curve is the very same unit tangent vector the surface then why/how is the unit normal vector to the curve not in the same direction as the surface's gradient vector?

Thank you