Weingarten derivational formulas

Formulas yielding the expansion of the derivative of the unit normal vector to a surface in terms of the first derivatives of the position vector of this surface. Let $\mathbf r = {\mathbf r } ( u, v)$ be the position vector of the surface, let $\mathbf n$ be the unit normal vector and let $E$, $F$, $G$, $L$, $M$, $N$ be the coefficients of the first and second fundamental forms of the surface, respectively; the Weingarten derivational formulas will then take the form

$$\mathbf n _ {u} = \frac{FM- GL }{EG - F ^ { 2 } } \mathbf r _ {u} + FL- \frac{EM}{EG- F ^ { 2 } } \mathbf r _ {v} ,$$

$$\mathbf n _ {v} = FN- \frac{GM}{EG- F ^ { 2 } } \mathbf r _ {u} + FM- \frac{EN}{EG- F ^ { 2 } } \mathbf r _ {v} .$$

The formulas were established in 1861 by J. Weingarten.

References

[1]	P.K. Rashevskii, "A course of differential geometry" , Moscow (1956) (In Russian)

References

[a1]	W. Blaschke, K. Leichtweiss, "Elementare Differentialgeometrie" , Springer (1973)
[a2]	N.J. Hicks, "Notes on differential geometry" , v. Nostrand (1965)