Weingarten derivational formulas

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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 be the position vector of the surface, let be the unit normal vector and let , , , , , be the coefficients of the first and second fundamental forms of the surface, respectively; the Weingarten derivational formulas will then take the form

The formulas were established in 1861 by J. Weingarten.


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



[a1] W. Blaschke, K. Leichtweiss, "Elementare Differentialgeometrie" , Springer (1973)
[a2] N.J. Hicks, "Notes on differential geometry" , v. Nostrand (1965)
How to Cite This Entry:
Weingarten derivational formulas. A.B. Ivanov (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098