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Derivation of the Gauss-Weingarten equations


Problem. Derive the Gauss-Weingarten equations:



             ole.gif

             ole1.gif

ole2.gif

             ole3.gif

             ole4.gif



Derivation. Before giving the derivation of the equations we first present some notation and a theorem that we shall need.


Determinant notation. We will use the notation


            |A B C|


to denote the determinant


             ole5.gif


whose columns are the components of the vectors


             ole6.gif





Theorem 1. If A, B, C are three linearly independent vectors in three-dimensional space and X is any other vector in 3-space then X can be expressed as a linear combination of vectors A, B and C i.e.


            X = αA + βB + γC .


Furthermore, the vectors X, A, B, C are related by the following identity:



1)        |A B C| X = |B C X| A + |C A X| B + |A B X| C




Proof. From a magic hat we pull the following set of three equations and assert them to be true.



ole7.gif  


Their validity follows from the fact for each equation, i = 1, 2, 3, the first row of the determinant is identical to one of the last three rows.


Let us now expand the determinant by the method of minors using the elements of the first row. We get

 

2)        |B C X| ai - |A C X| bi + |A B X| ci - |A B C| xi = 0                i = 1, 2, 3


which is equivalent to


3)        |A B C| X = |B C X| A + |C A X| B + |A B X| C



Thus since 1) is true and 3) is equivalent to 1), 3) is true.


End of Proof.



Corollary 1. If A, B, C are three linearly independent vectors in three-dimensional space and X is any other vector in 3-space then the vectors X, A, B, C are related by the following identity:


ole8.gif


This follows from the fact that for a determinant |A B C|


             ole9.gif



Derivation of Gauss equations. Let S be a simple surface element of class ole10.gif defined by the one-to-one mapping


            x = x(u, v)

            y = y(u, v)

            z = z(u, v)


of a region R of the uv-plane into xyz-space. Let ole11.gif


             ole12.gif


be the position vector to point P on the surface.


Let us now take the vectors ole13.gif as the basis for a coordinate system at point P on surface S and express ole14.gif as linear combinations of ole15.gif . Because of the identity


             ole16.gif


we can write 4) as


ole17.gif


We substitute into 5) using ole18.gif and obtain


  ole19.gif


Now ole20.gif is a vector normal to the surface at point P with a magnitude of ole21.gif . Thus


             ole22.gif


and


             ole23.gif


Let

 

ole24.gif


Substituting D into 6) we get


  ole25.gif


Substituting


             ole26.gif


into the first two terms we get



ole27.gif


Now successively substituting into 7) ole28.gif for ole29.gif we get



ole30.gif



where



             ole31.gif


 

ole32.gif


             ole33.gif




Expanding these coefficients and expressing them in terms of the fundamental coefficients E, F, G we obtain



             ole34.gif


ole35.gif


             ole36.gif



In deriving 10) we employ the following



             ole37.gif

 

             ole38.gif


             ole39.gif



which are the Christoffel symbols of the first kind. They are easily verified by taking the indicated partial derivatives of

 

             ole40.gif





Derivation of Weingarten equations. Taking the vectors ole41.gif as the basis for a coordinate system at point P on surface S we can express ole42.gif as linear combinations of ole43.gif as


             ole44.gif

             ole45.gif


Because ole46.gif and ole47.gif lie in the tangent plane, α3 = β3 = 0, and the equations are of the form


             ole48.gif

             ole49.gif


Let us now compute the values of the coefficients α1, α2, β1, β2. To that end, take the dot product of both sides of these equations, with first ole50.gif and then ole51.gif as follows:


             ole52.gif


These equations are equivalent to the following:


             ole53.gif


Solving the first two of these equations for α1, α2 and the second two for β1, β2 gives



             ole54.gif


             ole55.gif



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