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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


             ole.gif


whose columns are the components of the vectors


             ole1.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.



ole2.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:


ole3.gif


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


             ole4.gif



Derivation of Gauss equations. Let S be a simple surface element of class ole5.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 ole6.gif


             ole7.gif


be the position vector to point P on the surface.


Let us now take the vectors ole8.gif as the basis for a coordinate system at point P on surface S and express ole9.gif as linear combinations of ole10.gif . Because of the identity


             ole11.gif


we can write 4) as


ole12.gif


We substitute into 5) using ole13.gif and obtain


  ole14.gif


Now ole15.gif is a vector normal to the surface at point P with a magnitude of ole16.gif . Thus


             ole17.gif


and


             ole18.gif


Let

 

ole19.gif


Substituting D into 6) we get


  ole20.gif


Substituting


             ole21.gif


into the first two terms we get



ole22.gif


Now successively substituting into 7) ole23.gif for ole24.gif we get



ole25.gif



where



             ole26.gif


 

ole27.gif


             ole28.gif




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



             ole29.gif


ole30.gif


             ole31.gif



In computing 10) we employ the following



             ole32.gif

 

             ole33.gif


             ole34.gif



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

 

             ole35.gif





Derivation of Weingarten equations. Taking the vectors ole36.gif as the basis for a coordinate system at point P on surface S we can express ole37.gif as linear combinations of ole38.gif as


             ole39.gif

             ole40.gif


Because ole41.gif and ole42.gif lie in the tangent plane, α3 = β3 = 0, and the equations are of the form


             ole43.gif

             ole44.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 ole45.gif and then ole46.gif as follows:


             ole47.gif


These equations are equivalent to the following:


             ole48.gif


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



             ole49.gif


             ole50.gif


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