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
whose columns are the components of the vectors
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.
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:
This follows from the fact that for a determinant |A B C|
Derivation of Gauss equations. Let S be a simple surface element of class 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
be the position vector to point P on the surface.
Let us now take the vectors as the basis for a coordinate system at point P on surface S and express as linear combinations of . Because of the identity
we can write 4) as
We substitute into 5) using and obtain
Now is a vector normal to the surface at point P with a magnitude of . Thus
Substituting D into 6) we get
into the first two terms we get
Now successively substituting into 7) for we get
Expanding these coefficients and expressing them in terms of the fundamental coefficients E, F, G we obtain
In computing 10) we employ the following
which are the Christoffel symbols of the first kind. They are easily verified by taking the indicated partial derivatives of
Derivation of Weingarten equations. Taking the vectors as the basis for a coordinate system at point P on surface S we can express as linear combinations of as
Because and lie in the tangent plane, α3 = β3 = 0, and the equations are of the form
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 and then as follows:
These equations are equivalent to the following:
Solving the first two of these equations for α1, α2 and the second two for β1, β2 gives