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Derivation of the Gauss-Weingarten equations
Problem. Derive the Gauss-Weingarten equations:
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
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
and
Let
Substituting D into 6) we get
Substituting
into the first two terms we get
Now successively substituting into 7)
for
we get
where
Expanding these coefficients and expressing them in terms of the fundamental coefficients E, F, G we obtain
In deriving 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
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