Page 150 - Intro to Tensor Calculus

145

The tensor derivative of this equation with respect to the surface coordinates gives

i j
i
j
g ij n x + g ij n x α,β =0. (1.5.77)
β α
Substitute into equation (1.5.77) the relations from equations (1.5.57), (1.5.71) and (1.5.75) and show that

γ
b αβ = −a αγ η . (1.5.78)
β
γ
Solving the equation (1.5.78) for the coefficients η we find
β
γ αγ
η = −a b αβ . (1.5.79)
β
Now substituting equation (1.5.79) into the equation (1.5.75) produces the Weingarten formula

i
n i = −a γβ b γαx . (1.5.80)
,α β
This is a relation for the derivative of the unit normal in terms of the surface metric, curvature tensor and
surface tangents.
A third fundamental form of the surface is given by the quadratic form

α
C = c αβ du du β (1.5.81)

where c αβ is defined as the symmetric surface tensor

j
i
c αβ = g ij n n . (1.5.82)
,α
,β
By using the Weingarten formula in the equation (1.5.81) one can verify that
γδ
c αβ = a b αγ b βδ . (1.5.83)

Geodesic Coordinates

In a Cartesian coordinate system the metric tensor g ij is a constant and consequently the Christoffel
symbols are zero at all points of the space. This is because the Christoffel symbols are dependent upon
the derivatives of the metric tensor which is constant. If the space V N is not Cartesian then the Christoffel
symbols do not vanish at all points of the space. However, it is possible to find a coordinate system where
the Christoffel symbols will all vanish at a given point P of the space. Such coordinates are called geodesic
coordinates of the point P.
α
Consider a two dimensional surface with surface coordinates u and surface metric a αβ . If we transform
α
to some other two dimensional coordinate system, say ¯u with metric ¯a αβ , where the two coordinates are
related by transformation equations of the form

1
2
α
α
u = u (¯u , ¯u ), α =1, 2, (1.5.84)