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Extra info for Pade Approximants for Operators. Theory and Applications
Hyperbolic paraboloid where L is a reference length that is taken equal to one in the unit considered for this example. 3. 134) and all the other Christoffel symbols are equal to zero. 129) to compute the mixed components of the curvature tensor. 135) Note that b~ # bi. 136) Finally, the components of the third fundamental form are 32 2. 137) • The curvature tensor enjoys an additional symmetry property, which involves its covariant derivatives, namely 't/a,/3,A=1,2. 138) This is called the Codazzi Equation.
3. 134) and all the other Christoffel symbols are equal to zero. 129) to compute the mixed components of the curvature tensor. 135) Note that b~ # bi. 136) Finally, the components of the third fundamental form are 32 2. 137) • The curvature tensor enjoys an additional symmetry property, which involves its covariant derivatives, namely 't/a,/3,A=1,2. 138) This is called the Codazzi Equation. 138). 114). 118), we have o ba (3,).. /Lr:(3 - ba >',(3 - b(3/Lr:>. ba (3,).. - b(3/Lr:).. - bO:/Lr;>. ) bo:(3l)..
Corresponding to the direction considered. We call these specific directions the asymptotic directions of the surface. 143) bu = 0 in the new coordinate system. 143) and conclude that the curve obtained by intersecting the surface with the plane defined by the asymptotic direction and the vector normal to the surface has zero curvature at this point. e. 106), we can be more specific about the number of asymptotic directions. For a hyperbolic surface, there are exactly two asymptotic directions, which lie in between the directions of principal curvatures.