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LectureNotes12U - Math 497C Curves and Surfaces Fall 2004...

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Nov 11, 2004 1 Curves and Surfaces Fall 2004, PSU Lecture Notes 12 2.6 Gauss’s formulas, and ChristoFel Symbols Let X : U R 3 be a proper regular patch for a surface M , and set X i := D i X . Then { X 1 ,X 2 ,N } may be regarded as a moving bases of frame for R 3 similar to the Frenet Serret frames for curves. We should emphasize, however, two important di±erences: (i) there is no canonical choice of a moving bases for a surface or a piece of surface ( { X 1 ,X 2 ,N } depends on the choice of the chart X ); (ii) in general it is not possible to choose a patch X so that { X 1 ,X 2 ,N } is orthonormal (unless the Gaussian curvature of M vanishes everywhere). The following equations, the ²rst of which is known as Gauss’s formulas , may be regarded as the analog of Frenet-Serret formulas for surfaces: X ij = 2 X k =1 Γ k ij X k + l ij N, and N i = 2 X j =1 l j i X j . The coefficients Γ k ij are known as the ChristoFel symbols , and will be deter- mined below. Recall that l ij are just the coefficients of the second fundamen- tal form. To ²nd out what l j i are note that l ik = −h N, X ik i = h N i ,X k i = 2 X j =1 l j i h X j ,X k i = 2 X j =1 l j i g jk . Thus ( l ij )=( l j i )( g ij ) . So if we let ( g ij ):=( g ij ) 1 , then ( l j i )=( l ij )( g ij ) , which yields l j i = 2 X k =1 l ik g kj . 1
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This note was uploaded on 08/25/2011 for the course MATH 6456 taught by Professor Staff during the Spring '08 term at Georgia Tech.

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LectureNotes12U - Math 497C Curves and Surfaces Fall 2004...

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