tailieunhanh - Handbook of mathematics for engineers and scienteists part 62

Tham khảo tài liệu 'handbook of mathematics for engineers and scienteists part 62', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | . Theory of Surfaces 395 . Intrinsic Geometry of Surface . Intrinsic geometry and bending of surface. Suppose that two surfaces U and U are given and there is one-to-one correspondence between their points such that the length of each curve on U is equal to the length of the corresponding curve on U . Such a one-to-one mapping of U into U is called a bending of the surface U into the surface U and the surfaces U and U are said to be applicable. The correspondence between U and U is said to be isometric. The intrinsic geometry of a surface studies geometric constructions and quantities related to the surface that can be determined solely from the first quadratic form. The notions of length of a segment angle between two curves and area of part of a surface all belong in intrinsic geometry. on the opposite the curvature of a curve given on the surface by the equations u u t v v t cannot be found using only the first quadratic form and hence it does not belong in intrinsic geometry. . Index notation. Surface as Riemannian space. From now on in this chapter the following notation related to tensor analysis is used u1 u u2 v gii E gi2 g2i F g22 G bii L bi2 b2i M b22 N ri r r2 rv ri2 r 2i rvu ruv rn ruu r22 rw. In the new notation the first fundamental quadratic form becomes 22 E du2 2F dudv G dv2 EE ga3 dua du3 ga3 dua du3 and the second fundamental quadratic form is 22 L du2 2M dudv Ndv2 EE bag dua du3 ba 3 dua du3. The expressions for the coefficients of the first and second quadratic forms in the new notation become gij ri rj bj rj N where N is the unit normal vector to the surface i and j are equal to either i or 2. 396 Differential Geometry . Derivation formula. The Christoffel symbols Lkij of the first kind are defined to be the scalar products of the vectors rk and Fjj . rk rij k ij. The Christoffel symbols of the first kind satisfy the formula r _ 1 dfiki Nj dgij X k ij 2 duj dù duk This is one of the basic formulas in the theory