曲线坐标系 谢锡麟 计算度量张量的协变分量矩阵 (9p,9)(9gy,9y)(9y,9)a (9<,9)R3(g;9y)3(9,9)R 1+Ai A1A2 A1A3 A1A2 1+ A2 A2 A3 A1A3 A2A3 A 式中 A1: =or +C(r-or A2:=oy +s(wu-pu); 计算度量张量的逆变分量矩阵 (g2,g)3(g2,gy)3(g2,gs) (y)4|0g)(y) 3(9,g (9, 9 )R3 (9, 9)R3(9s, 9R A2 A1A 1 A3 Aa A2+A2 基于g2=g°gs,则有 999 91929 100 A2 0 AlA3 A1A2A3/(-A1A3-A2431+4+A2 AA1 基于=( ari(-),9 以及现有协变基与逆变基的特点,可有 1(,r3=0 ax2张量分析讲稿谢锡麟 曲线坐标系 谢锡麟 计算度量张量的协变分量矩阵 ( gij) ,   (gx , gx )R3 (gx , gy )R3 (gx , gζ )R3 (gy , gx )R3 (gy , gy )R3 (gy , gζ )R3 (gζ , gx )R3 (gζ , gy )R3 (gζ , gζ )R3   =   1 + A2 1 A1A2 A1A3 A1A2 1 + A2 2 A2A3 A1A3 A2A3 A2 3   , 式中    A1 := ϕx + ζ(ψx − ϕx); A2 := ϕy + ζ(ψy − ϕy); A3 := ψ − ϕ. 计算度量张量的逆变分量矩阵 ( g ij) ,   (g x , g x )R3 (g x , g y )R3 (g x , g ζ )R3 (g y , g x )R3 (g y , g y )R3 (g y , g ζ )R3 (g ζ , g x )R3 (g ζ , g y )R3 (g ζ , g ζ )R3   = ( gij)−1 = 1 A2 3   A2 3 0 −A1A3 0 A2 3 −A2A3 −A1A3 −A2A3 1 + A2 1 + A2 2   , 基于 g i = g isgs , 则有 ( g 1 g 2 g 3 ) = ( g1 g2 g3 ) (g ij) = 1 A2 3   1 0 0 0 1 0 A1 A2 A3     A2 3 0 −A1A3 0 A2 3 −A2A3 −A1A3 −A2A3 1 + A2 1 + A2 2   =   1 0 − A1 A3 0 1 − A2 A3 0 0 1 A3   . 基于 Γ k ij = ( ∂gj ∂xi (x), g k ) R3 , 以及现有协变基与逆变基的特点, 可有 Γ 3 ij = ( ∂gj ∂xi (x), g 3 ) R3 = 1 A3 ∂Aj ∂xi (x), Γ3 33 = 0. 9