曲线坐标系 谢锡麟 D Figure5:轴对称非规则圆管内流动示意 计算度量张量的逆变分量矩阵 )(,)(g,g)(g,9) (92,92) 1+2R2 R 2fR2(1+2n2)0-sR3R R ∈2R4 r2 s3R23 0∈2R4 基于g2=gsgs,则有 Rcos -ERsin n R cos n Rsin n Rcosn ERsin n 1 R R R张量分析讲稿谢锡麟 曲线坐标系 谢锡麟 x y z O R(z) θ H λ θ η O 1 2π H Dλθη (浤靜非) Figure 5: 轴对称非规则圆管内流动示意 计算度量张量的逆变分量矩阵 ( g ij) ,   (g ξ , g ξ )R3 (g ξ , g η )R3 (g ξ , g z )R3 (g η , g ξ )R3 (g η , g η )R3 (g η , g z )R3 (g z , g ξ )R3 (g z , g η )R3 (g z , g z )R3   = ( gij)−1 = 1 ξ 2R4   ξ 2R2 (1 + ξ 2R˙ 2 ) 0 −ξ 3R3R˙ 0 R2 0 −ξ 3R3R˙ 0 ξ 2R4   =   1 + ξ 2R˙ 2 R2 0 − ξR˙ R 0 1 ξ 2R2 0 − ξR˙ R 0 1   . 基于 g i = g isgs , 则有 ( g 1 g 2 g 3 ) = ( g1 g2 g3 ) (g ij) =   R cos η −ξR sin η ξR˙ cos η R sin η ξR cos η ξR˙ sin η 0 0 1     1 + ξ 2R˙ 2 R2 0 − ξR˙ R 0 1 ξ 2R2 0 − ξR˙ R 0 1   =   cos η R − sin η ξR 0 sin η R cos η ξR 0 − ξR˙ R 0 1   . 11