关键词: 张量分析, 几何方程, 平衡微分方程, 柱坐标系, 球坐标系, 弹性力学

Abstract: First, using the relations of the unit vectors between orthogonal curvilinear coordinates and Cartesian coordinates, and the invariability of the unit vectors in Cartesian coordinates, the gradient operator in both coordinates, and the partial derivative of the unit vectors to the coordinates are derived from the perspective of the transformation of the unit vectors. Then, using the fundamentals of the theorems of tensor field, the gradient of displacement vector and the divergence of stress tensor are derived by derivation. Last, the geometric equations of strain and the differential equations of equilibrium are derived in details in cylindrical coordinates and spherical coordinates from their forms expressed by tensor.

Key words: tensor analysis, geometric equations, equilibrium differential equations, cylindrical coordinates, spherical coordinates, elasticity