曲线正交坐标系下_∇^→_A^→与∇² _A^→的一种推导

A Derivation of _∇^→_A^→ and ∇² _A^→ in Curvilinear Orthogonal Coordinates

摘要: 电动力学教材一般仅仅在附录部分给出曲线正交坐标系下标量的梯度_∇^→Ψ、矢量的散度_∇^→•_A^→和旋度_∇^→×_A^→，以及标量的拉普拉斯算符∇²Ψ的一般表达式。在教学中我们发现，_∇^→ _A^→和∇² _A^→在曲线正交坐标系下的表达式也是很重要的。本文从微分几何的角度推导出三维欧氏空间中曲线正交坐标系下_∇^→ _A^→和∇² _A^→的一般表达式，以及它们在直角坐标系、柱坐标系和球坐标系的具体表达式，供感兴趣的学生和教师参考。

Abstract: In textbooks of electrodynamics, the formulas of the gradient _∇^→Ψ, divergence _∇^→•_A^→, curl _∇^→×_A^→，and Laplace operator ∇²Ψ in curvilinear orthogonal coordinates are often given in the appendix. However, in the teaching process, we find that the expressions of _∇^→ _A^→ and ∇² _A^→ in curvilinear orthogonal coordinates are also very important. In this paper, the general expressions of _∇^→ _A^→ and ∇² _A^→ in curvilinear orthogonal coordinates in three-dimensional Euclidean space are derived based on the method of differential geometry. And the formulas of _∇^→ _A^→ and ∇² _A^→ in the Cartesian coordinates, cylindrical coordinates and spherical coordinates are also given for convenience of interested students and teachers.