正交曲线坐标系下梯度、散度和旋度的一般计算方法

Calculating Gradient, Divergence, and Rotation under Orthogonal Curvilinear Coordinates

摘要: 在正交曲线坐标系下求标量场的梯度和矢量场的散度、旋度一般需要用到拉梅系数，其公式复杂难记。根据张量分析，标量场的梯度和矢量场的散度、旋度应该不依赖于坐标系，在不同的坐标系下应该有统一形式的一般求法。针对柱坐标系和球坐标系，通过简单的分析，给出了基于度规表达式的计算公式，一般性地解决了标量场的梯度和矢量场的散度、旋度的计算问题。本文的讨论，有助于理解拉梅系数产生的原因，更好地理解和认识梯度、散度和旋度的计算。通过对直角坐标系和一般曲线坐标系中结果的分析，澄清了正交曲线坐标系中柱坐标系和球坐标系的基底和对应的局域活动标架基底的关系。

Abstract: Lame coefficients must be considered in calculating gradient, divergence, rotation under ortho-gonal curvilinear coordinates. The expressions and equations are pretty complicated and hard to remember. According to tensor analysis, scalar field’s gradient and vector field’s divergence, rotation should be independent on coordinate system, and they can be calculated via a general method and equation. Thinking about the cylindrical coordinate system and spherical coordinate system, the concrete expressions of gradient, divergence, and rotation are obtained through the given metrics. The Lame coefficients can be understood, and the gradient, divergence, rotation conceptions can be clarified. Through the calculation process, we can distinguish the coordinate base of an orthogonal curvilinear coordinate system and the coordinate base of a local moving frame.