二维曲面理论引入数学物理方法的探讨

THE DISCUSSION ON INTRODUCING 2-DIMENSIONAL SURFACE THEORY INTO THE METHODS OF MATHEMATICAL PHYSICS

本文探讨如何将二维曲面理论引入到数学物理方法的教学当中,目的在于沟通从高等数学到一般微分几何知识之间的“鸿沟”。使得本科生在学习更抽象的微分几何前,具备一些直观的图像。文中主要利用嵌入到三维空间的曲面参数方程,计算给出二维曲面理论的第一、二基本形式,并且基于它们,建立起曲面上线元、面元、曲线夹角、曲率、测地线、平行移动的概念。运用这些概念,我们直接推导出在二维曲面上梯度、散度、旋度的数学表达式及对应的高斯、斯托克斯定理。这些式子经过简单推广便可以得到更高维曲线坐标系中梯度、散度、旋度的表达。作为曲面理论的一个直接应用,我们也讨论了如何将所得到曲面理论运用到肥皂膜上的流体,得到二维固定曲面上流体所满足的动力学方程,从而可以解释皂膜上的等厚干涉条纹的变化。最后我们介绍二维曲面理论到高维黎曼内蕴几何的推广以及在广义相对论上的应用。

This article mainly discusses how to introduce the 2-dimensional surface theory into the teaching of the methods of mathematical physics. We aim to communicate the "gap" between advanced mathematics and general differential geometry knowledge. It allows undergraduate students to have some intuitive images before learning more abstract differential geometry. The paper mainly uses the surface parameter equations embedded in the three-dimensional space to calculate the first and second basic forms of the 2-dimensional surface theory. Basing on them, the concept of the line elements, surface elements, the angle between curves, curvature, geodesics, parallel translation could be established. Using these concepts, we derive the mathematical formulas of gradient, divergence, and curl on a two-dimensional surface. We also obtain the corresponding Gauss、Stokes theorem. These formulas can be simply extended to obtain the expression of gradient, divergence, and curl in a higher-dimensional curvilinear coordinate system. As a direct application of the surface theory, we also discussed how to apply the surface theory to the fluid on the soap film to obtain the dynamic equation on the two-dimensional fixed surface, which can explain the motion of interference stripes of equal thickness on the soap film. Finally, we introduced the extension of two-dimensional surface theory to high-dimensional Riemannian intrinsic geometry and its application to general relativity.