圆域拉普拉斯方程定解问题的等价转换

Laplace’s Equation in the Polar Coordinates

对圆域拉普拉斯方程定解问题从直角坐标系到极坐标系的等价转换,当前教材中流行着分别侧重数学和物理的两种转换方法.本文从数学角度详细推导了这一等价转换过程,并讨论了两种方法的优缺点.结果表明,教师在讲授过程中宜利用数学分析进行等价转换,具有推理严谨、直观易懂的特点.通过数学分析补充了边界条件后,可进一步追加解释其物理意义,使学生能够从数学理论和实际物理意义两方面融会贯通地理解该定解问题,并较易推广至球形、柱形等物理问题.

For Laplace’s equation in the polar coordinates, there are two popular methods in the current textbooks based on mathematics and physics, respectively.This paper presents the polar form of Laplace’s equation in detail from the viewpoint of mathematics, and discusses the advantages and disadvantages for the two methods.We believe that the mathematical method has the advantages of rigorous reasoning and easy to understand.After supplementing the boundary conditions, instructors can interpret the physics meaning to help students understand this problem well from the views of mathematics and physics, and extend this method to the spherical or cylindrical coordinates setting.