“Helmholtz Differential Equation--Spherical Surface”的概念、定义、习题解题方法及翻译-数学辞典

　

单词

Helmholtz Differential Equation--Spherical Surface

释义

Helmholtz Differential Equation--Spherical Surface

On the surface of a Sphere, attempt Separation of Variables in Spherical Coordinates by writing

(1)

then the Helmholtz Differential Equation becomes

(2)

Dividing both sides by

,

(3)

which can now be separated by writing

(4)

The solution to this equation must be periodic, so

must be an Integer. The solution may then be defined either asa Complex function

(5)

for

, ...,

, or as a sum of Real sine and cosine functions

(6)

for

, ...,

. Plugging (4) into (3) gives

(7)

(8)

which is the Legendre Differential Equation for

with

(9)

giving

(10)

(11)

Solutions are therefore Legendre Polynomials with a Complex index. The general Complex solution is then

(12)

and the general Real solution is

(13)

Note that these solutions depend on only a single variable

. However, on the surface of a sphere, it is usual to expresssolutions in terms of the Spherical Harmonics derived for the 3-D spherical case, which depend onthe two variables

and

.

随便看

　

数学辞典收录了8975条数学词条，基本涵盖了常用数学知识及数学英语单词词组的翻译及用法，是数学学习的有利工具。

　

Copyright © 2000-2023 Newdu.com.com All Rights Reserved
更新时间：2025/5/23 19:49:53