“Laplace's Equation--Toroidal Coordinates”的概念、定义、习题解题方法及翻译-数学辞典

单词

Laplace's Equation--Toroidal Coordinates

释义

Laplace's Equation--Toroidal Coordinates

In Toroidal Coordinates, attempt Separation of Variables to Laplace's Equation by plugging in thetrial solution

(1)

then divide the result by

to obtain

(2)

The function

then separates with

, giving solution

(3)

Plugging

back in and dividing by

gives

(4)

The function

then separates with

, giving solution

(5)

Plugging

back in and multiplying by

gives

(6)

which can also be written

(7)

(Arfken 1970, pp. 114-115). Laplace's Equation is partially separable, although the Helmholtz DifferentialEquation is not.

Laplace's Equation is partially separable, although the Helmholtz Differential Equation is not.

References

Arfken, G. ``Toroidal Coordinates .'' §2.13 in Mathematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 112-114, 1970.

Byerly, W. E. An Elementary Treatise on Fourier's Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. New York: Dover, p. 264, 1959.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 666, 1953.

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。