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* 구면좌표계: <math>\psi=\psi(r,\theta,\phi)</math> | * 구면좌표계: <math>\psi=\psi(r,\theta,\phi)</math> | ||
: <math>\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial \psi}{\partial r}\right)+\frac{1}{r^2\sin\theta}\frac{\partial}{\partial \theta}\left(\frac{\partial \psi}{\partial \theta}\right)+\frac{1}{r^2\sin^2\theta}\frac{\partial^2 \psi}{\partial \phi^2}=0</math> | : <math>\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial \psi}{\partial r}\right)+\frac{1}{r^2\sin\theta}\frac{\partial}{\partial \theta}\left(\frac{\partial \psi}{\partial \theta}\right)+\frac{1}{r^2\sin^2\theta}\frac{\partial^2 \psi}{\partial \phi^2}=0</math> | ||
=== 2차원 좌표계 === | |||
* 직각좌표계: <math>\psi=\psi(x,y)</math> | |||
: <math>\frac{\partial^2 \psi}{\partial x^2}+\frac{\partial^2 \psi}{\partial y^2}=0</math> | |||
* 극좌표계: <math>\psi=\psi(r,\theta)</math> | |||
: <math>\frac{\partial^2\psi}{\partial r^2}+\frac{1}{r}\frac{\partial \psi}{\partial r}+\frac{1}{r^2}\frac{\partial^2\psi}{\partial \theta^2}=0</math> | |||
{{각주}} | {{각주}} | ||
[[분류:미분방정식]] | [[분류:미분방정식]] |
2015년 5월 17일 (일) 00:25 판
정의
다변수함수 [math]\displaystyle{ \psi=\psi(x_1,x_2,\cdots,x_n) }[/math]에 대한 편미분방정식
- [math]\displaystyle{ \triangledown^2 \psi=0 }[/math]
를 라플라스 방정식(Laplace equation)이라 하고, 라플라스 방정식을 만족하는 함수를 조화함수(Harmonic function)라고 한다.
공식
3차원 좌표계[1]
- 직각좌표계: [math]\displaystyle{ \psi=\psi(x,y,z) }[/math]
- [math]\displaystyle{ \frac{\partial^2 \psi}{\partial x^2}+\frac{\partial^2 \psi}{\partial y^2}+\frac{\partial^2 \psi}{\partial z^2}=0 }[/math]
- 원통좌표계: [math]\displaystyle{ \psi=\psi(r,\phi,z) }[/math]
- [math]\displaystyle{ \frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial \psi}{\partial r}\right)+\frac{1}{r^2}\frac{\partial^2\psi}{\partial \phi^2}+\frac{\partial^2\phi}{\partial z^2}=0 }[/math]
- 구면좌표계: [math]\displaystyle{ \psi=\psi(r,\theta,\phi) }[/math]
- [math]\displaystyle{ \frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial \psi}{\partial r}\right)+\frac{1}{r^2\sin\theta}\frac{\partial}{\partial \theta}\left(\frac{\partial \psi}{\partial \theta}\right)+\frac{1}{r^2\sin^2\theta}\frac{\partial^2 \psi}{\partial \phi^2}=0 }[/math]
2차원 좌표계
- 직각좌표계: [math]\displaystyle{ \psi=\psi(x,y) }[/math]
- [math]\displaystyle{ \frac{\partial^2 \psi}{\partial x^2}+\frac{\partial^2 \psi}{\partial y^2}=0 }[/math]
- 극좌표계: [math]\displaystyle{ \psi=\psi(r,\theta) }[/math]
- [math]\displaystyle{ \frac{\partial^2\psi}{\partial r^2}+\frac{1}{r}\frac{\partial \psi}{\partial r}+\frac{1}{r^2}\frac{\partial^2\psi}{\partial \theta^2}=0 }[/math]
각주
- ↑ Stephen T. Thornton · Jerry B. Marion (2011). 강석태 옮김. 『일반역학』(제5판). Cengage Learning. pp. 678-681. ISBN 9788962183009