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[[방정식,equation#s-11]] Poisson and Laplace
[[방정식,equation#s-12]] Poisson
[[방정식,equation#s-13]] Laplace
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 $\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0$

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//from 수학백과
라플라스 연산자([[라플라시안,Laplacian]])를 취했을 때, 0이 되는 [[함수,function]]를 찾는 방정식.

라플라스 방정식의 [[해,solution]]인 함수는 [[조화함수,harmonic_function]].

[[차원,dimension]]에 따라, (=0이 되는 LHS는 ???) 다음과 같이 표시.
 (2차원) $\nabla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}$
 (2차원) $\nabla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2}$
 (n차원) $\nabla^2 u = \frac{\partial^2 u}{\partial x_1^2} + \frac{\partial^2 u}{\partial x_2^2} + \cdots + \frac{\partial^2 u}{\partial x_n^2}$
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Laplace DE는 "(a particular case of the Helmholtz equation)". [* WpEn:Helmholtz_equation ]
rel. [[헬름홀츠_방정식,Helmholtz_equation]]
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[[https://terms.naver.com/entry.naver?docId=3405052&cid=47324&categoryId=47324 수학백과: 라플라스 방정식]]
https://mathworld.wolfram.com/LaplacesEquation.html
https://everything2.com/title/Laplace%2527s+equation

Up: [[편미분방정식,partial_differential_equation,PDE]]