# 格林恆等式

## 格林第一恆等式

$\int_\mathbb{U} \nabla\cdot\mathbf{F} \, \mathrm{d}V = \oint_{\partial \mathbb{U}} \mathbf{F}\cdot\mathbf{n}\, \mathrm{d}S$

$\int_\mathbb{U} (\psi \nabla^2 \phi+\nabla \phi \cdot \nabla \psi)\, \mathrm{d}V = \oint_{\partial \mathbb{U}} \psi{\partial \phi \over \partial n}\, \mathrm{d}S$

## 格林第二恆等式

$\int_\mathbb{U} \left( \psi \nabla^2 \phi - \phi \nabla^2 \psi\right)\, \mathrm{d}V = \oint_{\partial \mathbb{U}} \left( \psi {\partial \phi \over \partial n} - \phi {\partial \psi \over \partial n}\right)\, \mathrm{d}S$

## 格林第三恆等式

$\nabla^2 G(\mathbf{x},\mathbf{x}') = \delta(\mathbf{x} - \mathbf{x}')$

$G(\mathbf{x},\mathbf{x}')={-1 \over 4 \pi\|\mathbf{x} - \mathbf{x}' \|}$

$\psi(\mathbf{x} ) - \int_\mathbb{U} \left[ G(\mathbf{x},\mathbf{x}' ) \nabla'^{\,2} \psi(\mathbf{x}')\right]\, \mathrm{d}V'= \oint_{\partial \mathbb{U}} \left[\psi(\mathbf{x}') {\partial G(\mathbf{x},\mathbf{x}' ) \over \partial n'} - G(\mathbf{x},\mathbf{x}' ) {\partial \psi(\mathbf{x}') \over \partial n'} \right] \, \mathrm{d}S'$

$\nabla'^{\,2} \psi(\mathbf{x}')=0$

$\psi(\mathbf{x}) = \oint_{\partial \mathbb{U}} \left[\psi(\mathbf{x}') {\partial G(\mathbf{x},\mathbf{x}' ) \over \partial n'} - G(\mathbf{x},\mathbf{x}' ) {\partial \psi(\mathbf{x}') \over \partial n'} \right] \, \mathrm{d}S'$

## 參考文獻

1. ^ Strauss, Walter. Partial Differential Equations: An Introduction. Wiley.