# 拉普拉斯算子

（重定向自拉普拉斯算符

## 定义

${\displaystyle \Delta f=\nabla ^{2}f=\nabla \cdot \nabla f}$  ── (1)

${\displaystyle f}$  的拉普拉斯算子也是笛卡儿坐标系 ${\displaystyle x_{i}}$  中的所有非混合二阶偏导数

${\displaystyle \Delta f=\sum _{i=1}^{n}{\frac {\partial ^{2}f}{\partial x_{i}^{2}}}}$  ── (2)

${\displaystyle \Delta f=\mathrm {tr} (H(f)).\,\!}$

## 坐標表示式

### 二維空間

${\displaystyle \Delta f={\frac {\partial ^{2}f}{\partial x^{2}}}+{\frac {\partial ^{2}f}{\partial y^{2}}}}$

${\displaystyle \Delta f={1 \over r}{\partial \over \partial r}\left(r{\partial f \over \partial r}\right)+{1 \over r^{2}}{\partial ^{2}f \over \partial \theta ^{2}}}$

### 三維空間

${\displaystyle \Delta f={\frac {\partial ^{2}f}{\partial x^{2}}}+{\frac {\partial ^{2}f}{\partial y^{2}}}+{\frac {\partial ^{2}f}{\partial z^{2}}}.}$

${\displaystyle \Delta f={1 \over \rho }{\partial \over \partial \rho }\left(\rho {\partial f \over \partial \rho }\right)+{1 \over \rho ^{2}}{\partial ^{2}f \over \partial \theta ^{2}}+{\partial ^{2}f \over \partial z^{2}}.}$

${\displaystyle \Delta f={1 \over r^{2}}{\partial \over \partial r}\left(r^{2}{\partial f \over \partial r}\right)+{1 \over r^{2}\sin \theta }{\partial \over \partial \theta }\left(\sin \theta {\partial f \over \partial \theta }\right)+{1 \over r^{2}\sin ^{2}\theta }{\partial ^{2}f \over \partial \phi ^{2}}.}$

### N维空间

${\displaystyle \Delta f={\frac {\partial ^{2}f}{\partial r^{2}}}+{\frac {N-1}{r}}{\frac {\partial f}{\partial r}}+{\frac {1}{r^{2}}}\Delta _{S^{N-1}}f}$

## 恒等式

• 如果fg是两个函数，则它们的乘积的拉普拉斯算子为：
${\displaystyle \Delta (fg)=(\Delta f)g+2((\nabla f)\cdot (\nabla g))+f(\Delta g)}$

f是径向函数${\displaystyle f(r)}$ g球谐函数${\displaystyle Y_{lm}(\theta ,\phi )}$ ，是一个特殊情况。这个情况在许多物理模型中有所出现。${\displaystyle f(r)}$ 的梯度是一个径向向量，而角函数的梯度与径向向量相切，因此：

${\displaystyle 2(\nabla f(r))\cdot (\nabla Y_{lm}(\theta ,\phi ))=0}$

${\displaystyle \Delta Y_{\ell m}(\theta ,\phi )=-{\frac {\ell (\ell +1)}{r^{2}}}Y_{\ell m}(\theta ,\phi )}$

${\displaystyle \Delta (f(r)Y_{\ell m}(\theta ,\phi ))=\left({\frac {d^{2}f(r)}{dr^{2}}}+{\frac {2}{r}}{\frac {df(r)}{dr}}-{\frac {\ell (\ell +1)}{r^{2}}}f(r)\right)Y_{\ell m}(\theta ,\phi )}$

## 谱理论

${\displaystyle -\Delta f=\lambda f}$

## 推广

### 复杂空间上的实值函数

${\displaystyle \square ={\partial ^{2} \over \partial x^{2}}+{\partial ^{2} \over \partial y^{2}}+{\partial ^{2} \over \partial z^{2}}-{\frac {1}{c^{2}}}{\partial ^{2} \over \partial t^{2}}.}$

### 值域爲复杂空间

#### 向量值函數的拉普拉斯算子

${\displaystyle \nabla ^{2}\mathbf {A} =(\nabla ^{2}A_{x},\nabla ^{2}A_{y},\nabla ^{2}A_{z})}$

${\displaystyle \nabla ^{2}\mathbf {A} =\nabla (\nabla \cdot \mathbf {A} )-\nabla \times (\nabla \times \mathbf {A} )}$ ，也可用類似于拉普拉斯－德拉姆算子的方式定義，然後證明“旋度的旋度”向量恒等式．

## 参考文献

• Feynman, R, Leighton, R, and Sands, M. Chapter 12: Electrostatic Analogs. The Feynman Lectures on Physics. Volume 2. Addison-Wesley-Longman. 1970.
• Gilbarg, D and Trudinger, N. Elliptic partial differential equations of second order. Springer. 2001. ISBN 978-3540411604.
• Schey, H. M. Div, grad, curl, and all that. W W Norton & Company. 1996. ISBN 978-0393969979.