拉普拉斯算子

$\text{e} = \lim_{n\to\infty} \left(1+\frac{1}{n}\right)^n$

定义

$\Delta f = \nabla^2 f = \nabla \cdot \nabla f,$   (1)

f 的拉普拉斯算子也是笛卡儿坐标系$x_i$中的所有非混合二阶偏导数

$\Delta f = \sum_{i=1}^n \frac {\partial^2 f}{\partial x^2_i}.$   (2)

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

坐標表示式

二維空間

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

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

三維空間

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

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

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

N 维空间

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

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：
$\Delta(fg)=(\Delta f)g+2((\nabla f)\cdot(\nabla g))+f(\Delta g).$

f是径向函数$f(r)$g球谐函数$Y_{lm}(\theta,\phi)$，是一个特殊情况。这个情况在许多物理模型中有所出现。$f(r)$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：

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

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

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

推广

复杂空间上的实值函数

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

值域爲复杂空间

向量值函數的拉普拉斯算子

$\nabla^2 \mathbf{A} = (\nabla^2 A_x, \nabla^2 A_y, \nabla^2 A_z)$

$\nabla^2 \mathbf{A} = \nabla(\nabla \cdot \mathbf{A}) - \nabla \times (\nabla \times \mathbf{A})$

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.