# 拉普拉斯－贝尔特拉米算子

## 拉普拉斯－贝尔特拉米算子

$g$ 表示流形上的（伪）-度量张量，我们发现在局部坐标体积形式

$\mathrm{vol}_n := \sqrt{|g|} \;dx^1\wedge \ldots \wedge dx^n$

$\partial_i := \frac {\partial}{\partial x^i}$

$(\mbox{div} X) \; \mathrm{vol}_n := \mathcal{L}_X \mathrm{vol}_n$

$\mbox{div} X = \frac{1}{\sqrt{|g|}} \partial_i \left(\sqrt {|g|} X^i\right).$

$\langle \mbox{grad} f(x) , v_x \rangle = df(x)(v_x)$

$\left(\mbox{grad} f\right)^i = \partial^i f = g^{ij} \partial_j f.$

$\Delta f = \mbox{div grad} \; f = \frac{1}{\sqrt {|g|}} \partial_i \left(\sqrt{|g|} g^{ij} \partial_j f \right).$

$\Delta f = \partial_i \partial^i f + (\partial^i f) \partial_i \ln \sqrt{|g|}.$

$|g| = 1$，比如笛卡儿坐标下的欧几里得空间，容易得到

$\Delta f = \partial_i \partial^i f$

$\Delta f = g^{ij}\left(\frac{\partial^2 f}{\partial u^i\, \partial u^j} - \Gamma_{ij}^k \frac{\partial f}{\partial u^k} \right).$

$\int_M df(X) \;\mathrm{vol}_n = - \int_M f \mbox{div} X \;\mathrm{vol}_n$     （证明）

$\int_M f\Delta h \;\mathrm{vol}_n = - \int_M \langle \mbox{grad} f, \mbox{grad} h \rangle \;\mathrm{vol}_n = \int_M h\Delta f \;\mathrm{vol}_n$

### 利用共变导数

$H(f)_{ij}=H_f(X_i, X_j) =(\nabla df)(X_i, X_j)=(\nabla_{X_i} df)(X_j)= \nabla_{X_i}\nabla_{X_j} f - \nabla_{\nabla_{X_i}X_j} f$

$\Delta f = \sum_{ij} g^{ij} H(f)_{ij}.$

$\Delta f = \nabla^a \nabla_a f$

## 拉普拉斯－德拉姆算子

$\Delta= \mathrm{d}\delta+\delta\mathrm{d} = (\mathrm{d}+\delta)^2,\;$

### 性质

1. $\Delta(af + h) = a\,\Delta f + \Delta h\!$
2. $\Delta(fh) = f \,\Delta h + 2 (\partial_i f) (\partial^i h) + h\, \Delta f$    （证明）

## 例子

$\Delta_{S^{n-1}}f(x) = \Delta f(x/|x|)$

$\Delta f = r^{-n}\frac{\partial}{\partial r}\left(r^n\frac{\partial f}{\partial r}\right) + r^{-2}\Delta_{S^{n-1}}f.$

$\Delta f(t,\xi) = \sin^{1-n}t \frac{\partial}{\partial t}\left(\sin^{n-1}t\frac{\partial f}{\partial t}\right) + \sin^{-2}t\Delta_\xi f$

## 参考文献

• Flanders, H. Differential forms with applications to the physical sciences. Dover. 1989. ISBN 978-0486661698.
• Jürgen Jost, Riemannian Geometry and Geometric Analysis, (2002) Springer-Verlag, Berlin ISBN 3-540-42627-2 . (Provides a general introduction to curved surfaces).