# 拉普拉斯－贝尔特拉米算子

## 拉普拉斯－贝尔特拉米算子

${\displaystyle g}$ 表示流形上的（伪）-度量张量，我们发现在局部坐标体积形式

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

${\displaystyle \partial _{i}:={\frac {\partial }{\partial x^{i}}}}$

${\displaystyle ({\mbox{div}}X)\;\mathrm {vol} _{n}:={\mathcal {L}}_{X}\mathrm {vol} _{n}}$

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

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

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

${\displaystyle \Delta f={\mbox{div grad}}\;f={\frac {1}{\sqrt {|g|}}}\partial _{i}\left({\sqrt {|g|}}g^{ij}\partial _{j}f\right).}$

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

${\displaystyle |g|=1}$，比如笛卡儿坐标下的欧几里得空间，容易得到

${\displaystyle \Delta f=\partial _{i}\partial ^{i}f}$

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

${\displaystyle \int _{M}df(X)\;\mathrm {vol} _{n}=-\int _{M}f{\mbox{div}}X\;\mathrm {vol} _{n}}$     （证明）

${\displaystyle \int _{M}f\Delta h\;\mathrm {vol} _{n}=-\int _{M}\langle {\mbox{grad}}f,{\mbox{grad}}h\rangle \;\mathrm {vol} _{n}=\int _{M}h\Delta f\;\mathrm {vol} _{n}}$

### 利用共变导数

${\displaystyle H(f)_{ij}=H_{f}(X_{i},X_{j})=(\nabla df)(X_{i},X_{j})=(\nabla _{X_{i}}df)(X_{j})=\nabla _{X_{i}}\nabla _{X_{j}}f-\nabla _{\nabla _{X_{i}}X_{j}}f}$

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

${\displaystyle \Delta f=\nabla ^{a}\nabla _{a}f}$

## 拉普拉斯－德拉姆算子

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

### 性质

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

## 例子

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

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

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

## 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).