# 拉普拉斯算子

## 定義

${\displaystyle \nabla ^{2}}$

${\displaystyle =\nabla \cdot \nabla }$

${\displaystyle =(\mathbf {i} {\frac {\partial }{\partial x}}+\mathbf {j} {\frac {\partial }{\partial y}}+\mathbf {k} {\frac {\partial }{\partial z}})\cdot (\mathbf {i} {\frac {\partial }{\partial x}}+\mathbf {j} {\frac {\partial }{\partial y}}+\mathbf {k} {\frac {\partial }{\partial z}})}$

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

${\displaystyle \nabla ^{2}\mathbf {v} }$

${\displaystyle =\nabla ^{2}v_{x}\mathbf {i} +\nabla ^{2}v_{y}\mathbf {j} +\nabla ^{2}v_{z}\mathbf {k} }$

${\displaystyle =({\frac {\partial ^{2}v_{x}}{\partial x^{2}}}+{\frac {\partial ^{2}v_{x}}{\partial y^{2}}}+{\frac {\partial ^{2}v_{x}}{\partial z^{2}}})\mathbf {i} +({\frac {\partial ^{2}v_{y}}{\partial x^{2}}}+{\frac {\partial ^{2}v_{y}}{\partial y^{2}}}+{\frac {\partial ^{2}v_{y}}{\partial z^{2}}})\mathbf {j} +({\frac {\partial ^{2}v_{z}}{\partial x^{2}}}+{\frac {\partial ^{2}v_{z}}{\partial y^{2}}}+{\frac {\partial ^{2}v_{z}}{\partial z^{2}}})\mathbf {k} }$

## 應用

${\displaystyle {\frac {\partial ^{2}u}{\partial t^{2}}}=c^{2}\nabla ^{2}u}$