# 雙調和方程式

${\displaystyle \nabla ^{4}\varphi =0}$

${\displaystyle \nabla ^{2}\nabla ^{2}\varphi =0}$

${\displaystyle \Delta ^{2}\varphi =0}$

${\displaystyle \nabla ^{4}}$ 為四階的 Nabla算子，或是拉普拉斯算子的平方，稱為雙調和算子。 在 n 維座標下，以爱因斯坦求和约定可寫成

${\displaystyle \nabla ^{4}\varphi =\sum _{i=1}^{n}\sum _{j=1}^{n}\partial _{i}\partial _{i}\partial _{j}\partial _{j}\varphi .}$

${\displaystyle {\partial ^{4}\varphi \over \partial x^{4}}+{\partial ^{4}\varphi \over \partial y^{4}}+{\partial ^{4}\varphi \over \partial z^{4}}+2{\partial ^{4}\varphi \over \partial x^{2}\partial y^{2}}+2{\partial ^{4}\varphi \over \partial y^{2}\partial z^{2}}+2{\partial ^{4}\varphi \over \partial x^{2}\partial z^{2}}=0.}$

${\displaystyle \nabla ^{4}\left({1 \over r}\right)={3(15-8n+n^{2}) \over r^{5}}}$

${\displaystyle r={\sqrt {x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}}}.}$

n=3 和 n=5  才能行成雙調和方程式。

${\displaystyle {\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial }{\partial r}}\left({\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial \varphi }{\partial r}}\right)\right)\right)+{\frac {2}{r^{2}}}{\frac {\partial ^{4}\varphi }{\partial \theta ^{2}\partial r^{2}}}+{\frac {1}{r^{4}}}{\frac {\partial ^{4}\varphi }{\partial \theta ^{4}}}-{\frac {2}{r^{3}}}{\frac {\partial ^{3}\varphi }{\partial \theta ^{2}\partial r}}+{\frac {4}{r^{4}}}{\frac {\partial ^{2}\varphi }{\partial \theta ^{2}}}=0}$