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格林第一公式与第二公式

高斯定理 设空间闭区域 $\Omega$ 由分片光滑的闭曲面 $\Sigma$ 所围成, $\Sigma$的方向取外侧。函数 $P, Q, R$ 在 $\Omega$ 上有连续的一阶偏导数, 则:

$$
\begin{gathered}
\iiint_{\Omega}\left(\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z}\right) d x d y d z=\iint_{\Sigma} P d y d z+Q d z d x+R d x d y \\
=\iint_{\Sigma}[P \cos (n, x)+Q \cos (n, y)+R \cos (n, z)] d s
\end{gathered}
$$

由高斯公式可得:

$$
\begin{gathered}
\iiint_{\Omega}\left(\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z}\right) d V=\iint_{\Gamma}[P \cos (n, x)+Q \cos (n, y)+R \cos (n, z)] d s \\
\text { 令 } P=u \frac{\partial v}{\partial x}, \quad Q=u \frac{\partial v}{\partial y}, R=u \frac{\partial v}{\partial z}, \text { 求偏导数可得: } \\
\frac{\partial P}{\partial x}=\frac{\partial u}{\partial x} \frac{\partial v}{\partial x}+u \frac{\partial^2 v}{\partial x^2} \quad \frac{\partial Q}{\partial y}=\frac{\partial u}{\partial y} \frac{\partial v}{\partial y}+u \frac{\partial^2 v}{\partial y^2} \quad \frac{\partial R}{\partial z}=\frac{\partial u}{\partial z} \frac{\partial v}{\partial z}+u \frac{\partial^2 v}{\partial z^2}
\end{gathered}
$$

将求得的偏导数代入高斯公式可得:

$$
\begin{gathered}
\iiint_{\Omega}\left(\frac{\partial u}{\partial x} \frac{\partial v}{\partial x}+u \frac{\partial^2 v}{\partial x^2}+\frac{\partial u}{\partial y} \frac{\partial v}{\partial y}+u \frac{\partial^2 v}{\partial y^2}+\frac{\partial u}{\partial z} \frac{\partial v}{\partial z}+u \frac{\partial^2 v}{\partial z^2}\right) d V \\
=\iint_{\Gamma} u\left[\frac{\partial v}{\partial x} \cos (n, x)+\frac{\partial v}{\partial y} \cos (n, y)+\frac{\partial v}{\partial z} \cos (n, z)\right] d s \\
\iiint_{\Omega} u\left(\frac{\partial^2 v}{\partial x^2}+\frac{\partial^2 v}{\partial y^2}+\frac{\partial^2 v}{\partial z^2}\right) d V+\iiint_{\Omega}\left(\frac{\partial u}{\partial x} \frac{\partial v}{\partial x}+\frac{\partial u}{\partial y} \frac{\partial v}{\partial y}+\frac{\partial u}{\partial z} \frac{\partial v}{\partial z}\right) d V \\
=\iint_{\Gamma} u\left[\frac{\partial v}{\partial x} \cos (n, x)+\frac{\partial v}{\partial y} \cos (n, y)+\frac{\partial v}{\partial z} \cos (n, z)\right] d s
\end{gathered}
$$

因为 $\Delta v=\frac{\partial^2 v}{\partial x^2}+\frac{\partial^2 v}{\partial y^2}+\frac{\partial^2 v}{\partial z^2}$, 所以可得格林第一公式:

$$
\iiint_{\Omega} u \Delta v d V+\iiint_{\Omega}\left(\frac{\partial u}{\partial x} \frac{\partial v}{\partial x}+\frac{\partial u}{\partial y} \frac{\partial v}{\partial y}+\frac{\partial u}{\partial z} \frac{\partial v}{\partial z}\right) d V=\iint_{\Gamma} u \frac{\partial v}{\partial n} d s  ...(1)
$$

(1) 式称为第一格林公式。

令 $P=v \frac{\partial u}{\partial x}, Q=v \frac{\partial u}{\partial y}, R=v \frac{\partial u}{\partial z}$, 求偏导数可得:

$$
\frac{\partial P}{\partial x}=\frac{\partial v}{\partial x} \frac{\partial u}{\partial x}+v \frac{\partial^2 u}{\partial x^2} \quad \frac{\partial Q}{\partial y}=\frac{\partial v}{\partial y} \frac{\partial u}{\partial y}+v \frac{\partial^2 u}{\partial y^2} \quad \frac{\partial R}{\partial z}=\frac{\partial v}{\partial z} \frac{\partial u}{\partial z}+v \frac{\partial^2 u}{\partial z^2}
$$

将求得的偏导数代入高斯公式可得:

$$
\begin{gathered}
\iiint_{\Omega}\left(\frac{\partial v}{\partial x} \frac{\partial u}{\partial x}+v \frac{\partial^2 u}{\partial x^2}+\frac{\partial v}{\partial y} \frac{\partial u}{\partial y}+v \frac{\partial^2 u}{\partial y^2}+\frac{\partial v}{\partial z} \frac{\partial u}{\partial z}+v \frac{\partial^2 u}{\partial z^2}\right) d V \\
=\iint_{\Gamma} v\left[\frac{\partial u}{\partial x} \cos (n, x)+\frac{\partial u}{\partial y} \cos (n, y)+\frac{\partial u}{\partial z} \cos (n, z)\right] d s \\
\iiint_{\Omega} v\left(\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}+\frac{\partial^2 u}{\partial z^2}\right) d V+\iiint_{\Omega}\left(\frac{\partial v}{\partial x} \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y} \frac{\partial u}{\partial y}+\frac{\partial v}{\partial z} \frac{\partial u}{\partial z}\right) d V \\
=\iint_{\Gamma} v\left[\frac{\partial u}{\partial x} \cos (n, x)+\frac{\partial u}{\partial y} \cos (n, y)+\frac{\partial u}{\partial z} \cos (n, z)\right] d s
\end{gathered}
$$

因为 $\Delta u=\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}+\frac{\partial^2 u}{\partial z^2}$, 所以可得格林第一公式:

$$
\iiint_{\Omega} v \Delta u d V+\iiint_{\Omega}\left(\frac{\partial v}{\partial x} \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y} \frac{\partial u}{\partial y}+\frac{\partial v}{\partial z} \frac{\partial u}{\partial z}\right) d V=\iint_{\Gamma} v \frac{\partial u}{\partial n} d s  ...(2)
$$

由（1）-（2）式可得:

$$
\begin{gathered}
\iiint_{\Omega} u \Delta v d V+\iiint_{\Omega}\left(\frac{\partial u}{\partial x} \frac{\partial v}{\partial x}+\frac{\partial u}{\partial y} \frac{\partial v}{\partial y}+\frac{\partial u}{\partial z} \frac{\partial v}{\partial z}\right) d V-\iiint_{\Omega} v \Delta u d V \\
-\iiint_{\Omega}\left(\frac{\partial v}{\partial x} \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y} \frac{\partial u}{\partial y}+\frac{\partial v}{\partial z} \frac{\partial u}{\partial z}\right) d V=\iint_{\Gamma} u \frac{\partial v}{\partial n} d s-\iint_{\Gamma} v \frac{\partial u}{\partial n} d s \\
\iiint_{\Omega} u \Delta v d V-\iiint_{\Omega} v \Delta u d V=\iint_{\Gamma} u \frac{\partial v}{\partial n} d s-\iint_{\Gamma} v \frac{\partial u}{\partial n} d s \\
\iiint_{\Omega}(u \Delta v-v \Delta u) d V=\iint_{\Gamma}\left(u \frac{\partial v}{\partial n}-v \frac{\partial u}{\partial n}\right) d s
\end{gathered} ...(3)
$$

（3）式称为格林第二公式。

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