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拉普拉斯算子

## 拉普拉斯算子 
函数 $u$ 的梯度为 $\nabla u$ ，如果再求 $\nabla u$ 的散度，便成为 $\nabla \cdot(\nabla u) \equiv \nabla^2 u$ ，它是拉普拉斯算子 $\nabla^2$ 对 $u$ 的运算，简称拉普拉斯（Laplacian）。在偏微分方程中，$\nabla^2 u$ 扮演着十分重要的角色，特别是，我们在数学物理方法中将要讨论如下方程

$$
\begin{gathered}
\nabla^2 u=0 \quad \text { (拉普拉斯方程) } \\
\nabla^2 u+k^2 u=0 \quad \text { (亥姆霍兹方程) } \\
\frac{\partial^2 u}{\partial t^2}=a^2 \nabla^2 u \quad \text { (波动方程) } \\
\frac{\partial u}{\partial t}=a^2 \nabla^2 u \quad \text { (热传导方程) } \\
i \hbar \frac{\partial \psi}{\partial t}=-\frac{\hbar^2}{2 m} \nabla^2 \psi+V(r) \psi \quad \text { (薛定谔方程) }
\end{gathered}
$$

在这些方程中，$\nabla^2$ 可以是三维形式，也可以是二维形式

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

## 极坐标系下的拉普拉斯算子

极坐标系如图所示，它与直角坐标系的关系为
 ![图片](/uploads/2025-04/2ed230.jpg){width=300px}

$$
x=r \cos \theta, \quad y=r \sin \theta
$$

而

$$
r^2=x^2+y^2, \quad \tan \theta=\frac{y}{x}
$$

利用 $r^2=x^2+y^2$ 计算 $r$ 对 $x$ 的微商，得到

$$
2 r \frac{\partial r}{\partial x}=2 x \Rightarrow \frac{\partial r}{\partial x}=\frac{x}{r}
$$

再次对 $x$ 微商，得到

$$
\frac{\partial^2 r}{\partial x^2}=\frac{r-x \frac{\partial r}{\partial x}}{r^2}=\frac{r-x \frac{x}{r}}{r^2}=\frac{r^2-x^2}{r^3}=\frac{y^2}{r^3}
$$

利用 $\tan \theta=\frac{y}{x}$ 计算 $\theta$ 对 $x$ 的微商，得到

$$
\frac{\partial \theta}{\partial x}=\frac{1}{1+\left(\frac{y}{x}\right)^2}\left(-\frac{y}{x^2}\right)=-\frac{y}{r^2}
$$

再次对 $\theta$ 微商得

$$
\frac{\partial^2 \theta}{\partial x^2}=\frac{2 y}{r^3} \frac{\partial r}{\partial x}=\frac{2 x y}{r^4}
$$

现在对 $y$ 微分，按照类似的过程，我们得到

$$
\frac{\partial r}{\partial y}=\frac{y}{r}, \quad \frac{\partial^2 r}{\partial y^2}=\frac{x^2}{r^3}, \quad \frac{\partial \theta}{\partial y}=\frac{x}{r^2}, \quad \frac{\partial^2 \theta}{\partial y^2}=-\frac{2 x y}{r^4}
$$

从以上结果，容易推出下面两个关系式

$$
\begin{gathered}
\frac{\partial^2 \theta}{\partial x^2}+\frac{\partial^2 \theta}{\partial y^2}=0 \\
\frac{\partial r}{\partial x} \frac{\partial \theta}{\partial x}+\frac{\partial r}{\partial y} \frac{\partial \theta}{\partial y}=0
\end{gathered}
$$

现在我们可以着手将拉普拉斯算子变换到极坐标系。利用复合函数的微分法则，我们有

$$
\frac{\partial u}{\partial x}=\frac{\partial u}{\partial r} \frac{\partial r}{\partial x}+\frac{\partial u}{\partial \theta} \frac{\partial \theta}{\partial x}
$$

再次对 $x$ 微商得

$$
\begin{aligned}
\frac{\partial^2 u}{\partial x^2}= & \frac{\partial}{\partial x}\left(\frac{\partial u}{\partial r}\right) \frac{\partial r}{\partial x}+\frac{\partial u}{\partial r} \frac{\partial^2 r}{\partial x^2}+\frac{\partial}{\partial x}\left(\frac{\partial u}{\partial \theta}\right) \frac{\partial \theta}{\partial x}+\frac{\partial u}{\partial \theta} \frac{\partial^2 \theta}{\partial x^2} \\
= & \left(\frac{\partial^2 u}{\partial r^2} \frac{\partial r}{\partial x}+\frac{\partial^2 u}{\partial r \partial \theta} \frac{\partial \theta}{\partial x}\right) \frac{\partial r}{\partial x}+\frac{\partial u}{\partial r} \frac{\partial^2 r}{\partial x^2} \\
& +\left(\frac{\partial^2 u}{\partial r \partial \theta} \frac{\partial r}{\partial x}+\frac{\partial^2 u}{\partial \theta^2} \frac{\partial \theta}{\partial x}\right) \frac{\partial \theta}{\partial x}+\frac{\partial u}{\partial \theta} \frac{\partial^2 \theta}{\partial x^2} \\
= & \frac{\partial^2 u}{\partial r^2}\left(\frac{\partial r}{\partial x}\right)^2+2 \frac{\partial^2 u}{\partial r \partial \theta} \frac{\partial r}{\partial x} \frac{\partial \theta}{\partial x}+\frac{\partial u}{\partial r} \frac{\partial^2 r}{\partial x^2}+\frac{\partial^2 u}{\partial \theta^2}\left(\frac{\partial \theta}{\partial x}\right)^2+\frac{\partial u}{\partial \theta} \frac{\partial^2 \theta}{\partial x^2}
\end{aligned}
$$

对于 $y$ 有同样的结果

$$
\frac{\partial^2 u}{\partial y^2}=\frac{\partial^2 u}{\partial r^2}\left(\frac{\partial r}{\partial y}\right)^2+2 \frac{\partial^2 u}{\partial r \partial \theta} \frac{\partial r}{\partial y} \frac{\partial \theta}{\partial y}+\frac{\partial u}{\partial r} \frac{\partial^2 r}{\partial y^2}+\frac{\partial^2 u}{\partial \theta^2}\left(\frac{\partial \theta}{\partial y}\right)^2+\frac{\partial u}{\partial \theta} \frac{\partial^2 \theta}{\partial y^2}
$$

将式（1．2．26）与式（1．2．27）相加并注意式（1．2．23）和式（1．2．24），我们得到

$$
\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=\frac{\partial^2 u}{\partial r^2}\left[\left(\frac{\partial r}{\partial x}\right)^2+\left(\frac{\partial r}{\partial y}\right)^2\right]+2 \frac{\partial^2 u}{\partial r \partial \theta}\left[\frac{\partial r}{\partial x} \frac{\partial \theta}{\partial x}+\frac{\partial r}{\partial y} \frac{\partial \theta}{\partial y}\right]
$$

$$
\begin{aligned}
& +\frac{\partial u}{\partial r}\left[\frac{\partial^2 r}{\partial x^2}+\frac{\partial^2 r}{\partial y^2}\right]+\frac{\partial^2 u}{\partial \theta^2}\left[\left(\frac{\partial \theta}{\partial x}\right)^2+\left(\frac{\partial \theta}{\partial y}\right)^2\right]+\frac{\partial u}{\partial \theta}\left[\frac{\partial^2 \theta}{\partial x^2}+\frac{\partial^2 \theta}{\partial y^2}\right] \\
= & \frac{\partial^2 u}{\partial r^2}\left[\left(\frac{\partial r}{\partial x}\right)^2+\left(\frac{\partial r}{\partial y}\right)^2\right]+\frac{\partial u}{\partial r}\left[\frac{\partial^2 r}{\partial x^2}+\frac{\partial^2 r}{\partial y^2}\right] \\
& +\frac{\partial^2 u}{\partial \theta^2}\left[\left(\frac{\partial \theta}{\partial x}\right)^2+\left(\frac{\partial \theta}{\partial y}\right)^2\right]
\end{aligned}  ...(1.2.28)
$$

进一步利用式（1．2．20）～式（1．2．22），则式（1．2．28）变为

$$
\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=\frac{\partial^2 u}{\partial r^2}\left(\frac{x^2}{r^2}+\frac{y^2}{r^2}\right)+\frac{\partial u}{\partial r}\left(\frac{x^2}{r^3}+\frac{y^2}{r^3}\right)+\frac{\partial^2 u}{\partial \theta^2}\left(\frac{x^2}{r^4}+\frac{y^2}{r^4}\right)
$$

利用 $r^2=x^2+y^2$ ，我们得到极坐标系中的拉普拉斯

$$
\nabla^2 u=\frac{\partial^2 u}{\partial r^2}+\frac{1}{r} \frac{\partial u}{\partial r}+\frac{1}{r^2} \frac{\partial^2 u}{\partial \theta^2}
$$

它还可以写为更加简洁的形式

$$
\boxed{
\nabla^2 u=\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial u}{\part

其他版本
【高等数学】理解：旋度

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