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## 1. 拉普拉斯算子作用于矢量

### 1.1 课本中的定义

$$\displaystyle f$$为二阶可微的实函数，那么有:

$f = ^2f = f$

$^2f=_{i=1}^n{}$

### 1.2 疑问

$\nabla^2 \boldsymbol E + \omega ^2\mu\epsilon\boldsymbol E =0$

### 1.3 真正的定义

Vector Laplacian A vector Laplacian can be defined for a vector $$\mathbf{A}$$ by $\nabla^{2} \mathbf{A}=\nabla(\nabla \cdot \mathbf{A})-\nabla \times(\nabla \times \mathbf{A})$ where the notation $$\dot{ }$$ is sometimes used to distinguish the vector Laplacian from the scalar Laplacian $$\nabla^{2}$$ (Moon and Spencer $$1988, \mathrm{p} .3$$ ). In tensor notation, $$\mathbf{A}$$ is written $$A_{\mu}$$, and the identity becomes

\begin{aligned} \nabla^{2} A_{\mu} &=A_{\mu ; \lambda} ; \lambda \\ &=\left(g^{\lambda x} A_{\mu ; \lambda}\right)_{; \kappa} \\ &=g^{\lambda} \kappa_{; \kappa} A_{\mu ; \lambda}+g^{\lambda x} A_{\mu ; \lambda x} \end{aligned} > A tensor Laplacian may be similarly defined. > In cylindrical coordinates, the vector Laplacian is given by

$\nabla^2 \boldsymbol T = \nabla^2(T_x,T_y,T_z) = (\nabla^2 T_x)\hat x + (\nabla^2 T_y)\hat y + (\nabla^2 T_z)\hat z$

### 2.1 Hessian矩阵的定义

Hessian矩阵通常定义如下: $\nabla^2f = \mathbf{H}=\left[\begin{array}{cccc} \frac{\partial^{2} f}{\partial x_{1}^{2}} & \frac{\partial^{2} f}{\partial x_{1} \partial x_{2}} & \cdots & \frac{\partial^{2} f}{\partial x_{1} \partial x_{n}} \\ \frac{\partial^{2} f}{\partial x_{2} \partial x_{1}} & \frac{\partial^{2} f}{\partial x_{2}^{2}} & \cdots & \frac{\partial^{2} f}{\partial x_{2} \partial x_{n}} \\ \frac{\partial^{2} f}{\partial x_{n} \partial x_{1}} & \frac{\partial^{2} f}{\partial x_{n} \partial x_{2}} & \cdots & \frac{\partial^{2} f}{\partial x_{n}^{2}} \end{array}\right]$

### 2.2 疑问

$$\nabla^2$$作用于矢量时，如$$\nabla^2 \boldsymbol E$$，所得结果为一向量

### 2.3 符号的混淆

Hessian matrix

While $$\nabla^{2}$$ usually represents the Laplacian, sometimes $$\nabla^{2}$$ also represents the Hessian matrix. The former refers to the inner product of $$\nabla$$, while the latter refers to the dyadic product of $$\nabla$$ : $\nabla^{2}=\nabla \cdot \nabla^{T}$ So whether $$\nabla^{2}$$ refers to a Laplacian or a Hessian matrix depends on the context.

$\nabla=\left[\begin{array}{l} \frac{\partial }{\partial x_{1}} \\ \frac{\partial }{\partial x_{2}} \\ \cdots \\ \frac{\partial }{\partial x_{n}} \\ \end{array}\right]$