向量微分算符公式

利用度规简单推导出向量微分算符(Nabla算符)

\nabla = (\frac{\partial}{\partial x} \frac{\partial}{\partial y} \frac{\partial}{\partial z})

在极坐标和球坐标下的公式。

柱坐标下的向量微分算符

柱坐标变换

{\begin{aligned} x & = r \cos θ \\ y & = r \sin θ \\ z & = z \end{aligned}

在极坐标 $(r, θ)$ 下，微元

\begin{aligned} (d s)^{2} & = (d x)^{2} + (d y)^{2} + (d z)^{2} \\ = (\cos θ d r + r \sin θ d θ)^{2} + (\sin θ d r - r \cos θ d θ)^{2} + (d z)^{2} \\ = (d r)^{2} + r^{2} (d θ)^{2} + (d z)^{2} \end{aligned}

故度规张量在极坐标下的矩阵形式为

G = [\begin{array}{ccc} 1 & 0 & 0 \\ 0 & r^{2} & 0 \\ 0 & 0 & 1 \end{array}]

于是向量微分算符

\nabla = \sum_{i} \frac{1}{\sqrt{g_{i i}}} \frac{\partial}{\partial e_{i}} \hat{e_{i}} = \frac{\partial}{\partial r} \hat{r} + \frac{1}{r} \frac{\partial}{\partial θ} \hat{θ} + \frac{\partial}{\partial z} \hat{z}

WARNING

注意在其他坐标系下求散度(div)不是简单地将 $\nabla$ 和矢量 $\vec{V}$ 进行点乘。

具体求法见正交曲面坐标系的微分算符部分。

度规系数

h_{1} = 1, h_{2} = r, h_{3} = 1

微分算符表达式

\nabla ψ = \frac{\partial ψ}{\partial r} \hat{r} + \frac{1}{r} \frac{\partial ψ}{\partial θ} \hat{θ} + \frac{\partial ψ}{\partial z} \hat{z}

\nabla \cdot \vec{f} = \frac{1}{r} \frac{\partial}{\partial r} (r f_{r}) + \frac{1}{r} \frac{\partial f_{ψ}}{\partial θ} + \frac{\partial f_{z}}{\partial z}

\nabla \times \vec{f} = \frac{1}{r} | \begin{array}{ccc} \hat{r} & r \hat{θ} & \hat{z} \\ \frac{\partial}{\partial r} & \frac{\partial}{\partial θ} & \frac{\partial}{\partial z} \\ f_{r} & r f_{θ} & f_{z} \end{array} |

\nabla^{2} ψ = \frac{1}{r} \frac{\partial}{\partial r} (r \frac{\partial ψ}{\partial r}) + \frac{1}{r^{2}} \frac{\partial^{2} ψ}{\partial θ^{2}} + \frac{\partial^{2} ψ}{\partial z^{2}}

同上易得度规张量在球坐标 $(r, θ, φ)$ 下的矩阵形式，各个坐标如下：

G = [\begin{array}{ccc} 1 & 0 & 0 \\ 0 & r^{2} & 0 \\ 0 & 0 & r^{2} \sin^{2} θ \end{array}]

微分向量算符

\nabla = \sum_{i} \frac{1}{\sqrt{g_{i i}}} \frac{\partial}{\partial e_{i}} \hat{e_{i}} = \frac{\partial}{\partial r} \hat{r} + \frac{1}{r} \frac{\partial}{\partial θ} \hat{θ} + \frac{1}{r \sin θ} \frac{\partial}{\partial φ} \hat{φ}

度规系数

h_{1} = 1, h_{2} = r, h_{3} = r \sin θ

微分算符表达式

\nabla ψ = \frac{\partial ψ}{\partial r} \hat{r} + \frac{1}{r} \frac{\partial ψ}{\partial θ} \hat{θ} + \frac{1}{r \sin θ} \frac{\partial ψ}{\partial ϕ} \hat{ϕ}

\nabla \cdot \vec{f} = \frac{1}{r^{2}} \frac{\partial}{\partial r} (r^{2} f_{r}) + \frac{1}{r \sin θ} \frac{\partial}{\partial θ} (\sin θ f_{θ}) + \frac{1}{r \sin θ} \frac{\partial f_{ϕ}}{\partial ϕ}

\nabla \times \vec{f} = \frac{1}{r^{2} \sin θ} | \begin{array}{ccc} \hat{r} & r \hat{θ} & r \sin θ \hat{ϕ} \\ \frac{\partial}{\partial r} & \frac{\partial}{\partial θ} & \frac{\partial}{\partial ϕ} \\ f_{r} & r f_{θ} & r \sin θ f_{ϕ} \end{array} |

\nabla^{2} ψ = \frac{1}{r^{2}} \frac{\partial}{\partial r} (r^{2} \frac{\partial ψ}{\partial r}) + \frac{1}{r^{2} \sin θ} \frac{\partial}{\partial θ} (\sin θ \frac{\partial ψ}{\partial θ}) + \frac{1}{r^{2} \sin^{2} θ} \frac{\partial^{2} ψ}{\partial ϕ^{2}}

关于向量微分算符 $\nabla$ 的相关运算法则如下：

其中 $ϕ, ψ$ 代表标量场， $\vec{f}, \vec{g}$ 代表矢量场。

\nabla (ϕ ψ) = ϕ \nabla ψ + ψ \nabla ϕ

\nabla \cdot (ϕ \vec{f}) = (\nabla ϕ) \cdot \vec{f} + ϕ \nabla \cdot \vec{f}

\nabla \times (ϕ \vec{f}) = (\nabla ϕ) \times \vec{f} + ϕ \nabla \times \vec{f}

\nabla \cdot (\vec{f} \times \vec{g}) = (\nabla \times \vec{f}) \cdot \vec{g} - \vec{f} \cdot (\nabla \times \vec{g})

\nabla \times (\vec{f} \times \vec{g}) = (\vec{g} \cdot \nabla) \vec{f} + (\nabla \cdot \vec{g}) \vec{f} - (\vec{f} \cdot \nabla) \vec{g} - (\nabla \cdot \vec{f}) \vec{g}

\nabla (\vec{f} \cdot \vec{g}) = \vec{f} \times (\nabla \times \vec{g}) + (\vec{f} \cdot \nabla) \vec{g} + \vec{g} \times (\nabla \times \vec{f}) + (\vec{g} \cdot \nabla) \vec{f}

INFO

上面的公式只需要同时考虑 $\nabla$ 同时作为一个矢量和一个微分算符即可导出。

INFO

矢量三矢积公式：

(\vec{a} \times \vec{b}) \times \vec{c} = (\vec{a} \cdot \vec{c}) \vec{b} - (\vec{b} \cdot \vec{c}) \vec{a}

\vec{c} \times (\vec{a} \times \vec{b}) = (\vec{b} \cdot \vec{c}) \vec{a} - (\vec{a} \cdot \vec{c}) \vec{b}