《矢量分析与场论》笔记

参考：①《矢量分析与场论（第四版）》谢树艺。②《糕等电动力学》——03正交曲线坐标系 - 知乎。

第一章矢量分析

要点

$\vec A\cdot \vec B$ $\vec A\times \vec B$ 同极限运算可交换
$\vec A\cdot \vec B$ $\vec A\times \vec B$ 的导数运算与数的乘法类似
Lagrange 中值定理：
$\vec{A} (b) - \vec{A} (a) = (b - a) (A_{x}^{'} (ξ_{1}) i + A_{y}^{'} (ξ_{2}) j + A_{z}^{'} (ξ) k)$
$\xi_1,\xi_2,\xi_3$ 常不相等。

第二章场论

有势场

有势场 $A$ $\exists ~u~~{\rm s.t.}~~ \vec A=\nabla u$

定理：线单连域内有势场 = 无旋场

$\rt(\nabla A)=0$ ；无旋场=>有势场：Stocks公式

管形场

定义 $\dv \vec A=0$ 。管形场即无源场。

定理：面单连域内管形场为另一个场的旋度场

$\dv(\rt A)=0$ ;

$\vec B=(\int_{x_0}^xQ\dd z-\int_{y_0}^yR(..z_0)\dd y)\i-\int_{z_0}^zP\dd z\j$

调和场

定义：既是管形场由是有势场。

$A$ $u$ $\dv (\nabla u)=0$ ，即拉普拉斯方程：

\frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}} + \frac{\partial^{2} u}{\partial z^{2}} = 0

$u$ 被称为调和函数。

$\nabla$

$A\cdot \nabla=A_x\part_x+A_y\part_y+A_z\part _z$ ，可作用于标量和矢量。

有如下公式：

\begin{aligned} \nabla (u v) = u \nabla v + v \nabla u \\ \nabla (A \cdot B) = A \times (\nabla \times B) + (A \cdot \nabla) B + B \times (\nabla \times A) + (B \cdot \nabla) A \\ \nabla f (u) = f^{'} (u) \nabla u \\ \nabla f (u, v) = f_{u} \nabla u + f_{v} \nabla v \\ \nabla \cdot (u A) = u \nabla \cdot A + \nabla u \cdot A \\ \nabla \cdot (A \times B) = B \cdot \nabla \times A - A \cdot \nabla \times B \\ \nabla \cdot (\nabla \times A) = 0 \\ \nabla \times (u A) = u \nabla \times A + \nabla u \times A \\ \nabla \times (A \times B) = (B \cdot \nabla) A - (A \cdot \nabla) B - B (\nabla \cdot A) + A (\nabla \cdot B) \\ \nabla \times (\nabla \times A) = \nabla (\nabla \cdot A) - Δ A (Δ A = Δ A_{x} i + Δ A_{y} j + Δ A_{z} k) \\ \nabla \times (\nabla u) = 0 \end{aligned}

$\nabla$ 的前面。

有如下推论：

\begin{matrix} \nabla r = {\hat{e}}_{r} \\ \nabla \times \vec{r} = 0 \\ \nabla \times (f (r) \vec{r}) = 0 \end{matrix}

第四章梯度、散度、旋度与调和量在正交曲线坐标系中的表示式

$(x,y,z)\leftrightarrow (q_1,q_2,q_3)$ $q_1=q_1(x,y,z),...$ $q_1(x,y,z) ={\rm Const.}$ $q_2(x,y,z)={\rm Const.}$ $q_3$ 的坐标曲线。坐标曲线的正切向单位矢量即为单位矢量（此单位矢量是在直角坐标系意义下的）。正交曲线坐标系中处处三个单位矢量相互垂直。

拉梅系数

$(q_1,q_2,q_3)\rightarrow(q_1+\dd q_1,q_2,q_3)$ 的微小坐标曲线段满足：

\begin{aligned} d s_{1} & = \sqrt{d x^{2} + d y^{2} + d z^{2}} \\ = \sqrt{(\frac{\partial x}{\partial q_{1}} d q_{1})^{2} + (\frac{\partial y}{\partial q_{1}} d q_{1})^{2} + (\frac{\partial z}{\partial q_{1}} d q_{1})^{2}} \\ = \sqrt{(\frac{\partial x}{\partial q_{1}})^{2} + (\frac{\partial y}{\partial q_{1}})^{2} + (\frac{\partial z}{\partial q_{1}})^{2}} d q_{1} \end{aligned}

拉梅系数 $H_i=\sqrt{\big({\part x\over \part q_1}\big)^2+\big({\part y\over \part q_1}\big)^2+\big({\part z\over \part q_1}\big)^2}$ $\dd s_i=H_i\dd q_i$ 。

${\part(x,y,z)\over\part(q_1,q_2,q_3)}=H_1H_2H_3$

正交曲线坐标系的单位矢量

${\part \vec r\over \part q_i}={\part x\over \part q_i}\i+{\part y\over \part q_i}\j+{\part z\over \part q_i}\k$ $\left | {\part \vec r\over \part q_i} \right |=H_i$ 。故有

{\hat{e}}_{i} = \frac{1}{H_{i}} \frac{\partial \vec{r}}{\partial q_{i}} = \frac{1}{H_{i}} \frac{\partial x}{\partial q_{i}} i + \frac{1}{H_{i}} \frac{\partial y}{\partial q_{i}} j + \frac{1}{H_{i}} \frac{\partial z}{\partial q_{i}} k

${1\over H_i}{\part x\over \part q_i}$ $\hat e_i,\i$ 的夹角，于是又有：

i = \frac{1}{H_{1}} \frac{\partial x}{\partial q_{1}} {\hat{e}}_{1} + \frac{1}{H_{2}} \frac{\partial x}{\partial q_{2}} {\hat{e}}_{2} + \frac{1}{H_{3}} \frac{\partial x}{\partial q_{3}} {\hat{e}}_{3}

各量在正交曲线坐标系中的表示

$\nabla u(q_1,q_2,q_3)$ 满足：

\begin{aligned} \nabla u (q_{1}, q_{2}, q_{3}) & = \frac{\partial u}{\partial {\hat{e}}_{1}} {\hat{e}}_{1} + \frac{\partial u}{\partial {\hat{e}}_{2}} {\hat{e}}_{2} + \frac{\partial u}{\partial {\hat{e}}_{3}} {\hat{e}}_{3} \\ = \frac{1}{H_{1}} \frac{\partial u}{\partial q_{1}} {\hat{e}}_{1} + \frac{1}{H_{2}} \frac{\partial u}{\partial q_{2}} {\hat{e}}_{2} + \frac{1}{H_{3}} \frac{\partial u}{\partial q_{3}} {\hat{e}}_{3} \end{aligned}

根据②提供的几何手段，可以得到

\begin{matrix} \nabla \cdot A = \frac{1}{H_{1} H_{2} H_{3}} (\frac{\partial (H_{2} H_{3} A_{1})}{\partial q_{1}} + \frac{\partial (H_{1} H_{3} A_{2})}{\partial q_{2}} + \frac{\partial (H_{1} H_{2} A_{3})}{\partial q_{1}}) \\ \begin{aligned} \nabla \times A = & \frac{1}{H_{2} H_{3}} (\frac{\partial (H_{3} A_{3})}{\partial q_{2}} - \frac{\partial (H_{2} A_{2})}{\partial q_{3}}) {\hat{e}}_{1} + \\ \frac{1}{H_{1} H_{3}} (\frac{\partial (H_{1} A_{1})}{\partial q_{3}} - \frac{\partial (H_{3} A_{3})}{\partial q_{1}}) {\hat{e}}_{2} + \\ \frac{1}{H_{1} H_{2}} (\frac{\partial (H_{2} A_{2})}{\partial q_{1}} - \frac{\partial (H_{1} A_{1})}{\partial q_{2}}) {\hat{e}}_{3} \\ = & \frac{1}{H_{1} H_{2} H_{3}} | \begin{array}{ccc} H_{1} {\hat{e}}_{1} & H_{2} {\hat{e}}_{2} & H_{3} {\hat{e}}_{3} \\ \frac{\partial}{\partial q_{1}} & \frac{\partial}{\partial q_{2}} & \frac{\partial}{\partial q_{3}} \\ H_{1} A_{1} & H_{2} A_{2} & H_{3} A_{3} \end{array} | \end{aligned} \\ Δ u = \nabla \cdot (\nabla u) = \frac{1}{H_{1} H_{2} H_{3}} (\begin{matrix} \frac{\partial}{\partial q_{1}} (\frac{H_{2} H_{3}}{H_{1}} \frac{\partial u}{\partial q_{1}}) + \frac{\partial}{\partial q_{2}} (\frac{H_{1} H_{3}}{H_{2}} \frac{\partial u}{\partial q_{2}}) \\ + \frac{\partial}{\partial q_{3}} (\frac{H_{1} H_{2}}{H_{3}} \frac{\partial u}{\partial q_{3}}) \end{matrix}) \end{matrix}

严格推导过程利用如下结论：

\begin{matrix} \frac{\partial {\hat{e}}_{1}}{\partial q_{1}} = - \frac{{\hat{e}}_{2}}{H_{2}} \frac{\partial H_{1}}{\partial q_{2}} - \frac{{\hat{e}}_{3}}{H_{3}} \frac{\partial H_{1}}{\partial q_{3}} \\ \frac{\partial {\hat{e}}_{1}}{\partial q_{2}} = \frac{{\hat{e}}_{2}}{H_{1}} \frac{\partial H_{2}}{\partial q_{1}} \\ \frac{\partial {\hat{e}}_{1}}{\partial q_{3}} = \frac{{\hat{e}}_{3}}{H_{1}} \frac{\partial H_{3}}{\partial q_{1}} \\ . . . \end{matrix}

《矢量分析与场论》笔记

第一章 矢量分析

第二章 场论

有势场

管形场

调和场

第三章 哈密顿算子 ∇\nabla