普遍意义下矢量的散度和旋度表达式以及它们的矩阵形式的推导

现有任意一个三维矢量A⃗\vec AA（高维矢量的情况类似）：
(1)A⃗=A1e^q1+A2e^q2+A3e^q3\vec A=A_1\hat e_{q_1}+A_2\hat e_{q_2}+A_3\hat e_{q_3}\tag 1 A=A1e^q1+A2e^q2+A3e^q3(1)
e^qi(i=1,2,3)\hat e_{q_i}(i=1,2,3)e^qi(i=1,2,3)为空间中的坐标单位矢量，满足：
(2)e^qi⋅e^qj={0,i≠j1,i=j\hat e_{q_i}\cdot\hat e_{q_j}=\left\{ \begin{aligned} 0 &, & i\ne j \\ 1 &, & i=j \end{aligned} \right.\tag 2 e^qi⋅e^qj={01,,i̸=ji=j(2)
(3)e^q1×e^q2=e^q3e^q3×e^q1=e^q2e^q2×e^q3=e^q1\hat e_{q_1}\times\hat e_{q_2}=\hat e_{q_3}\\\hat e_{q_3}\times\hat e_{q_1}=\hat e_{q_2}\\\hat e_{q_2}\times\hat e_{q_3}=\hat e_{q_1}\tag 3e^q1×e^q2=e^q3e^q3×e^q1=e^q2e^q2×e^q3=e^q1(3)
在此坐标系下，位矢全微分：
(4)dr⃗=∑i=13hidqie^qid\vec r=\sum^3_{i=1}h_idq_i\hat e_{q_i}\tag 4dr=i=1∑3hidqie^qi(4)
另有，无限小距离平方为：
(5)(ds)2=(dr⃗)2=∑i=13(hidqi)2(ds)^2=(d\vec r)^2=\sum^3_{i=1}(h_idq_i)^2\tag 5(ds)2=(dr)2=i=1∑3(hidqi)2(5)
hih_ihi为每一坐标方向上的距离微元与该坐标微元的比值，称为该坐标体系在此坐标上的度量系数：
(6)hi=dr⃗⋅e^qidqih_i=\frac{d\vec r\cdot\hat e_{q_i}}{dq_i}\tag 6hi=dqidr⋅e^qi(6)
于是，在该坐标体系中，每个坐标方向上的距离微元为：
(7)dli=hidqidl_i=h_idq_i\tag 7dli=hidqi(7)
面积微元为：
(8)dSi=hjhkdqjdqk，i，j，k互不相等dS_i=h_jh_kdq_jdq_k，i，j，k互不相等\tag 8dSi=hjhkdqjdqk，i，j，k互不相等(8)
体积微元为：
(9)dV=h1h2h3dq1dq2dq3dV=h_1h_2h_3dq_1dq_2dq_3 \tag 9dV=h1h2h3dq1dq2dq3(9)

一、散度

矢量A⃗\vec AA在体积元dVdVdV表面上的通量为：

(10)dΦ=−A1h2h3dq2dq3+(A1+∂A1∂q1dq1)(h2+∂h2∂q1dq1)(h3+∂h3∂q1dq1)dq2dq3−A2h1h3dq1dq3+(A2+∂A2∂q2dq2)(h1+∂h1∂q2dq2)(h3+∂h3∂q2dq2)dq1dq3−A3h1h2dq1dq2+(A3+∂A3∂q3dq3)(h1+∂h1∂q3dq3)(h2+∂h2∂q3dq3)dq1dq2=A1h2∂h3∂q1dq1dq2dq3+A1h3∂h2∂q1dq1dq2dq3+∂A1∂q1h2h3dq1dq2dq3+A2h1∂h3∂q2dq1dq2dq3+A2h3∂h1∂q2dq1dq2dq3+∂A2∂q2h1h3dq1dq2dq3+A3h1∂h2∂q3dq1dq2dq3+A3h2∂h1∂q3dq1dq2dq3+∂A3∂q3h1h2dq1dq2dq3=[∂(A1h2h3)∂q1+∂(A2h3h1)∂q2+∂(A3h1h2)∂q3]dq1dq2dq3\begin{aligned} d\Phi &=-A_1h_2h_3dq_2dq_3+(A_1+\frac{\partial A_1}{\partial q_1}dq_1)(h_2+\frac{\partial h_2}{\partial q_1}dq_1)(h_3+\frac{\partial h_3}{\partial q_1}dq_1)dq_2dq_3\\ &-A_2h_1h_3dq_1dq_3+(A_2+\frac{\partial A_2}{\partial q_2}dq_2)(h_1+\frac{\partial h_1}{\partial q_2}dq_2)(h_3+\frac{\partial h_3}{\partial q_2}dq_2)dq_1dq_3\\ &-A_3h_1h_2dq_1dq_2+(A_3+\frac{\partial A_3}{\partial q_3}dq_3)(h_1+\frac{\partial h_1}{\partial q_3}dq_3)(h_2+\frac{\partial h_2}{\partial q_3}dq_3)dq_1dq_2\\ &=A_1h_2\frac{\partial h_3}{\partial q_1}dq_1dq_2dq_3+A_1h_3\frac{\partial h_2}{\partial q_1}dq_1dq_2dq_3+\frac{\partial A_1}{\partial q_1}h_2h_3dq_1dq_2dq_3\\ &+A_2h_1\frac{\partial h_3}{\partial q_2}dq_1dq_2dq_3+A_2h_3\frac{\partial h_1}{\partial q_2}dq_1dq_2dq_3+\frac{\partial A_2}{\partial q_2}h_1h_3dq_1dq_2dq_3\\ &+A_3h_1\frac{\partial h_2}{\partial q_3}dq_1dq_2dq_3+A_3h_2\frac{\partial h_1}{\partial q_3}dq_1dq_2dq_3+\frac{\partial A_3}{\partial q_3}h_1h_2dq_1dq_2dq_3\\ &=[\frac{\partial (A_1h_2h_3)}{\partial q_1}+\frac{\partial (A_2h_3h_1)}{\partial q_2}+\frac{\partial (A_3h_1h_2)}{\partial q_3}]dq_1dq_2dq_3 \end{aligned}\tag {10} dΦ=−A1h2h3dq2dq3+(A1+∂q1∂A1dq1)(h2+∂q1∂h2dq1)(h3+∂q1∂h3dq1)dq2dq3−A2h1h3dq1dq3+(A2+∂q2∂A2dq2)(h1+∂q2∂h1dq2)(h3+∂q2∂h3dq2)dq1dq3−A3h1h2dq1dq2+(A3+∂q3∂A3dq3)(h1+∂q3∂h1dq3)(h2+∂q3∂h2dq3)dq1dq2=A1h2∂q1∂h3dq1dq2dq3+A1h3∂q1∂h2dq1dq2dq3+∂q1∂A1h2h3dq1dq2dq3+A2h1∂q2∂h3dq1dq2dq3+A2h3∂q2∂h1dq1dq2dq3+∂q2∂A2h1h3dq1dq2dq3+A3h1∂q3∂h2dq1dq2dq3+A3h2∂q3∂h1dq1dq2dq3+∂q3∂A3h1h2dq1dq2dq3=[∂q1∂(A1h2h3)+∂q2∂(A2h3h1)+∂q3∂(A3h1h2)]dq1dq2dq3(10)
（以上忽略微元幂次在4次及以上的高阶项）
(11)∇⋅A⃗=dΦdV=1h1h2h3[∂(A1h2h3)∂q1+∂(A2h3h1)∂q2+∂(A3h1h2)∂q3]\nabla \cdot \vec A=\frac{d\Phi}{dV} =\frac{1}{h_1h_2h_3}[\frac{\partial (A_1h_2h_3)}{\partial q_1}+\frac{\partial (A_2h_3h_1)}{\partial q_2}+\frac{\partial (A_3h_1h_2)}{\partial q_3}]\tag {11}∇⋅A=dVdΦ=h1h2h31[∂q1∂(A1h2h3)+∂q2∂(A2h3h1)+∂q3∂(A3h1h2)](11)

二、旋度

矢量A⃗\vec AA在各个坐标平面上微环的环流为：
(12)dΓ1=A2h2dq2+(A3+∂A3∂q2dq2)(h3+∂h3∂q2dq2)dq3−(A2+∂A2∂q3dq3)(h2+∂h2∂q3dq3)dq2−A3h3dq3=A3∂h3∂q2dq2dq3+h3∂A3∂q2dq2dq3−A2∂h2∂q3dq2dq3−h2∂A2∂q3dq2dq3=[∂(A3h3)∂q2−∂(A2h2)∂q3]dq2dq3\begin{aligned} d\Gamma_1 &=A_2h_2dq_2+(A_3+\frac{\partial A_3}{\partial q_2}dq_2)(h_3+\frac{\partial h_3}{\partial q_2}dq_2)dq_3-(A_2+\frac{\partial A_2}{\partial q_3}dq_3)(h_2+\frac{\partial h_2}{\partial q_3}dq_3)dq_2-A_3h_3dq_3\\ &=A_3\frac{\partial h_3}{\partial q_2}dq_2dq_3+h_3\frac{\partial A_3}{\partial q_2}dq_2dq_3-A_2\frac{\partial h_2}{\partial q_3}dq_2dq_3-h_2\frac{\partial A_2}{\partial q_3}dq_2dq_3\\ &=[\frac{\partial (A_3h_3)}{\partial q_2}-\frac{\partial (A_2h_2)}{\partial q_3}]dq_2dq_3 \end{aligned}\tag {12} dΓ1=A2h2dq2+(A3+∂q2∂A3dq2)(h3+∂q2∂h3dq2)dq3−(A2+∂q3∂A2dq3)(h2+∂q3∂h2dq3)dq2−A3h3dq3=A3∂q2∂h3dq2dq3+h3∂q2∂A3dq2dq3−A2∂q3∂h2dq2dq3−h2∂q3∂A2dq2dq3=[∂q2∂(A3h3)−∂q3∂(A2h2)]dq2dq3(12)
同理可得：
(13)dΓ2=[∂(A1h1)∂q3−∂(A3h3)∂q1]dq1dq3d\Gamma_2=[\frac{\partial (A_1h_1)}{\partial q_3}-\frac{\partial (A_3h_3)}{\partial q_1}]dq_1dq_3\tag {13} dΓ2=[∂q3∂(A1h1)−∂q1∂(A3h3)]dq1dq3(13)
(14)dΓ3=[∂(A2h2)∂q1−∂(A1h1)∂q2]dq1dq2d\Gamma_3=[\frac{\partial (A_2h_2)}{\partial q_1}-\frac{\partial (A_1h_1)}{\partial q_2}]dq_1dq_2\tag {14} dΓ3=[∂q1∂(A2h2)−∂q2∂(A1h1)]dq1dq2(14)
（以上忽略微元幂次在3次及以上的高阶项）
矢量A⃗\vec AA在各个坐标方向上的环流密度为：
(15)rot1A⃗=dΓ1dS1=[∂(A3h3)∂q2−∂(A2h2)∂q3]dq2dq3h2h3dq2dq3=1h2h3[∂(A3h3)∂q2−∂(A2h2)∂q3]rot_1\vec A=\frac{d\Gamma_1}{dS_1}=\frac{[\frac{\partial (A_3h_3)}{\partial q_2}-\frac{\partial (A_2h_2)}{\partial q_3}]dq_2dq_3}{h_2h_3dq_2dq_3}=\frac{1}{h_2h_3}[\frac{\partial (A_3h_3)}{\partial q_2}-\frac{\partial (A_2h_2)}{\partial q_3}]\tag {15} rot1A=dS1dΓ1=h2h3dq2dq3[∂q2∂(A3h3)−∂q3∂(A2h2)]dq2dq3=h2h31[∂q2∂(A3h3)−∂q3∂(A2h2)](15)
(16)rot2A⃗=dΓ2dS2=[∂(A1h1)∂q3−∂(A3h3)∂q1]dq1dq3h1h3dq1dq3=1h1h3[∂(A1h1)∂q3−∂(A3h3)∂q1]rot_2\vec A=\frac{d\Gamma_2}{dS_2}=\frac{[\frac{\partial (A_1h_1)}{\partial q_3}-\frac{\partial (A_3h_3)}{\partial q_1}]dq_1dq_3}{h_1h_3dq_1dq_3}=\frac{1}{h_1h_3}[\frac{\partial (A_1h_1)}{\partial q_3}-\frac{\partial (A_3h_3)}{\partial q_1}]\tag{16}rot2A=dS2dΓ2=h1h3dq1dq3[∂q3∂(A1h1)−∂q1∂(A3h3)]dq1dq3=h1h31[∂q3∂(A1h1)−∂q1∂(A3h3)](16)
(17)rot3A⃗=dΓ3dS3=[∂(A2h2)∂q1−∂(A1h1)∂q1]dq1dq2h1h2dq1dq2=1h1h2[∂(A2h2)∂q1−∂(A1h1)∂q2]rot_3\vec A=\frac{d\Gamma_3}{dS_3}=\frac{[\frac{\partial (A_2h_2)}{\partial q_1}-\frac{\partial (A_1h_1)}{\partial q_1}]dq_1dq_2}{h_1h_2dq_1dq_2}=\frac{1}{h_1h_2}[\frac{\partial (A_2h_2)}{\partial q_1}-\frac{\partial (A_1h_1)}{\partial q_2}]\tag{17}rot3A=dS3dΓ3=h1h2dq1dq2[∂q1∂(A2h2)−∂q1∂(A1h1)]dq1dq2=h1h21[∂q1∂(A2h2)−∂q2∂(A1h1)](17)
故矢量A⃗\vec AA的旋度为：
(18)∇×A⃗=e^q1rot1A⃗+e^q2rot2A⃗+e^q3rot3A⃗=1h2h3[∂(A3h3)∂q2−∂(A2h2)∂q3]e^q1+1h1h3[∂(A1h1)∂q3−∂(A3h3)∂q1]e^q2+1h1h2[∂(A2h2)∂q1−∂(A1h1)∂q2]e^q3=1h1h2h3∣h1e^q1h2e^q2h3e^q3∂∂q1∂∂q2∂∂q3h1A1h2A2h3A3∣\begin{aligned} \nabla\times \vec A &=\hat e_{q_1}rot_1\vec A+\hat e_{q_2}rot_2\vec A+\hat e_{q_3}rot_3\vec A\\ &=\frac{1}{h_2h_3}[\frac{\partial (A_3h_3)}{\partial q_2}-\frac{\partial (A_2h_2)}{\partial q_3}]\hat e_{q_1}+\frac{1}{h_1h_3}[\frac{\partial (A_1h_1)}{\partial q_3}-\frac{\partial (A_3h_3)}{\partial q_1}]\hat e_{q_2}+\frac{1}{h_1h_2}[\frac{\partial (A_2h_2)}{\partial q_1}-\frac{\partial (A_1h_1)}{\partial q_2}]\hat e_{q_3}\\ &=\frac{1}{h_1h_2h_3}\left |\begin{matrix} h_1\hat e_{q_1} & h_2\hat e_{q_2} & h_3\hat e_{q_3} \\\\ \frac{\partial}{\partial q_1} & \frac{\partial}{\partial q_2} & \frac{\partial}{\partial q_3}\\\\ h_1A_1 & h_2A_2 & h_3A_3 \end{matrix} \right | \end{aligned}\tag {18}∇×A=e^q1rot1A+e^q2rot2A+e^q3rot3A=h2h31[∂q2∂(A3h3)−∂q3∂(A2h2)]e^q1+h1h31[∂q3∂(A1h1)−∂q1∂(A3h3)]e^q2+h1h21[∂q1∂(A2h2)−∂q2∂(A1h1)]e^q3=h1h2h31∣∣∣∣∣∣∣∣∣∣h1e^q1∂q1∂h1A1h2e^q2∂q2∂h2A2h3e^q3∂q3∂h3A3∣∣∣∣∣∣∣∣∣∣(18)

三、散度以及旋度的矩阵形式

(19)A⃗=[A1A2A3]\vec A= \begin{bmatrix} A_1\\\\ A_2\\\\ A_3 \end{bmatrix}\tag{19} A=⎣⎢⎢⎢⎢⎡A1A2A3⎦⎥⎥⎥⎥⎤(19)
(20)∇⋅A⃗=1h1h2h3[∂∂q1∂∂q2∂∂q3]T[h2h3000h3h1000h1h2][A1A2A3]\nabla \cdot\vec A=\frac{1}{h_1h_2h_3} \begin{bmatrix} \frac{\partial}{\partial q_1}\\\\ \frac{\partial}{\partial q_2}\\\\ \frac{\partial}{\partial q_3} \end{bmatrix}^T \begin{bmatrix} h_2h_3 & 0 & 0\\\\ 0 & h_3h_1 & 0\\\\ 0 & 0 & h_1h_2 \end{bmatrix} \begin{bmatrix} A_1\\\\A_2\\\\A_3 \end{bmatrix}\tag{20} ∇⋅A=h1h2h31⎣⎢⎢⎢⎢⎡∂q1∂∂q2∂∂q3∂⎦⎥⎥⎥⎥⎤T⎣⎢⎢⎢⎢⎡h2h3000h3h1000h1h2⎦⎥⎥⎥⎥⎤⎣⎢⎢⎢⎢⎡A1A2A3⎦⎥⎥⎥⎥⎤(20)
(21)∇×A⃗=1h1h2h3[h1000h2000h3][0−∂∂q3∂∂q2∂∂q30−∂∂q1−∂∂q2∂∂q10][h1000h2000h3][A1A2A3]\nabla \times\vec A=\frac{1}{h_1h_2h_3} \begin{bmatrix} h_1 & 0 & 0\\\\ 0 & h_2 & 0\\\\ 0 & 0 & h_3 \end{bmatrix} \begin{bmatrix} 0 & -\frac{\partial}{\partial q_3} & \frac{\partial}{\partial q_2}\\\\ \frac{\partial}{\partial q_3} & 0 & -\frac{\partial}{\partial q_1}\\\\ -\frac{\partial}{\partial q_2} & \frac{\partial}{\partial q_1} & 0 \end{bmatrix} \begin{bmatrix} h_1 & 0 & 0\\\\ 0 & h_2 & 0\\\\ 0 & 0 & h_3 \end{bmatrix} \begin{bmatrix} A_1\\\\A_2\\\\A_3 \end{bmatrix}\tag{21} ∇×A=h1h2h31⎣⎢⎢⎢⎢⎡h1000h2000h3⎦⎥⎥⎥⎥⎤⎣⎢⎢⎢⎢⎡0∂q3∂−∂q2∂−∂q3∂0∂q1∂∂q2∂−∂q1∂0⎦⎥⎥⎥⎥⎤⎣⎢⎢⎢⎢⎡h1000h2000h3⎦⎥⎥⎥⎥⎤⎣⎢⎢⎢⎢⎡A1A2A3⎦⎥⎥⎥⎥⎤(21)
附：算子运算规则（举例说明）：
ϕ(x,y,z)\phi(x,y,z)ϕ(x,y,z)与ψ(x,y,z)\psi(x,y,z)ψ(x,y,z)是两个函数，算子∇\nabla∇（∇=(e^x∂∂x+e^y∂∂y+e^z∂∂z)\nabla=(\hat e_x\frac{\partial}{\partial x}+\hat e_y\frac{\partial}{\partial y}+\hat e_z\frac{\partial}{\partial z})∇=(e^x∂x∂+e^y∂y∂+e^z∂z∂)）的运算规则如下：
(22)ϕ(x,y,z)∇ψ(x,y,z)=ϕ∂ψ∂xe^x+ϕ∂ψ∂ye^y+ϕ∂ψ∂ze^z\phi(x,y,z)\nabla\psi(x,y,z)=\phi\frac{\partial\psi}{\partial x}\hat e_x+\phi\frac{\partial\psi}{\partial y}\hat e_y+\phi\frac{\partial\psi}{\partial z}\hat e_z\tag{22}ϕ(x,y,z)∇ψ(x,y,z)=ϕ∂x∂ψe^x+ϕ∂y∂ψe^y+ϕ∂z∂ψe^z(22)