現有任意一個三維矢量 A ⃗ \vec A A
（高維矢量的情況類似）：
(1) A ⃗ = A 1 e ^ q 1 + A 2 e ^ q 2 + A 3 e ^ q 3 \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 ^ q i ( i = 1 , 2 , 3 ) \hat e_{q_i}(i=1,2,3) e^qi(i=1,2,3)為空間中的坐标機關矢量，滿足：
(2) e ^ q i ⋅ e ^ q j = { 0 , i ≠ j 1 , 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 ^ q 1 × e ^ q 2 = e ^ q 3 e ^ q 3 × e ^ q 1 = e ^ q 2 e ^ q 2 × e ^ q 3 = e ^ q 1 \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 3 e^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) d r ⃗ = ∑ i = 1 3 h i d q i e ^ q i d\vec r=\sum^3_{i=1}h_idq_i\hat e_{q_i}\tag 4 dr
=i=1∑3hidqie^qi(4)
另有，無限小距離平方為：
(5) ( d s ) 2 = ( d r ⃗ ) 2 = ∑ i = 1 3 ( h i d q i ) 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)
h i h_i hi為每一坐标方向上的距離微元與該坐标微元的比值，稱為該坐标體系在此坐标上的度量系數：
(6) h i = d r ⃗ ⋅ e ^ q i d q i h_i=\frac{d\vec r\cdot\hat e_{q_i}}{dq_i}\tag 6 hi=dqidr
⋅e^qi(6)
于是，在該坐标體系中，每個坐标方向上的距離微元為：
(7) d l i = h i d q i dl_i=h_idq_i\tag 7 dli=hidqi(7)
面積微元為：
(8) d S i = h j h k d q j d q k ， i ， j ， k 互 不 相 等 dS_i=h_jh_kdq_jdq_k，i，j，k互不相等\tag 8 dSi=hjhkdqjdqk，i，j，k互不相等(8)
體積微元為：
(9) d V = h 1 h 2 h 3 d q 1 d q 2 d q 3 dV=h_1h_2h_3dq_1dq_2dq_3 \tag 9 dV=h1h2h3dq1dq2dq3(9)
一、散度
矢量 A ⃗ \vec A A
在體積元 d V dV dV表面上的通量為：
(10) d Φ = − A 1 h 2 h 3 d q 2 d q 3 + ( A 1 + ∂ A 1 ∂ q 1 d q 1 ) ( h 2 + ∂ h 2 ∂ q 1 d q 1 ) ( h 3 + ∂ h 3 ∂ q 1 d q 1 ) d q 2 d q 3 − A 2 h 1 h 3 d q 1 d q 3 + ( A 2 + ∂ A 2 ∂ q 2 d q 2 ) ( h 1 + ∂ h 1 ∂ q 2 d q 2 ) ( h 3 + ∂ h 3 ∂ q 2 d q 2 ) d q 1 d q 3 − A 3 h 1 h 2 d q 1 d q 2 + ( A 3 + ∂ A 3 ∂ q 3 d q 3 ) ( h 1 + ∂ h 1 ∂ q 3 d q 3 ) ( h 2 + ∂ h 2 ∂ q 3 d q 3 ) d q 1 d q 2 = A 1 h 2 ∂ h 3 ∂ q 1 d q 1 d q 2 d q 3 + A 1 h 3 ∂ h 2 ∂ q 1 d q 1 d q 2 d q 3 + ∂ A 1 ∂ q 1 h 2 h 3 d q 1 d q 2 d q 3 + A 2 h 1 ∂ h 3 ∂ q 2 d q 1 d q 2 d q 3 + A 2 h 3 ∂ h 1 ∂ q 2 d q 1 d q 2 d q 3 + ∂ A 2 ∂ q 2 h 1 h 3 d q 1 d q 2 d q 3 + A 3 h 1 ∂ h 2 ∂ q 3 d q 1 d q 2 d q 3 + A 3 h 2 ∂ h 1 ∂ q 3 d q 1 d q 2 d q 3 + ∂ A 3 ∂ q 3 h 1 h 2 d q 1 d q 2 d q 3 = [ ∂ ( A 1 h 2 h 3 ) ∂ q 1 + ∂ ( A 2 h 3 h 1 ) ∂ q 2 + ∂ ( A 3 h 1 h 2 ) ∂ q 3 ] d q 1 d q 2 d q 3 \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次及以上的高階項）
(11) ∇ ⋅ A ⃗ = d Φ d V = 1 h 1 h 2 h 3 [ ∂ ( A 1 h 2 h 3 ) ∂ q 1 + ∂ ( A 2 h 3 h 1 ) ∂ q 2 + ∂ ( A 3 h 1 h 2 ) ∂ q 3 ] \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 A ⃗ \vec A A
在各個坐标平面上微環的環流為：
(12) d Γ 1 = A 2 h 2 d q 2 + ( A 3 + ∂ A 3 ∂ q 2 d q 2 ) ( h 3 + ∂ h 3 ∂ q 2 d q 2 ) d q 3 − ( A 2 + ∂ A 2 ∂ q 3 d q 3 ) ( h 2 + ∂ h 2 ∂ q 3 d q 3 ) d q 2 − A 3 h 3 d q 3 = A 3 ∂ h 3 ∂ q 2 d q 2 d q 3 + h 3 ∂ A 3 ∂ q 2 d q 2 d q 3 − A 2 ∂ h 2 ∂ q 3 d q 2 d q 3 − h 2 ∂ A 2 ∂ q 3 d q 2 d q 3 = [ ∂ ( A 3 h 3 ) ∂ q 2 − ∂ ( A 2 h 2 ) ∂ q 3 ] d q 2 d q 3 \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) d Γ 2 = [ ∂ ( A 1 h 1 ) ∂ q 3 − ∂ ( A 3 h 3 ) ∂ q 1 ] d q 1 d q 3 d\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) d Γ 3 = [ ∂ ( A 2 h 2 ) ∂ q 1 − ∂ ( A 1 h 1 ) ∂ q 2 ] d q 1 d q 2 d\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次及以上的高階項）
矢量 A ⃗ \vec A A
在各個坐标方向上的環流密度為：
(15) r o t 1 A ⃗ = d Γ 1 d S 1 = [ ∂ ( A 3 h 3 ) ∂ q 2 − ∂ ( A 2 h 2 ) ∂ q 3 ] d q 2 d q 3 h 2 h 3 d q 2 d q 3 = 1 h 2 h 3 [ ∂ ( A 3 h 3 ) ∂ q 2 − ∂ ( A 2 h 2 ) ∂ q 3 ] 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) r o t 2 A ⃗ = d Γ 2 d S 2 = [ ∂ ( A 1 h 1 ) ∂ q 3 − ∂ ( A 3 h 3 ) ∂ q 1 ] d q 1 d q 3 h 1 h 3 d q 1 d q 3 = 1 h 1 h 3 [ ∂ ( A 1 h 1 ) ∂ q 3 − ∂ ( A 3 h 3 ) ∂ q 1 ] 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) r o t 3 A ⃗ = d Γ 3 d S 3 = [ ∂ ( A 2 h 2 ) ∂ q 1 − ∂ ( A 1 h 1 ) ∂ q 1 ] d q 1 d q 2 h 1 h 2 d q 1 d q 2 = 1 h 1 h 2 [ ∂ ( A 2 h 2 ) ∂ q 1 − ∂ ( A 1 h 1 ) ∂ q 2 ] 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 A ⃗ \vec A A
的旋度為：
(18) ∇ × A ⃗ = e ^ q 1 r o t 1 A ⃗ + e ^ q 2 r o t 2 A ⃗ + e ^ q 3 r o t 3 A ⃗ = 1 h 2 h 3 [ ∂ ( A 3 h 3 ) ∂ q 2 − ∂ ( A 2 h 2 ) ∂ q 3 ] e ^ q 1 + 1 h 1 h 3 [ ∂ ( A 1 h 1 ) ∂ q 3 − ∂ ( A 3 h 3 ) ∂ q 1 ] e ^ q 2 + 1 h 1 h 2 [ ∂ ( A 2 h 2 ) ∂ q 1 − ∂ ( A 1 h 1 ) ∂ q 2 ] e ^ q 3 = 1 h 1 h 2 h 3 ∣ h 1 e ^ q 1 h 2 e ^ q 2 h 3 e ^ q 3 ∂ ∂ q 1 ∂ ∂ q 2 ∂ ∂ q 3 h 1 A 1 h 2 A 2 h 3 A 3 ∣ \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、散度以及旋度的矩陣形式
(19) A ⃗ = [ A 1 A 2 A 3 ] \vec A= \begin{bmatrix} A_1\\\\ A_2\\\\ A_3 \end{bmatrix}\tag{19} A
=⎣⎢⎢⎢⎢⎡A1A2A3⎦⎥⎥⎥⎥⎤(19)
(20) ∇ ⋅ A ⃗ = 1 h 1 h 2 h 3 [ ∂ ∂ q 1 ∂ ∂ q 2 ∂ ∂ q 3 ] T [ h 2 h 3 0 0 0 h 3 h 1 0 0 0 h 1 h 2 ] [ A 1 A 2 A 3 ] \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) ∇ × A ⃗ = 1 h 1 h 2 h 3 [ h 1 0 0 0 h 2 0 0 0 h 3 ] [ 0 − ∂ ∂ q 3 ∂ ∂ q 2 ∂ ∂ q 3 0 − ∂ ∂ q 1 − ∂ ∂ q 2 ∂ ∂ q 1 0 ] [ h 1 0 0 0 h 2 0 0 0 h 3 ] [ A 1 A 2 A 3 ] \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：算子運算規則（舉例說明）：
ϕ ( x , y , z ) \phi(x,y,z) ϕ(x,y,z)與 ψ ( x , y , z ) \psi(x,y,z) ψ(x,y,z)是兩個函數，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∂)）的運算規則如下：
(22) ϕ ( x , y , z ) ∇ ψ ( x , y , z ) = ϕ ∂ ψ ∂ x e ^ x + ϕ ∂ ψ ∂ y e ^ y + ϕ ∂ ψ ∂ z e ^ 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)
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