数学物理方法的前几节课讲了怎么在
R 3 R^3 R 3 空间中进行向量分析，其中涉及到了很多与微积分有关的知识，所以特意写这一篇。
数学物理方法这几节课给我印象最深的就是它简化了点乘和叉乘运算的表达形式，主要有以下两种简化方法：爱因斯坦求和约定和Levi-Civita符号。
爱因斯坦求和约定可以将向量的表达形式简化，也可以针对向量点乘，或者说是标量积。Kronecher delta符号 δ i j = { 0 , i ≠ j 1 , i = j \delta_{ij}=\begin{cases}0,&i\neq j \\1,&i=j\end{cases} δ i j = { 0 , 1 , i ̸ = j i = j e 1 , e 2 , e 3 \bold e_1,\bold e_2,\bold e_3 e 1 , e 2 , e 3
简化三维向量表达 A = A 1 e 1 + A 2 e 2 + A 3 e 3 \bold A=A_1\bold e_1+A_2\bold e_2+A_3\bold e_3 A = A 1 e 1 + A 2 e 2 + A 3 e 3 A \bold A A A = A i e i = A 1 e 1 + A 2 e 2 + A 3 e 3 \bold A=A_i\bold e_i=A_1\bold e_1+A_2\bold e_2+A_3\bold e_3 A = A i e i = A 1 e 1 + A 2 e 2 + A 3 e 3
简化向量标量积 A = A 1 e 1 + A 2 e 2 + A 3 e 3 \bold A=A_1\bold e_1+A_2\bold e_2+A_3\bold e_3 A = A 1 e 1 + A 2 e 2 + A 3 e 3 B = B 1 e 1 + B 2 e 2 + B 3 e 3 \bold B=B_1\bold e_1+B_2\bold e_2+B_3\bold e_3 B = B 1 e 1 + B 2 e 2 + B 3 e 3
由以上的向量简化形式，我们可以继续推导：A ⋅ B = A i e i ⋅ B j e j = A i B j e i ⋅ e j = A i B j δ i j = A i B i \bold A\cdot\bold B=A_i\bold e_i\cdot B_j\bold e_j=A_iB_j\bold e_i\cdot\bold e_j=A_iB_j\delta_{ij}=A_iB_i A ⋅ B = A i e i ⋅ B j e j = A i B j e i ⋅ e j = A i B j δ i j = A i B i A ⋅ B = A 1 B 1 + A 2 B 2 + A 3 B 3 \bold A\cdot\bold B=A_1B_1+A_2B_2+A_3B_3 A ⋅ B = A 1 B 1 + A 2 B 2 + A 3 B 3
向量的叉乘或向量积可以通过Levi-Civita符号进行简化。Levi-Civita符号简化三阶行列式 ε i j k = { 1 , Even Permutation of indexes − 1 , Odd Permutation of indexes 0 , 2 of indexes are equal \varepsilon_{ijk}=\begin{cases}1,&\text{Even Permutation of indexes}\\ -1,&\text{Odd Permutation of indexes}\\ 0,& \text{2 of indexes are equal}\end{cases} ε i j k = ⎩ ⎪ ⎨ ⎪ ⎧ 1 , − 1 , 0 , Even Permutation of indexes Odd Permutation of indexes 2 of indexes are equal ε i j k = 1 \varepsilon_{ijk}=1 ε i j k = 1 ε i j k = − 1 \varepsilon_{ijk}=-1 ε i j k = − 1
现有两个向量做向量积，其表达式为：A × B = ∣ e 1 e 2 e 3 A 1 A 2 A 3 B 1 B 2 B 3 ∣ = e 1 ( A 2 B 3 − A 3 B 2 ) + e 2 ( A 3 B 1 − A 1 B 3 ) + e 3 ( A 1 B 2 − A 2 B 1 ) \bold A\times\bold B=\begin{vmatrix}\bold e_1&\bold e_2&\bold e_3\\ A_1&A_2&A_3 \\ B_1&B_2&B_3\end{vmatrix}=\bold e_1(A_2B_3-A_3B_2)+\bold e_2(A_3B_1-A_1B_3)+\bold e_3(A_1B_2-A_2B_1) A × B = ∣ ∣ ∣ ∣ ∣ ∣ e 1 A 1 B 1 e 2 A 2 B 2 e 3 A 3 B 3 ∣ ∣ ∣ ∣ ∣ ∣ = e 1 ( A 2 B 3 − A 3 B 2 ) + e 2 ( A 3 B 1 − A 1 B 3 ) + e 3 ( A 1 B 2 − A 2 B 1 ) A × B = ϵ i j k e i A j B k \bold A\times\bold B=\epsilon_{ijk}\bold e_i A_jB_k A × B = ϵ i j k e i A j B k e 1 , e 2 , e 3 \bold e_1,\bold e_2,\bold e_3 e 1 , e 2 , e 3 ε i j k e i A j B k = ε j i k e j A i B k = ε k j i e k A j B i . . . \varepsilon_{ijk}\bold e_i A_jB_k=\varepsilon_{jik}\bold e_j A_iB_k=\varepsilon_{kji}\bold e_k A_jB_i\ ... ε i j k e i A j B k = ε j i k e j A i B k = ε k j i e k A j B i . . .
介绍完了向量乘法的两种表达形式的简化，在正式推导坐标之间的变换之前，我们还需要把球坐标和柱坐标的基向量与直角坐标系下的坐标相对应，也就是找到各组基之间的函数关系。曲线坐标系中基向量常见的表达形式是u 1 , u 2 , u 3 u_1, u_2, u_3 u 1 , u 2 , u 3
表达柱坐标系的三个方向的字母有所不同，本文采用r , θ , z r,\ \theta,\ z r , θ , z { x ( r , θ , z ) = r cos θ y ( r , θ , z ) = r sin θ z ( r , θ , z ) = z \begin{cases}x(r,\theta,z)=r\cos\theta\\ y(r,\theta,z)=r\sin\theta\\ z(r,\theta,z)=z\end{cases} ⎩ ⎪ ⎨ ⎪ ⎧ x ( r , θ , z ) = r cos θ y ( r , θ , z ) = r sin θ z ( r , θ , z ) = z { r ( x , y , z ) = x 2 + y 2 θ ( x , y , z ) = tan − 1 y x z ( x , y , z ) = z \begin{cases}r(x,y,z)=\sqrt{x^2+y^2}\\ \theta(x,y,z)=\tan^{-1}\frac{y}{x}\\ z(x,y,z)=z\end{cases} ⎩ ⎪ ⎨ ⎪ ⎧ r ( x , y , z ) = x 2 + y 2 θ ( x , y , z ) = tan − 1 x y z ( x , y , z ) = z
表达球坐标的三个方向的字母有所不同，本文采用ρ , ϕ , θ \rho,\ \phi,\ \theta ρ , ϕ , θ { x ( ρ , ϕ , θ ) = ρ sin ϕ cos θ y ( ρ , ϕ , θ ) = ρ sin ϕ sin θ z ( ρ , ϕ , θ ) = ρ cos ϕ \begin{cases}x(\rho,\phi, \theta)=\rho\sin\phi\cos\theta\\ y(\rho,\phi, \theta)=\rho\sin\phi\sin\theta\\ z(\rho,\phi, \theta)=\rho\cos\phi\end{cases} ⎩ ⎪ ⎨ ⎪ ⎧ x ( ρ , ϕ , θ ) = ρ sin ϕ cos θ y ( ρ , ϕ , θ ) = ρ sin ϕ sin θ z ( ρ , ϕ , θ ) = ρ cos ϕ
线元用d r 2 dr^2 d r 2 d r 2 = d r ⋅ d r = d x i d x i = d x 1 2 + d x 2 2 + d x 3 2 dr^2=d\bold r\cdot d\bold r=dx_idx_i=dx^2_1+dx^2_2+dx^2_3 d r 2 = d r ⋅ d r = d x i d x i = d x 1 2 + d x 2 2 + d x 3 2
弧长的微分形式为（三个方向分别对应一种微分表达）：{ d s 1 = d x d s 2 = d y d s 3 = d z \begin{cases}ds_1=dx\\ ds_2=dy\\ds_3=dz\end{cases} ⎩ ⎪ ⎨ ⎪ ⎧ d s 1 = d x d s 2 = d y d s 3 = d z
面元有三种，即分别在三个方向上(x,y,z)进行投影：{ d σ 12 = d x d y , Projection on x-y plane d σ 23 = d y d z , Projection on y-z plane d σ 13 = d x d z , Projection on x-z plane \begin{cases}d\sigma_{12}=dxdy,&\text{Projection on x-y plane}\\ d\sigma_{23}=dydz,&\text{Projection on y-z plane}\\d\sigma_{13}=dxdz,&\text{Projection on x-z plane}\end{cases} ⎩ ⎪ ⎨ ⎪ ⎧ d σ 1 2 = d x d y , d σ 2 3 = d y d z , d σ 1 3 = d x d z , Projection on x-y plane Projection on y-z plane Projection on x-z plane
面元有三种，即分别在三个方向上进行投影：d V = d s 1 d s 2 d s 3 = d x d y d z dV=ds_1ds_2ds_3=dxdydz d V = d s 1 d s 2 d s 3 = d x d y d z
本节介绍如何从直角坐标下的三种元变换到曲线坐标，多公式预警！
通用的方法是设曲线坐标系的三个方向为u 1 , u 2 , u 3 u_1,u_2,u_3 u 1 , u 2 , u 3 x 1 , x 2 , x 3 x_1,x_2,x_3 x 1 , x 2 , x 3 u i u_i u i { x 1 = x 1 ( u 1 , u 2 , u 3 ) x 2 = x 2 ( u 1 , u 2 , u 3 ) x 3 = x 3 ( u 1 , u 2 , u 3 ) \begin{cases}x_1=x^1(u_1,u_2,u_3)\\x_2=x^2(u_1,u_2,u_3)\\x_3=x^3(u_1,u_2,u_3)\end{cases} ⎩ ⎪ ⎨ ⎪ ⎧ x 1 = x 1 ( u 1 , u 2 , u 3 ) x 2 = x 2 ( u 1 , u 2 , u 3 ) x 3 = x 3 ( u 1 , u 2 , u 3 ) 注意这里的上标并不代表次幂，只是一个记号。 线元 就变成了：d r 2 = d x i d x i = d x 1 2 + d x 2 2 + d x 3 2 = ∂ x i ∂ u j d u j ∂ x i ∂ u k d u k dr^2=dx_idx_i=dx^2_1+dx^2_2+dx^2_3=\frac{\partial x^i}{\partial u_j}du_j\frac{\partial x^i}{\partial u_k}du_k d r 2 = d x i d x i = d x 1 2 + d x 2 2 + d x 3 2 = ∂ u j ∂ x i d u j ∂ u k ∂ x i d u k d x i = ∂ x i ∂ u 1 d u 1 + ∂ x i ∂ u 2 d u 2 + ∂ x i ∂ u 3 d u 3 dx_i=\frac{\partial x^i}{\partial u_1}du_1+\frac{\partial x^i}{\partial u_2}du_2+\frac{\partial x^i}{\partial u_3}du_3 d x i = ∂ u 1 ∂ x i d u 1 + ∂ u 2 ∂ x i d u 2 + ∂ u 3 ∂ x i d u 3
令g i j = ∂ x i ∂ u j ∂ x i ∂ u k g_{ij}=\frac{\partial x^i}{\partial u_j}\frac{\partial x^i}{\partial u_k} g i j = ∂ u j ∂ x i ∂ u k ∂ x i d r 2 = g i j d u i d u j dr^2=g_{ij}du_idu_j d r 2 = g i j d u i d u j g i j = { g i i i = j 0 i ≠ j g_{ij}=\begin{cases}g_{ii}&i=j\\0&i\neq j\end{cases} g i j = { g i i 0 i = j i ̸ = j 曲线坐标系下线元的形式 ：d r 2 = g 11 ( d u 1 ) 2 + g 22 ( d u 2 ) 2 + g 33 ( d u 3 ) 2 dr^2=g_{11}(du_1)^2+g_{22}(du_2)^2+g_{33}(du_3)^2 d r 2 = g 1 1 ( d u 1 ) 2 + g 2 2 ( d u 2 ) 2 + g 3 3 ( d u 3 ) 2
则每一个弧长微分 为：{ d s 1 = g 11 d u 1 = h 1 d u 1 d s 2 = g 22 d u 2 = h 2 d u 2 d s 3 = g 33 d u 3 = h 3 d u 3 \begin{cases}ds_1=\sqrt{g_{11}}du_1=h_1du_1\\ ds_2=\sqrt{g_{22}}du_2=h_2du_2\\ ds_3=\sqrt{g_{33}}du_3=h_3du_3\end{cases} ⎩ ⎪ ⎨ ⎪ ⎧ d s 1 = g 1 1 d u 1 = h 1 d u 1 d s 2 = g 2 2 d u 2 = h 2 d u 2 d s 3 = g 3 3 d u 3 = h 3 d u 3
面元 为：{ d σ 12 = h 1 h 2 d u 1 d u 2 d σ 23 = h 2 h 3 d u 2 d u 3 d σ 13 = h 1 h 3 d u 1 d u 3 , \begin{cases}d\sigma_{12}=h_1h_2du_1du_2\\ d\sigma_{23}=h_2h_3du_2du_3\\d\sigma_{13}=h_1h_3du_1du_3,\end{cases} ⎩ ⎪ ⎨ ⎪ ⎧ d σ 1 2 = h 1 h 2 d u 1 d u 2 d σ 2 3 = h 2 h 3 d u 2 d u 3 d σ 1 3 = h 1 h 3 d u 1 d u 3 ,
体元 为:d V = d s 1 d s 2 d s 3 = h 1 h 2 h 3 d u 1 d u 2 d u 3 dV=ds_1ds_2ds_3=h_1h_2h_3du_1du_2du_3 d V = d s 1 d s 2 d s 3 = h 1 h 2 h 3 d u 1 d u 2 d u 3
本章第二节已经说明了直角坐标和柱坐标的转换函数了。d r 2 = d r ⋅ d r = ∂ x i ∂ u j d u j ∂ x i ∂ u k d u k dr^2=d\bold r\cdot d\bold r=\frac{\partial x^i}{\partial u_j}du_j\frac{\partial x^i}{\partial u_k}du_k d r 2 = d r ⋅ d r = ∂ u j ∂ x i d u j ∂ u k ∂ x i d u k (1) d r 2 = ( ∂ x ∂ r d r + ∂ x ∂ θ d θ ) 2 + ( ∂ y ∂ r d r + ∂ y ∂ θ d θ ) 2 + ( ∂ z ∂ z d z ) 2 dr^2=(\frac{\partial x}{\partial r}dr+\frac{\partial x}{\partial \theta}d\theta)^2+(\frac{\partial y}{\partial r}dr+\frac{\partial y}{\partial \theta}d\theta)^2+(\frac{\partial z}{\partial z}dz)^2\tag{1} d r 2 = ( ∂ r ∂ x d r + ∂ θ ∂ x d θ ) 2 + ( ∂ r ∂ y d r + ∂ θ ∂ y d θ ) 2 + ( ∂ z ∂ z d z ) 2 ( 1 ) (2) ∂ x ∂ r = cos θ ; ∂ x ∂ θ = − r sin θ ; ∂ y ∂ r = sin θ ; ∂ y ∂ θ = r cos θ \frac{\partial x}{\partial r}=\cos\theta;\frac{\partial x}{\partial \theta}=-r\sin\theta;\frac{\partial y}{\partial r}=\sin\theta;\frac{\partial y}{\partial \theta}=r\cos\theta\tag{2} ∂ r ∂ x = cos θ ; ∂ θ ∂ x = − r sin θ ; ∂ r ∂ y = sin θ ; ∂ θ ∂ y = r cos θ ( 2 ) { g r r = 1 g θ θ = r 2 g z z = 1 , { h r = 1 h θ = r h z = 1 \begin{cases}g_{rr}=1\\ g_{\theta\theta}=r^2\\ g_{zz}=1\end{cases},\ \begin{cases}h_{r}=1\\ h_{\theta}=r\\ h_{z}=1\end{cases} ⎩ ⎪ ⎨ ⎪ ⎧ g r r = 1 g θ θ = r 2 g z z = 1 , ⎩ ⎪ ⎨ ⎪ ⎧ h r = 1 h θ = r h z = 1
最后是直角坐标和球坐标的转换了。d r 2 = d r ⋅ d r = ∂ x i ∂ u j d u j ∂ x i ∂ u k d u k dr^2=d\bold r\cdot d\bold r=\frac{\partial x^i}{\partial u_j}du_j\frac{\partial x^i}{\partial u_k}du_k d r 2 = d r ⋅ d r = ∂ u j ∂ x i d u j ∂ u k ∂ x i d u k (3) d r 2 = ( ∂ x ∂ ρ d ρ + ∂ x ∂ ϕ d ϕ + ∂ x ∂ θ d θ ) 2 + ( ∂ y ∂ ρ d ρ + ∂ y ∂ ϕ d ϕ + ∂ y ∂ θ d θ ) 2 + ( ∂ z ∂ ρ d ρ + ∂ z ∂ ϕ d ϕ ) 2 dr^2=(\frac{\partial x}{\partial \rho}d\rho+\frac{\partial x}{\partial \phi}d\phi+\frac{\partial x}{\partial \theta}d\theta)^2+(\frac{\partial y}{\partial \rho}d\rho+\frac{\partial y}{\partial \phi}d\phi+\frac{\partial y}{\partial \theta}d\theta)^2+(\frac{\partial z}{\partial \rho}d\rho+\frac{\partial z}{\partial \phi}d\phi)^2\ \tag{3} d r 2 = ( ∂ ρ ∂ x d ρ + ∂ ϕ ∂ x d ϕ + ∂ θ ∂ x d θ ) 2 + ( ∂ ρ ∂ y d ρ + ∂ ϕ ∂ y d ϕ + ∂ θ ∂ y d θ ) 2 + ( ∂ ρ ∂ z d ρ + ∂ ϕ ∂ z d ϕ ) 2 ( 3 ) ∂ x ∂ ρ = sin ϕ cos θ ; ∂ x ∂ ϕ = − ρ cos ϕ cos θ ; ∂ x ∂ θ = − ρ sin ϕ sin θ ; \frac{\partial x}{\partial \rho}=\sin\phi\cos\theta;\frac{\partial x}{\partial \phi}=-\rho\cos\phi\cos\theta;\frac{\partial x}{\partial \theta}=-\rho\sin\phi\sin\theta;
∂ ρ ∂ x = sin ϕ cos θ ; ∂ ϕ ∂ x = − ρ cos ϕ cos θ ; ∂ θ ∂ x = − ρ sin ϕ sin θ ; ∂ y ∂ ρ = sin ϕ sin θ ; ∂ y ∂ ϕ = ρ cos ϕ sin θ ; ∂ y ∂ θ = ρ sin ϕ cos θ \frac{\partial y}{\partial \rho}=\sin\phi\sin\theta;\frac{\partial y}{\partial \phi}=\rho\cos\phi\sin\theta;\frac{\partial y}{\partial \theta}=\rho\sin\phi\cos\theta ∂ ρ ∂ y = sin ϕ sin θ ; ∂ ϕ ∂ y = ρ cos ϕ sin θ ; ∂ θ ∂ y = ρ sin ϕ cos θ (4) ∂ z ∂ ρ = cos ϕ ; ∂ z ∂ ϕ = − ρ sin ϕ \frac{\partial z}{\partial \rho}=\cos\phi;\frac{\partial z}{\partial \phi}=-\rho\sin\phi\tag{4} ∂ ρ ∂ z = cos ϕ ; ∂ ϕ ∂ z = − ρ sin ϕ ( 4 ) { h ρ = 1 h ϕ = ρ h θ = ρ sin ϕ \begin{cases}h_{\rho}=1\\ h_{\phi}=\rho\\ h_{\theta}=\rho\sin\phi\end{cases} ⎩ ⎪ ⎨ ⎪ ⎧ h ρ = 1 h ϕ = ρ h θ = ρ sin ϕ
线元 { d s r = d r d s θ = r d θ d s z = d z \begin{cases}ds_r=dr\\ds_\theta=rd\theta\\ ds_z=dz\end{cases} ⎩ ⎪ ⎨ ⎪ ⎧ d s r = d r d s θ = r d θ d s z = d z 面元 { d σ r θ = d s r d s θ = r d r d θ d σ θ z = d s θ d s z = r d θ d z d σ r z = d s r d s z = d r d z \begin{cases}d\sigma_{r\theta}=ds_rds_\theta =rdrd\theta\\d\sigma_{\theta z}=ds_\theta ds_z =rd\theta dz\\ d\sigma_{rz}=ds_rds_z=drdz\end{cases} ⎩ ⎪ ⎨ ⎪ ⎧ d σ r θ = d s r d s θ = r d r d θ d σ θ z = d s θ d s z = r d θ d z d σ r z = d s r d s z = d r d z 体元 d V = d s r d s θ d s z = r d r d θ d z dV=ds_rds_\theta ds_z=rdrd\theta dz d V = d s r d s θ d s z = r d r d θ d z
线元 { d s ρ = d ρ d s ϕ = ρ d ϕ d s θ = ρ sin θ d θ \begin{cases}ds_\rho=d\rho\\
ds_\phi=\rho d\phi\\
ds_\theta=\rho\sin\theta d\theta
\end{cases} ⎩ ⎪ ⎨ ⎪ ⎧ d s ρ = d ρ d s ϕ = ρ d ϕ d s θ = ρ sin θ d θ 面元 { d σ ρ ϕ = d s ρ d s ϕ = ρ d ρ d ϕ d σ ϕ θ = ρ 2 sin θ d ϕ d θ d σ ρ θ = ρ sin θ d ρ d θ \begin{cases}d\sigma_{\rho\phi}=ds_\rho ds_\phi =\rho d\rho d\phi\\
d\sigma_{\phi\theta}=\rho^2\sin\theta d\phi d\theta \\
d\sigma_{\rho\theta}=\rho\sin\theta d\rho d\theta\end{cases} ⎩ ⎪ ⎨ ⎪ ⎧ d σ ρ ϕ = d s ρ d s ϕ = ρ d ρ d ϕ d σ ϕ θ = ρ 2 sin θ d ϕ d θ d σ ρ θ = ρ sin θ d ρ d θ 体元 d V = d s ρ d s ϕ d s θ = ρ 2 sin ϕ d ρ d ϕ d θ dV=ds_\rho ds_\phi ds_\theta=\rho^2\sin\phi d \rho d \phi d\theta d V = d s ρ d s ϕ d s θ = ρ 2 sin ϕ d ρ d ϕ d θ
以上是曲线坐标系中最基本的几个表达式。本节在上文内容的基础上引出柱坐标和球坐标以及曲线坐标系下梯度，散度，旋度的表达形式。
直角坐标系中某一标量场f ( x , y , z ) f(x,y,z) f ( x , y , z ) ∇ f = ∂ f ∂ x e i + ∂ f ∂ y e j + ∂ f ∂ z e k \nabla f=\frac{\partial f}{\partial x}\bold e_i+\frac{\partial f}{\partial y}\bold e_j+\frac{\partial f}{\partial z}\bold e_k ∇ f = ∂ x ∂ f e i + ∂ y ∂ f e j + ∂ z ∂ f e k ∇ f ⋅ d r = d f \nabla f\cdot d\bold r=df ∇ f ⋅ d r = d f
如果是任意正交曲线坐标系下的标量场f ( u 1 , u 2 , u 3 ) f(u_1,u_2,u_3) f ( u 1 , u 2 , u 3 ) d f = ∂ f ∂ u i e u i df=\frac{\partial f}{\partial u_i}\bold e_{u_i} d f = ∂ u i ∂ f e u i d f = ∇ f i d s i = ∇ f i h i d u i df=\nabla f_i ds_i =\nabla f_i h_idu_i d f = ∇ f i d s i = ∇ f i h i d u i ∂ f ∂ u i = ∇ f i h i d u i → ∇ f i = 1 h i ∂ f ∂ u i \frac{\partial f}{\partial u_i}=\nabla f_i h_idu_i\ \to \ \nabla f_i=\frac{1}{h_i}\frac{\partial f}{\partial u_i} ∂ u i ∂ f = ∇ f i h i d u i → ∇ f i = h i 1 ∂ u i ∂ f ∇ f = 1 h 1 ∂ f ∂ u 1 e u 1 + 1 h 2 ∂ f ∂ u 2 e u 2 + 1 h 3 ∂ f ∂ u 3 e u 3 \nabla f=\frac{1}{h_1}\frac{\partial f}{\partial u_1}\bold e_{u_1}+\frac{1}{h_2}\frac{\partial f}{\partial u_2}\bold e_{u_2}+\frac{1}{h_3}\frac{\partial f}{\partial u_3}\bold e_{u_3} ∇ f = h 1 1 ∂ u 1 ∂ f e u 1 + h 2 1 ∂ u 2 ∂ f e u 2 + h 3 1 ∂ u 3 ∂ f e u 3
曲线坐标系下的哈密顿算子 ∇ \nabla ∇
∇ = 1 h 1 ∂ ∂ u 1 e u 1 + 1 h 2 ∂ ∂ u 2 e u 2 + 1 h 3 ∂ ∂ u 3 e u 3 \nabla =\frac{1}{h_1}\frac{\partial }{\partial u_1}\bold e_{u_1}+\frac{1}{h_2}\frac{\partial }{\partial u_2}\bold e_{u_2}+\frac{1}{h_3}\frac{\partial }{\partial u_3}\bold e_{u_3} ∇ = h 1 1 ∂ u 1 ∂ e u 1 + h 2 1 ∂ u 2 ∂ e u 2 + h 3 1 ∂ u 3 ∂ e u 3
假设曲线坐标系下有一向量场A \bold A A div A = lim Δ V → 0 1 Δ V ∯ S A ⋅ d σ \text{div}\bold A=\lim_{\Delta V\to 0}\frac{1}{\Delta V}\oiint_S\bold A\cdot d \bold\sigma div A = Δ V → 0 lim Δ V 1 ∬ S A ⋅ d σ e 1 \bold e_1 e 1 ∂ ∂ u 1 ( A 1 h 2 h 3 ) d u 1 d u 2 d u 3 \frac{\partial }{\partial u_1}(A_1h_2h_3)du_1du_2du_3 ∂ u 1 ∂ ( A 1 h 2 h 3 ) d u 1 d u 2 d u 3 d V = h 1 h 2 h 3 u 1 u 2 u 3 dV=h_1h_2h_3u_1u_2u_3 d V = h 1 h 2 h 3 u 1 u 2 u 3 e 1 \bold e_1 e 1 1 h 1 h 2 h 3 ∂ ∂ u 1 ( A 1 h 2 h 3 ) \frac{1}{h_1h_2h_3}\frac{\partial }{\partial u_1}(A_1h_2h_3) h 1 h 2 h 3 1 ∂ u 1 ∂ ( A 1 h 2 h 3 )
整体的散度表达式 为：∇ ⋅ A = 1 h 1 h 2 h 3 [ ∂ ∂ u 1 ( A 1 h 2 h 3 ) + ∂ ∂ u 2 ( A 2 h 3 h 1 ) + ∂ ∂ u 3 ( A 3 h 1 h 2 ) ] \nabla\cdot\bold A= \frac{1}{h_1h_2h_3}\begin{bmatrix}\frac{\partial }{\partial u_1}(A_1h_2h_3)+\frac{\partial }{\partial u_2}(A_2h_3h_1)+\frac{\partial }{\partial u_3}(A_3h_1h_2)\end{bmatrix} ∇ ⋅ A = h 1 h 2 h 3 1 [ ∂ u 1 ∂ ( A 1 h 2 h 3 ) + ∂ u 2 ∂ ( A 2 h 3 h 1 ) + ∂ u 3 ∂ ( A 3 h 1 h 2 ) ]
当h 1 = h 2 = h 3 = 1 h_1=h_2=h_3=1 h 1 = h 2 = h 3 = 1
旋度的定义 A \bold A A curl A ≡ ∇ × A = lim Δ σ → 0 1 Δ σ ∮ C A ⋅ d l \text{curl}\bold A\equiv\nabla\times\bold A=\lim_{\Delta \sigma\to 0}\frac{1}{\Delta\sigma}\oint_C \bold A\cdot d\bold l curl A ≡ ∇ × A = Δ σ → 0 lim Δ σ 1 ∮ C A ⋅ d l
推导思路： ∇ × A = ∣ e u 1 h 1 e u 1 h 2 e u 1 h 3 ∂ ∂ u 1 ∂ ∂ u 2 ∂ ∂ u 3 h 1 A 1 h 2 A 2 h 3 A 3 ∣ \nabla\times\bold A=\begin{vmatrix}\bold e_{u_1}h_1&\bold e_{u_1}h_2&\bold e_{u_1}h_3\\
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\frac{\partial}{\partial u_1}&\frac{\partial}{\partial u_2}&\frac{\partial}{\partial u_3}\\ \\
h_1A_1&h_2A_2&h_3A_3
\end{vmatrix} ∇ × A = ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ e u 1 h 1 ∂ u 1 ∂ h 1 A 1 e u 1 h 2 ∂ u 2 ∂ h 2 A 2 e u 1 h 3 ∂ u 3 ∂ h 3 A 3 ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣
球坐标和柱坐标下的散度和旋度表达式只需带入相应的参数h h h 。
