本文搬运自Axeho’s blog ，作了一些补充。
Author : Axeho
Link : 关于向量场的微积分
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Hamilton 算符
基本定义与物理含义
在直角坐标系下，定义如下算子：
∇ = ∂ ∂ x 1 e 1 + ∂ ∂ x 2 e 2 + ∂ ∂ x 3 e 3 \nabla = \frac{\partial }{\partial x_1}\boldsymbol e_1 + \frac{\partial }{\partial x_2}\boldsymbol e_{2} + \frac{\partial }{\partial x_3}\boldsymbol e_3
∇ = ∂ x 1 ∂ e 1 + ∂ x 2 ∂ e 2 + ∂ x 3 ∂ e 3
并称为 H a m i l t o n ~\mathrm{Hamilton}~ Hamilton N a b l a ~\mathrm{Nabla}~ Nabla ∇ ~\nabla~ ∇
∂ ∂ x i ′ = ∂ ∂ x j ⋅ ∂ x j ∂ x i ′ = T j i T ∂ ∂ x j = T i j ∂ ∂ x j \frac{\partial}{\partial x_i^{\prime}}=\frac{\partial}{\partial x_j} \cdot \frac{\partial x_j}{\partial x_i^{\prime}}=T_{j i}^{\mathrm{T}} \frac{\partial}{\partial x_j}=T_{i j} \frac{\partial}{\partial x_j}
∂ x i ′ ∂ = ∂ x j ∂ ⋅ ∂ x i ′ ∂ x j = T ji T ∂ x j ∂ = T ij ∂ x j ∂
随坐标的正交变换而作相同系数的正交变换，具有相同的转动性质，所以既是微分算子，又是矢量（注意，这里的矢量和线性空间中抽象向量的定义不同，它连带一个变换）。
∇ ~\nabla~ ∇ φ ( x , y , z ) ~\varphi(x,y,z)~ φ ( x , y , z )
∇ φ = g r a d φ = ∂ φ ∂ x 1 e 1 + ∂ φ ∂ x 2 e 2 + ∂ φ ∂ x 3 e 3 \nabla\varphi = \boldsymbol{\mathrm{grad}}~\varphi = \frac{\partial \varphi}{\partial x_1}\boldsymbol e_1 + \frac{\partial \varphi}{\partial x_2}\boldsymbol e_2 + \frac{\partial \varphi}{\partial x_3}\boldsymbol e_3
∇ φ = grad φ = ∂ x 1 ∂ φ e 1 + ∂ x 2 ∂ φ e 2 + ∂ x 3 ∂ φ e 3
可以看出，数量场的梯度是一个向量场。 ∇ ~\nabla~ ∇
d i v v = ∇ ⋅ v = ∂ P ∂ x 1 + ∂ Q ∂ x 2 + ∂ R ∂ x 3 \mathrm{div}~\boldsymbol v = \nabla\cdot\boldsymbol v = \frac{\partial P}{\partial x_1} + \frac{\partial Q}{\partial x_2} + \frac{\partial R}{\partial x_3}
div v = ∇ ⋅ v = ∂ x 1 ∂ P + ∂ x 2 ∂ Q + ∂ x 3 ∂ R
可以看出，向量场的散度是一个数量场。 ∇ ~\nabla~ ∇
r o t v = ∇ × v = ( ∂ R ∂ y − ∂ Q ∂ z ) e 1 + ( ∂ P ∂ z − ∂ R ∂ x ) e 2 + ( ∂ Q ∂ x − ∂ P ∂ y ) e 3 \boldsymbol{\mathrm{rot}}~\boldsymbol v = \nabla\times\boldsymbol v =
\left(
\frac{\partial R}{\partial y} -
\frac{\partial Q}{\partial z}
\right)
\boldsymbol e_ 1+
\left(
\frac{\partial P}{\partial z} -
\frac{\partial R}{\partial x}
\right)
\boldsymbol e_2 +
\left(
\frac{\partial Q}{\partial x} -
\frac{\partial P}{\partial y}
\right)
\boldsymbol e_3
rot v = ∇ × v = ( ∂ y ∂ R − ∂ z ∂ Q ) e 1 + ( ∂ z ∂ P − ∂ x ∂ R ) e 2 + ( ∂ x ∂ Q − ∂ y ∂ P ) e 3
可以看出，向量场的旋度是一个向量场。
r o t v = ∇ × v = ∣ e 1 e 2 e 3 ∂ ∂ x ∂ ∂ y ∂ ∂ z P Q R ∣ \boldsymbol{\mathrm{rot}}~\boldsymbol v = \nabla\times\boldsymbol v =
\begin{vmatrix}
\boldsymbol e_1 &
\boldsymbol e_2 &
\boldsymbol e_3 \\[6pt]
\frac{\partial }{\partial x} &
\frac{\partial }{\partial y} &
\frac{\partial }{\partial z} \\[6pt]
P & Q & R
\end{vmatrix}
rot v = ∇ × v = ∣ ∣ e 1 ∂ x ∂ P e 2 ∂ y ∂ Q e 3 ∂ z ∂ R ∣ ∣
可以验证以下结果
r o t g r a d φ = ∇ × ∇ φ = 0 d i v r o t a = ∇ ⋅ ( ∇ × a ) = 0 \begin{aligned}
\boldsymbol{\mathrm{rot~grad}}~\varphi &= \nabla\times\nabla\varphi = \boldsymbol 0 \\
\mathrm{div}~\boldsymbol{\mathrm{rot}}~\boldsymbol a &= \nabla\cdot(\nabla\times \boldsymbol a) = 0
\end{aligned}
rot grad φ div rot a = ∇ × ∇ φ = 0 = ∇ ⋅ ( ∇ × a ) = 0
上式表明，梯度的旋度为零向量，旋度的散度为零。
我们知道，梯度，旋度和散度有着几何或是物理意义。梯度的方向是数量场增加最快的方向，大小则是该方向的斜率大小。散度描述了向量场的“源”或“漏”，向量从此处发散或汇聚到此处，散度的大小即是某种通量。旋度则描述了向量场的旋转情况，旋度的大小是涡量的最大值，方向则是旋转轴的指向。
由此，我们可以比较直观的理解梯度的旋度和旋度的散度问题。若梯度的旋度不为零，则意味着梯度场中存在涡量。梯度的方向是增加最快的方向，若在场中沿各个向量可以连接出闭合曲线，那么意味着原数量场的最大值即是其最小值，这在数量场不是常量时是不可能的。
而旋度的散度为零，则表明向量场各点处涡量（环量除以面积取极限）的旋转轴不会汇聚到或发散自一点。倘若转轴汇聚到或发散自一点，那么涡量会相互抵消，就不存在这些转轴了。
引入求和约定
引入爱因斯坦求和约定，Hamilton \text{Hamilton} Hamilton
∇ = ∂ i e i \nabla = \partial_i \bm{e}_i
∇ = ∂ i e i
我们约定，在一个单项式中重复出现的下标默认为对该下标求和 e i \bm{e}_i e i i i i
标量场 的梯度：
∇ φ = ∂ i φ e i \nabla \varphi = \partial_i \varphi\,\bm{e}_i
∇ φ = ∂ i φ e i
向量场 的散度：
∇ ⋅ φ = ∂ i e i ⋅ φ j e j = ∂ i φ j δ i j = ∂ i φ i \nabla \cdot \bm{\varphi} = \partial_i \bm{e}_i \cdot \varphi_j \bm{e}_j = \partial_i\varphi_j \delta_{ij} = \partial_i\varphi_i
∇ ⋅ φ = ∂ i e i ⋅ φ j e j = ∂ i φ j δ ij = ∂ i φ i
向量场 的旋度：
∇ × φ = ϵ i j k ∂ j φ k e i \nabla \times \bm{\varphi} = \epsilon_{ijk}\partial_j \varphi_k \,\bm{e}_i
∇ × φ = ϵ ijk ∂ j φ k e i
可以看到，如果使用求和符号，就可以不用时刻检查作用对象究竟是向量场还是标量场，只需把标量场写作 φ j e j \varphi_j \bm{e}_j φ j e j
这是数学史上的一大发现，若不信的话，可以试着返回那不使用这方法的古板日子。
——阿尔伯特·爱因斯坦
求和约定以其表示的简洁而著称，但是初学者使用起来非常容易眼花。此处黑体也表示向量或者向量场，请读者一定要拿出草稿纸写一写，认真比较它们的异同。
对 ∇ ⋅ ( ∇ × φ ) = 0 \nabla\cdot (\nabla\times \bm{\varphi}) = 0 ∇ ⋅ ( ∇ × φ ) = 0
∇ ⋅ ( ∇ × φ ) = ∂ i e i ⋅ ϵ l m n ∂ m φ n e l = ϵ l m n ∂ i ∂ m δ i l φ n = ϵ l m n ∂ l ∂ m φ n \begin{aligned}
\nabla\cdot (\nabla\times \bm{\varphi}) &= \partial_i \bm{e}_i \cdot \epsilon_{lmn}\partial_m \varphi_n \,\bm{e}_l\\
&= \epsilon_{lmn} \partial_i\partial_m \delta_{il} \varphi_n \\
&= \epsilon_{lmn} \partial_l\partial_m \varphi_n
\end{aligned}
∇ ⋅ ( ∇ × φ ) = ∂ i e i ⋅ ϵ l mn ∂ m φ n e l = ϵ l mn ∂ i ∂ m δ i l φ n = ϵ l mn ∂ l ∂ m φ n
由于 ∂ l ∂ m \partial_l\partial_m ∂ l ∂ m m , n m,n m , n ϵ l m n \epsilon_{lmn} ϵ l mn m , n m,n m , n 这相当于奇函数在对称区间积分 ，所以求和为 0 0 0
又如，证明 ∇ ( A ⋅ B ) = ( B ⋅ ∇ ) A + B × ( ∇ × A ) + ( A ⋅ ∇ ) B + A × ( ∇ × B ) \nabla(\boldsymbol{A} \cdot \boldsymbol{B})=(\boldsymbol{B} \cdot \nabla) \boldsymbol{A}+\boldsymbol{B} \times(\nabla \times \boldsymbol{A})+(\boldsymbol{A} \cdot \nabla) \boldsymbol{B}+\boldsymbol{A} \times(\nabla \times \boldsymbol{B}) ∇ ( A ⋅ B ) = ( B ⋅ ∇ ) A + B × ( ∇ × A ) + ( A ⋅ ∇ ) B + A × ( ∇ × B )
等式右边前两项的 i i i
= B j ∂ j A i + ε i j k B j ( ∇ × A ) k = B j ∂ j A i + ε i j k B j ε k l m ∂ l A m = B j ∂ j A i + ( δ i l δ j m − δ i m δ j l ) B j ∂ l A m = B j ∂ j A i + B j ∂ i A j − B j ∂ j A i = B j ∂ i A j , \begin{aligned}
&=B_j \partial_j A_i+\varepsilon_{i j k} B_j(\nabla \times A)_k \\
&=B_j \partial_j A_i+\varepsilon_{i j k} B_j \varepsilon_{k l m} \partial_l A_m \\
&=B_j \partial_j A_i+\left(\delta_{i l} \delta_{j m}-\delta_{i m} \delta_{j l}\right) B_j \partial_l A_m \\
&=B_j \partial_j A_i+B_j \partial_i A_j-B_j \partial_j A_i=B_j \partial_i A_j,
\end{aligned}
= B j ∂ j A i + ε ijk B j ( ∇ × A ) k = B j ∂ j A i + ε ijk B j ε k l m ∂ l A m = B j ∂ j A i + ( δ i l δ jm − δ im δ j l ) B j ∂ l A m = B j ∂ j A i + B j ∂ i A j − B j ∂ j A i = B j ∂ i A j ,
同理可得左边 i i i
再如，一个电偶极子处于原点，已知电势 U U U
U = 1 4 π ε 0 p ⋅ r r 3 E = − ∇ U = − 1 4 π ε 0 ∇ ( p ⋅ r r 3 ) = − 1 4 π ε 0 ∇ ( p ⋅ r ) r 3 − p ⋅ r 4 π ε 0 ∇ ( 1 r 3 ) = − 1 4 π ε 0 ∂ i ( p j r j ) e i r 3 + 3 p ⋅ r 4 π ε 0 r 4 ∇ r = − 1 4 π ε 0 p j δ i j e i r 3 + 3 p ⋅ r 4 π ε 0 r 4 r r = − 1 4 π ε 0 p r 3 + 3 ( p ⋅ r ) r 4 π ε 0 r 5 \begin{aligned}
U &=\frac{1}{4 \pi \varepsilon_0} \frac{\boldsymbol{p} \cdot \boldsymbol{r}}{r^3} \\
\boldsymbol{E} &=-\nabla U=-\frac{1}{4 \pi \varepsilon_0} \nabla\left(\frac{\boldsymbol{p} \cdot \boldsymbol{r}}{r^3}\right) \\
&=-\frac{1}{4 \pi \varepsilon_0} \frac{\nabla(\boldsymbol{p} \cdot \boldsymbol{r})}{r^3}-\frac{\boldsymbol{p} \cdot \boldsymbol{r}}{4 \pi \varepsilon_0} \nabla\left(\frac{1}{r^3}\right) \\
&=-\frac{1}{4 \pi \varepsilon_0} \frac{\partial_i\left(p_j r_j\right) \boldsymbol{e}_i}{r^3}+\frac{3 \boldsymbol{p} \cdot \boldsymbol{r}}{4 \pi \varepsilon_0 r^4} \nabla r \\
&=-\frac{1}{4 \pi \varepsilon_0} \frac{p_j \delta_{i j} \boldsymbol{e}_i}{r^3}+\frac{3 \boldsymbol{p} \cdot \boldsymbol{r}}{4 \pi \varepsilon_0 r^4} \frac{\boldsymbol{r}}{r}\\
&=-\frac{1}{4 \pi \varepsilon_0} \frac{\boldsymbol{p}}{r^3}+\frac{3(\boldsymbol{p} \cdot \boldsymbol{r}) \boldsymbol{r}}{4 \pi \varepsilon_0 r^5}
\end{aligned}
U E = 4 π ε 0 1 r 3 p ⋅ r = − ∇ U = − 4 π ε 0 1 ∇ ( r 3 p ⋅ r ) = − 4 π ε 0 1 r 3 ∇ ( p ⋅ r ) − 4 π ε 0 p ⋅ r ∇ ( r 3 1 ) = − 4 π ε 0 1 r 3 ∂ i ( p j r j ) e i + 4 π ε 0 r 4 3 p ⋅ r ∇ r = − 4 π ε 0 1 r 3 p j δ ij e i + 4 π ε 0 r 4 3 p ⋅ r r r = − 4 π ε 0 1 r 3 p + 4 π ε 0 r 5 3 ( p ⋅ r ) r
其中 ∂ i ( p j r j ) e i = p j δ i j e i \partial_i\left(p_j r_j\right) \boldsymbol{e}_i = p_j \delta_{i j} \boldsymbol{e}_i ∂ i ( p j r j ) e i = p j δ ij e i p \boldsymbol{p} p
Green定理，Gauss定理与Stocks定理
设 D ~D~ D L ~L~ L L = ∂ D ~L = \partial D~ L = ∂ D v = P ( x , y ) i + Q ( x , y ) j ~\boldsymbol{v} = P(x,y)\boldsymbol{i} + Q(x,y)\boldsymbol{j}~ v = P ( x , y ) i + Q ( x , y ) j D ~D~ D G r e e n ~\mathrm{Green}~ Green
∮ ∂ D P d x + Q d y = ∬ D ( ∂ Q ∂ x − ∂ P ∂ y ) d x d y \oint_{\partial D}P~\mathrm{d}x + Q~\mathrm{d}y = \iint_D\left({\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}}\right)\mathrm{d}x\mathrm{d}y
∮ ∂ D P d x + Q d y = ∬ D ( ∂ x ∂ Q − ∂ y ∂ P ) d x d y
其中曲线积分的方向为 L = ∂ D ~L = \partial D~ L = ∂ D
设 D ~D~ D D ~D~ D D ~D~ D v ~\boldsymbol{v}~ v D ~D~ D
向量场 v ~\boldsymbol v~ v D ~D~ D L ~L~ L
∮ L v ⋅ d r = 0 \oint_L \boldsymbol{v}\cdot\mathrm{d}\boldsymbol r = 0
∮ L v ⋅ d r = 0
或者说 v ~\boldsymbol v~ v D ~D~ D
向量场 v ~\boldsymbol v~ v φ ( x , y ) ~\varphi(x,y)~ φ ( x , y )
v = g r a d φ ( x , y ) = ∇ φ ( x , y ) \boldsymbol v = \boldsymbol{\mathrm{grad}}~\varphi(x,y) = \nabla\varphi(x,y)
v = grad φ ( x , y ) = ∇ φ ( x , y )
且这样的函数 φ ( x , y ) ~\varphi(x, y)~ φ ( x , y )
向量场 v ~\boldsymbol v~ v Q , P ~Q, P~ Q , P
∂ Q ∂ x − ∂ P ∂ y = 0 \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 0
∂ x ∂ Q − ∂ y ∂ P = 0
设 v = P ( x , y , z ) i + Q ( x , y , z ) j + R ( x , y , z ) k ~\boldsymbol v = P(x,y,z)\boldsymbol i + Q(x,y,z)\boldsymbol j + R(x,y,z)\boldsymbol k~ v = P ( x , y , z ) i + Q ( x , y , z ) j + R ( x , y , z ) k V ~V~ V V ~V~ V V ~V~ V X ~X~ X Y ~Y~ Y Z ~Z~ Z G a u s s ~\mathrm{Gauss}~ Gauss
◯ ∫ ∫ S P d y d z + Q d z d x + R d x d y = ∭ V ( ∂ P ∂ x + ∂ Q ∂ y + ∂ R ∂ z ) d x d y d z \bigcirc\!\!\!\!\!\!\!\!\int\!\!\!\!\!\int_S
{P~\mathrm dy\mathrm dz + Q~\mathrm dz\mathrm dx + R~\mathrm dx\mathrm dy} = \iiint_V\left({\frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}}\right)~\mathrm dx\mathrm dy \mathrm dz
◯ ∫ ∫ S P d y d z + Q d z d x + R d x d y = ∭ V ( ∂ x ∂ P + ∂ y ∂ Q + ∂ z ∂ R ) d x d y d z
或
◯ ∫ ∫ S v ⋅ d S = ∭ V ∇ ⋅ v d V = ∭ V d i v v d V \bigcirc\!\!\!\!\!\!\!\!\int\!\!\!\!\!\int_S{\boldsymbol v\cdot\mathrm d\boldsymbol S} = \iiint_V\nabla\cdot\boldsymbol v\mathrm dV = \iiint_V\mathrm{div}~\boldsymbol v~\mathrm dV
◯ ∫ ∫ S v ⋅ d S = ∭ V ∇ ⋅ v d V = ∭ V div v d V
其中 S ~S~ S V ~V~ V S = ∂ V ~S = \partial V~ S = ∂ V V ~V~ V
G a u s s ~\mathrm{Gauss}~ Gauss
设 v = P ( x , y , z ) i + Q ( x , y , z ) j + R ( x , y , z ) k ~\boldsymbol v = P(x,y,z)\boldsymbol i + Q(x,y,z)\boldsymbol j + R(x,y,z)\boldsymbol k~ v = P ( x , y , z ) i + Q ( x , y , z ) j + R ( x , y , z ) k V ~V~ V S ~S~ S L ~L~ L L ~L~ L S ~S~ S L = ∂ S ~L = \partial S~ L = ∂ S S t o c k s ~\mathrm{Stocks}~ Stocks
∮ L P d x + Q d y + R d z = ∬ S ( ∂ R ∂ y − ∂ Q ∂ z ) d y d z + ( ∂ P ∂ z − ∂ R ∂ x ) d z d x + ( ∂ Q ∂ x − ∂ P ∂ y ) d x d y \begin{aligned}
&\oint_L P~\mathrm dx + Q~\mathrm dy + R~\mathrm dz \\
&= \iint_S\left({\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}}\right)~\mathrm dy\mathrm dz + \left({\frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}}\right)~\mathrm dz\mathrm dx + \left({\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}}\right)~\mathrm dx\mathrm dy
\end{aligned}
∮ L P d x + Q d y + R d z = ∬ S ( ∂ y ∂ R − ∂ z ∂ Q ) d y d z + ( ∂ z ∂ P − ∂ x ∂ R ) d z d x + ( ∂ x ∂ Q − ∂ y ∂ P ) d x d y
或
∮ L v ⋅ d r = ∬ S r o t v ⋅ d S = ∬ S ∇ × v ⋅ d S \oint_L\boldsymbol v\cdot \mathrm d\boldsymbol r = \iint_S\boldsymbol{\mathrm{rot}}~\boldsymbol v\cdot \mathrm d\boldsymbol S = \iint_S \nabla\times \boldsymbol v\cdot \mathrm d\boldsymbol S
∮ L v ⋅ d r = ∬ S rot v ⋅ d S = ∬ S ∇ × v ⋅ d S
其中 L ~L~ L S ~S~ S L ~L~ L S ~S~ S
S t o c k s ~\mathrm{Stocks}~ Stocks S t o c k s ~\mathrm{Stocks}~ Stocks G r e e n ~\mathrm{Green}~ Green G r e e n ~\mathrm{Green}~ Green S t o c k s ~\mathrm{Stocks}~ Stocks N e w t o n − L e i b n i z ~\mathrm{Newton-Leibniz}~ Newton − Leibniz
∫ a b f ( x ) d x = F ( b ) − F ( a ) , F ′ ( x ) = f ( x ) \int_a^b f(x)~\mathrm dx = F(b) - F(a),\quad F'(x) = f(x)
∫ a b f ( x ) d x = F ( b ) − F ( a ) , F ′ ( x ) = f ( x )
的右边看成零维的“积分”（即在 [ a , b ] ~[a,b]~ [ a , b ] ∂ [ a , b ] = { a , b } ~\partial [a,b] = \{a,b\}~ ∂ [ a , b ] = { a , b } F ( a ) ~F(a)~ F ( a ) G r e e n ~\mathrm{Green}~ Green G a u s s ~\mathrm{Gauss}~ Gauss S t o c k s ~\mathrm{Stocks}~ Stocks N e w t o n − L e i b n i z ~\mathrm{Newton-Leibniz}~ Newton − Leibniz
其他形式的曲线、曲面积分
本节将会涉及以下几种形式的曲线曲面积分
∫ L φ d r = ∫ L φ τ d s ∫ L d r × v = ∫ L τ × v d s ∬ S φ d S = ∬ S φ n d S ∬ S d S × v = ∬ S n × v d S \begin{aligned}
\int_L \varphi~\mathrm{d}\boldsymbol r
&=
\int_L \varphi\boldsymbol \tau~\mathrm{d}s \\
\int_L \mathrm{d}\boldsymbol r \times \boldsymbol v
&=
\int_L \boldsymbol\tau\times\boldsymbol v~\mathrm{d}s \\
\iint_S \varphi~\mathrm{d}\boldsymbol S
&=
\iint_S \varphi\boldsymbol n~\mathrm{d}S \\
\iint_S \mathrm{d}\boldsymbol S\times \boldsymbol v
&=
\iint_S \boldsymbol n\times\boldsymbol v~\mathrm{d}S
\end{aligned}
∫ L φ d r ∫ L d r × v ∬ S φ d S ∬ S d S × v = ∫ L φ τ d s = ∫ L τ × v d s = ∬ S φ n d S = ∬ S n × v d S
其中 τ ~\boldsymbol\tau~ τ n ~\boldsymbol n~ n S ~S~ S G a u s s ~\mathrm{Gauss}~ Gauss S t o c k s ~\mathrm{Stocks}~ Stocks
设 S ~S~ S S ~S~ S ∂ S ~\partial S~ ∂ S
∮ ∂ S φ d r = ∬ S d S × ∇ φ ∮ ∂ S d r × v = ∬ S ( d S × ∇ ) × v \begin{aligned}
\oint_{\partial S}\varphi~\mathrm{d}\boldsymbol r &= \iint_S\mathrm{d}\boldsymbol S\times\nabla\varphi \\
\oint_{\partial S}\mathrm{d}\boldsymbol r\times\boldsymbol v &= \iint_S(\mathrm{d}\boldsymbol S\times\nabla)\times\boldsymbol v
\end{aligned}
∮ ∂ S φ d r ∮ ∂ S d r × v = ∬ S d S × ∇ φ = ∬ S ( d S × ∇ ) × v
设 V ~V~ V V ~V~ V ∂ V ~\partial V~ ∂ V
◯ ∫ ∫ ∂ V φ d S = ∭ V ∇ φ d V ◯ ∫ ∫ ∂ V d S × v = ∭ V ∇ × v d V \begin{aligned}
\bigcirc\!\!\!\!\!\!\!\!\int\!\!\!\!\!\int_{\partial V}\varphi~\mathrm{d}\boldsymbol S &= \iiint_V\nabla\varphi~\mathrm{d}V \\
\bigcirc\!\!\!\!\!\!\!\!\int\!\!\!\!\!\int_{\partial V}\mathrm{d}\boldsymbol S\times\boldsymbol v &= \iiint_V\nabla\times\boldsymbol v~\mathrm{d}V
\end{aligned}
◯ ∫ ∫ ∂ V φ d S ◯ ∫ ∫ ∂ V d S × v = ∭ V ∇ φ d V = ∭ V ∇ × v d V
根据以上两个定理，我们可以给出梯度、散度和旋度的积分表示。设 φ ~\varphi~ φ v ~\boldsymbol v~ v P ~P~ P
∇ φ = lim V → P [ 1 σ ( V ) ◯ ∫ ∫ S φ d S ] ∇ ⋅ v = lim V → P [ 1 σ ( V ) ◯ ∫ ∫ S v ⋅ d S ] ∇ × v = lim V → P [ 1 σ ( V ) ◯ ∫ ∫ S d S × v ] \begin{aligned}
\nabla\varphi &= \lim_{V \to P}\left[\frac{1}{\sigma(V)}\bigcirc\!\!\!\!\!\!\!\!\int\!\!\!\!\!\int_S\varphi~\mathrm{d}\boldsymbol S\right] \\[9pt]
\nabla\cdot\boldsymbol v &= \lim_{V \to P}\left[\frac{1}{\sigma(V)}\bigcirc\!\!\!\!\!\!\!\!\int\!\!\!\!\!\int_S\boldsymbol v\cdot\mathrm{d}\boldsymbol S\right] \\[9pt]
\nabla\times\boldsymbol v &= \lim_{V \to P}\left[\frac{1}{\sigma(V)}\bigcirc\!\!\!\!\!\!\!\!\int\!\!\!\!\!\int_S\mathrm{d}\boldsymbol S\times\boldsymbol v\right]
\end{aligned}
∇ φ ∇ ⋅ v ∇ × v = V → P lim [ σ ( V ) 1 ◯ ∫ ∫ S φ d S ] = V → P lim [ σ ( V ) 1 ◯ ∫ ∫ S v ⋅ d S ] = V → P lim [ σ ( V ) 1 ◯ ∫ ∫ S d S × v ]
极限 V → P ~V \to P~ V → P V ~V~ V P ~P~ P σ ( V ) ~{\sigma(V)}~ σ ( V ) V ~V~ V S = ∂ V ~\boldsymbol S = \partial V~ S = ∂ V V ~V~ V
曲线坐标系中的矢量分析
我们把一个无限小的位移矢量写为
d l = f d u u ^ + g d v v ^ + h d w w ^ \mathrm{d}\boldsymbol{l} = f\mathrm{d}u\hat{\boldsymbol{u}} + g\mathrm{d}v\hat{\boldsymbol{v}} + h\mathrm{d}w\hat{\boldsymbol{w}}
d l = f d u u ^ + g d v v ^ + h d w w ^
其中 f , g , h ~f, g, h~ f , g , h f = g = h = 1 ~f = g = h = 1~ f = g = h = 1 f = 1 , g = r , h = r sin θ ~f = 1, g = r, h = r\sin\theta~ f = 1 , g = r , h = r sin θ ( u , v , w ) ~(u, v, w)~ ( u , v , w ) ( u + d u , v + d v , w + d w ) ~(u + \mathrm{d}u, v + \mathrm{d}v, w + \mathrm{d}w)~ ( u + d u , v + d v , w + d w ) t ( u , v , w ) ~t(u, v, w)~ t ( u , v , w )
d t = ∂ t ∂ u d u + ∂ t ∂ v d v + ∂ t ∂ w d w \mathrm{d}t = \frac{\partial t}{\partial u}\mathrm{d}u + \frac{\partial t}{\partial v}\mathrm{d}v + \frac{\partial t}{\partial w}\mathrm{d}w
d t = ∂ u ∂ t d u + ∂ v ∂ t d v + ∂ w ∂ t d w
另一方面，我们也可以把它写成点积的形式，即
d t = ∇ t ⋅ d l = ( ∇ t ) u f d u + ( ∇ t ) v g d v + ( ∇ t ) w h d w \mathrm{d}t = \nabla t \cdot \mathrm{d}\boldsymbol{l} = (\nabla t)_u f \mathrm{d}u + (\nabla t)_v g \mathrm{d}v + (\nabla t)_w h \mathrm{d}w
d t = ∇ t ⋅ d l = ( ∇ t ) u f d u + ( ∇ t ) v g d v + ( ∇ t ) w h d w
对比可知
( ∇ t ) u f = 1 f ∂ t ∂ u ( ∇ t ) v g = 1 g ∂ t ∂ v ( ∇ t ) w h = 1 h ∂ t ∂ w (\nabla t)_u f = \frac{1}{f} \frac{\partial t}{\partial u}
(\nabla t)_v g = \frac{1}{g} \frac{\partial t}{\partial v}
(\nabla t)_w h = \frac{1}{h} \frac{\partial t}{\partial w}
( ∇ t ) u f = f 1 ∂ u ∂ t ( ∇ t ) v g = g 1 ∂ v ∂ t ( ∇ t ) w h = h 1 ∂ w ∂ t
故
∇ t = 1 f ∂ t ∂ u u ^ + 1 g ∂ t ∂ v v ^ + 1 h ∂ t ∂ w w ^ \nabla t = \frac{1}{f} \frac{\partial t}{\partial u}\hat{\boldsymbol{u}} + \frac{1}{g} \frac{\partial t}{\partial v}\hat{\boldsymbol{v}} + \frac{1}{h} \frac{\partial t}{\partial w}\hat{\boldsymbol{w}}
∇ t = f 1 ∂ u ∂ t u ^ + g 1 ∂ v ∂ t v ^ + h 1 ∂ w ∂ t w ^
接下来是梯度、旋度以及拉普拉斯算子。
A ( u , v , w ) = A u u ^ + A v v ^ + A w w ^ \boldsymbol{A}(u, v, w) = A_u\hat{u} + A_v\hat{v} + A_w\hat{w}
A ( u , v , w ) = A u u ^ + A v v ^ + A w w ^
无限小的体积元
d τ = d l u d l v d l w = ( f g h ) d u d v d w \mathrm{d}\tau = \mathrm{d}l_u \mathrm{d}l_v \mathrm{d}l_w = (fgh)\mathrm{d}u\mathrm{d}v\mathrm{d}w
d τ = d l u d l v d l w = ( f g h ) d u d v d w
散度为
∇ ⋅ A = 1 f g h [ ∂ ∂ u ( g h A u ) + ∂ ∂ v ( f h A v ) + ∂ ∂ w ( f g A w ) ] \nabla \cdot \boldsymbol{A} = \frac{1}{fgh}\left[\frac{\partial}{\partial u}(gh A_u) + \frac{\partial}{\partial v}(fh A_v) + \frac{\partial}{\partial w}(fg A_w)\right]
∇ ⋅ A = f g h 1 [ ∂ u ∂ ( g h A u ) + ∂ v ∂ ( f h A v ) + ∂ w ∂ ( f g A w ) ]
无限小的面积元（以与 u v ~uv~ uv
d a = ( f g ) d u d v w ^ \mathrm{d}\boldsymbol{a} = (fg)\mathrm{d}u\mathrm{d}v\hat{\boldsymbol{w}}
d a = ( f g ) d u d v w ^
散度为
∇ × A = 1 g h [ ∂ ∂ v ( h A w ) − ∂ ∂ w ( g A v ) ] u ^ + 1 f h [ ∂ ∂ w ( f A u ) − ∂ ∂ u ( h A w ) ] v ^ + 1 f g [ ∂ ∂ u ( g A v ) − ∂ ∂ v ( f A u ) ] w ^ \nabla\times\boldsymbol{A} = \frac{1}{gh}\left[ \frac{\partial}{\partial v}(h A_w) - \frac{\partial}{\partial w}(g A_v) \right]\hat{\boldsymbol{u}} + \frac{1}{fh}\left[ \frac{\partial}{\partial w}(f A_u) - \frac{\partial}{\partial u}(h A_w) \right]\hat{\boldsymbol{v}} + \frac{1}{fg}\left[ \frac{\partial}{\partial u}(g A_v) - \frac{\partial}{\partial v}(f A_u) \right]\hat{\boldsymbol{w}}
∇ × A = g h 1 [ ∂ v ∂ ( h A w ) − ∂ w ∂ ( g A v ) ] u ^ + f h 1 [ ∂ w ∂ ( f A u ) − ∂ u ∂ ( h A w ) ] v ^ + f g 1 [ ∂ u ∂ ( g A v ) − ∂ v ∂ ( f A u ) ] w ^
由上面的结论可以写出拉普拉斯算子作用的结果
∇ 2 t = 1 f g h [ ∂ ∂ u ( g h f ∂ t ∂ u ) + ∂ ∂ v ( f h g ∂ t ∂ v ) + ∂ ∂ w ( f g h ∂ t ∂ w ) ] \nabla^2 t = \frac{1}{fgh}\left[ \frac{\partial}{\partial u}\left( \frac{gh}{f}\frac{\partial t}{\partial u} \right) + \frac{\partial}{\partial v}\left( \frac{fh}{g}\frac{\partial t}{\partial v} \right) + \frac{\partial}{\partial w}\left( \frac{fg}{h}\frac{\partial t}{\partial w} \right) \right]
∇ 2 t = f g h 1 [ ∂ u ∂ ( f g h ∂ u ∂ t ) + ∂ v ∂ ( g f h ∂ v ∂ t ) + ∂ w ∂ ( h f g ∂ w ∂ t ) ]
矢量恒等式
最后本博文记录一些矢量恒等式，方便随时查阅。 A ⋅ ( B × C ) = B ⋅ ( C × A ) = C ⋅ ( A × B ) ~\boldsymbol A \cdot (\boldsymbol B \times \boldsymbol C) = \boldsymbol B \cdot (\boldsymbol C \times \boldsymbol A) = \boldsymbol C \cdot (\boldsymbol A \times \boldsymbol B)~ A ⋅ ( B × C ) = B ⋅ ( C × A ) = C ⋅ ( A × B )
A × ( B × C ) = B ( A ⋅ C ) − C ( A ⋅ B ) ~\boldsymbol A \times (\boldsymbol B \times \boldsymbol C) = \boldsymbol B (\boldsymbol A \cdot \boldsymbol C) - \boldsymbol C (\boldsymbol A \cdot \boldsymbol B)~ A × ( B × C ) = B ( A ⋅ C ) − C ( A ⋅ B )
积规则
∇ ( f g ) = f ( ∇ g ) + g ( ∇ f ) ~\nabla(fg) = f(\nabla g) + g(\nabla f)~ ∇ ( f g ) = f ( ∇ g ) + g ( ∇ f )
∇ ( A ⋅ B ) = A × ( × B ) + B × ( ∇ × A ) + ( A ⋅ ∇ ) B + ( B ⋅ ∇ ) A ~\nabla(\boldsymbol A \cdot \boldsymbol B) = \boldsymbol A \times (\boldsymbol \times B) + \boldsymbol B \times (\nabla \times \boldsymbol A) + (\boldsymbol A \cdot \nabla)\boldsymbol B + (\boldsymbol B \cdot \nabla)\boldsymbol A~ ∇ ( A ⋅ B ) = A × ( × B ) + B × ( ∇ × A ) + ( A ⋅ ∇ ) B + ( B ⋅ ∇ ) A
∇ ⋅ ( f A ) = f ( ∇ ⋅ A ) + A ⋅ ( ∇ f ) ~\nabla \cdot (f\boldsymbol A) = f(\nabla \cdot \boldsymbol A) + \boldsymbol A \cdot (\nabla f)~ ∇ ⋅ ( f A ) = f ( ∇ ⋅ A ) + A ⋅ ( ∇ f )
∇ ⋅ ( A × B ) = B ⋅ ( ∇ × A ) − A ⋅ ( ∇ × B ) ~\nabla \cdot (\boldsymbol A \times \boldsymbol B) = \boldsymbol B \cdot (\nabla \times \boldsymbol A) - \boldsymbol A \cdot (\nabla \times \boldsymbol B)~ ∇ ⋅ ( A × B ) = B ⋅ ( ∇ × A ) − A ⋅ ( ∇ × B )
∇ × ( f A ) = f ( ∇ × A ) − A × ( ∇ f ) ~\nabla \times (f\boldsymbol A) = f(\nabla \times \boldsymbol A) - \boldsymbol A \times (\nabla f)~ ∇ × ( f A ) = f ( ∇ × A ) − A × ( ∇ f )
∇ × ( A × B ) = ( B ⋅ ∇ ) A − ( A ⋅ ∇ ) B + A ( ∇ ⋅ B ) − B ( ∇ ⋅ A ) ~\nabla \times (\boldsymbol A \times \boldsymbol B) = (\boldsymbol B \cdot \nabla)\boldsymbol A - (\boldsymbol A \cdot \nabla)\boldsymbol B + \boldsymbol A(\nabla \cdot \boldsymbol B) - \boldsymbol B (\nabla \cdot \boldsymbol A)~ ∇ × ( A × B ) = ( B ⋅ ∇ ) A − ( A ⋅ ∇ ) B + A ( ∇ ⋅ B ) − B ( ∇ ⋅ A )
二阶导数
∇ ⋅ ( ∇ × A ) = 0 ~\nabla \cdot(\nabla \times \boldsymbol A) = 0~ ∇ ⋅ ( ∇ × A ) = 0
∇ × ( ∇ f ) = 0 ~\nabla \times (\nabla f) = 0~ ∇ × ( ∇ f ) = 0
∇ × ( ∇ × A ) = ∇ ( ∇ ⋅ A ) − ∇ 2 A ~\nabla \times (\nabla \times \boldsymbol A) = \nabla (\nabla \cdot \boldsymbol A) - \nabla^2 \boldsymbol A~ ∇ × ( ∇ × A ) = ∇ ( ∇ ⋅ A ) − ∇ 2 A
