本文介绍了流体力学中张量的基础知识，包括场论的概念，如标量场和矢量场的定义，以及梯度、散度和旋度的计算。讨论了无源场、位势场和无旋场的特性，并阐述了高斯定理和斯托克斯定理。此外，还涵盖了张量表示法，包括梯度、散度、旋度和拉普拉斯算子在曲线坐标系中的表达式。内容深入浅出，适合研究生学习流体课程时作为参考。
研究生的流体课程都需要张量
看了吴望一的流体力学、吴介之的Vortical Flows和Stephen B. Pope的Turbulence Flows，感觉Turbulence Flows的附录A、B介绍的最好，应该足够了
吴望一：流体力学（上）：第一章：场论 笔记
吴望一，流体力学
第一章 场论和张量初步
1.1 场的定义及分类
- 标量场： ϕ = ϕ ( r , t ) = ϕ ( x , y , z , t ) \phi=\phi(\mathbf{r},t)=\phi(x,y,z,t) ϕ=ϕ(r,t)=ϕ(x,y,z,t)
- 矢量场： a = a ( r , t ) = a ( x , y , z , t ) \mathbf{a}=\mathbf{a}(\mathbf{r},t)=\mathbf{a}(x,y,z,t) a=a(r,t)=a(x,y,z,t)
- 均匀场： ϕ ( t ) , a ( t ) \phi(t),\mathbf{a}(t) ϕ(t),a(t)
- 定常场： ϕ ( r ) , a ( r ) \phi(\mathbf{r}),\mathbf{a}(\mathbf{r}) ϕ(r),a(r)
1.2 场的几何表示
- 标量场，等位面，
ϕ
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t
0
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=
c
o
n
s
t
=
ϕ
0
\phi(\mathbf{r},t_0)=\mathrm{const}=\phi_0
ϕ(r,t0)=const=ϕ0
等位面相互位置、疏密程度看标量函数的变化状况，法向方向变化 - 矢量场，矢量线，线上每一点切线方向与该点的矢量方向重合，矢量线的疏密程度估计矢量在各点大小
a × d r = 0 d x a x ( x , y , z , t ) = d y a y ( x , y , z , t ) = d z a z ( x , y , z , t ) \mathbf{a}\times d\mathbf{r}=0\qquad\dfrac{dx}{a_x(x,y,z,t)}=\dfrac{dy}{a_y(x,y,z,t)}=\dfrac{dz}{a_z(x,y,z,t)} a×dr=0ax(x,y,z,t)dx=ay(x,y,z,t)dy=az(x,y,z,t)dz
1.3 梯度——标量场不均匀性的量度
- 求等位线方向导数 ∂ φ ∂ n \dfrac{\partial\varphi}{\partial\mathbf{n}} ∂n∂φ，其他方向的方向导数： ∂ φ ∂ s = ∂ φ ∂ n cos ( n , s ) \dfrac{\partial\varphi}{\partial\mathbf{s}}=\dfrac{\partial\varphi}{\partial\mathbf{n}}\cos(\mathbf{n},\mathbf{s}) ∂s∂φ=∂n∂φcos(n,s)，投影
- 梯度： g r a d φ = ∂ φ ∂ x i + ∂ φ ∂ y j + ∂ φ ∂ z k = ∇ φ \mathrm{grad}\varphi=\dfrac{\partial\varphi}{\partial x}\mathbf{i}+\dfrac{\partial\varphi}{\partial y}\mathbf{j}+\dfrac{\partial\varphi}{\partial z}\mathbf{k}=\nabla\varphi gradφ=∂x∂φi+∂y∂φj+∂z∂φk=∇φ
1.4 矢量a通过S面的通量-矢量a的散度-高斯定理
- 散度： d i v a = ∂ a x ∂ x + ∂ a y ∂ y + ∂ a z ∂ z = ∇ ⋅ a \mathrm{div}\mathbf{a}=\dfrac{\partial a_x}{\partial x}+\dfrac{\partial a_y}{\partial y}+\dfrac{\partial a_z}{\partial z}=\nabla\cdot\mathbf{a} diva=∂x∂ax+∂y∂ay+∂z∂az=∇⋅a
- Gauss
∫ ∂ V n ⋅ f d S = ∫ V ∇ ⋅ f d V \int_{\partial V}\mathbf{n}\cdot\mathbf{f}\,dS=\int_V \nabla\cdot\mathbf{f}\,dV ∫∂Vn⋅fdS=∫V∇⋅fdV
1.5 无源场及其性质
1.6 矢量a沿回线的环量-矢量a的旋度-斯托克斯定理
- Stokes
∮ C f ⋅ d x = ∫ S n ⋅ ( ∇ × f ) d S = ∫ S ( n × ∇ ) ⋅ f d S \oint_C \mathbf{f}\cdot d\mathbf{x}=\int_S\mathbf{n}\cdot(\nabla\times\mathbf{f})dS=\int_S(\mathbf{n}\times\nabla)\cdot\mathbf{f}dS ∮Cf⋅dx=∫Sn⋅(∇×f)dS=∫S(n×∇)⋅fdS - 旋度： r o t a = ∇ × a \mathrm{rot}\mathbf{a}=\nabla\times\mathbf{a} rota=∇×a
1.7 无旋场及其性质：位势场
1.8 基本运算公式
- ∇ F ( φ ) = F ′ ( φ ) ∇ φ ∇ φ ( r ) = φ ′ ( r ) r r \nabla F(\varphi)=F'(\varphi)\nabla\varphi\qquad\nabla\varphi(\mathbf{r})=\varphi'(\mathbf{r})\dfrac{\mathbf{r}}{r} ∇F(φ)=F′(φ)∇φ∇φ(r)=φ′(r)rr
- ∇ ⋅ ( φ a ) = φ ∇ ⋅ a + ∇ φ ⋅ a \nabla\cdot(\varphi\mathbf{a})=\varphi\nabla\cdot\mathbf{a}+\nabla\varphi\cdot\mathbf{a} ∇⋅(φa)=φ∇⋅a+∇φ⋅a
- ∇ ⋅ ( a × b ) = b ⋅ ( ∇ × a ) − a ⋅ ( ∇ × b ) \nabla\cdot(\mathbf{a}\times\mathbf{b})=\mathbf{b}\cdot(\nabla\times\mathbf{a})-\mathbf{a}\cdot(\nabla\times\mathbf{b}) ∇⋅(a×b)=b⋅(∇×a)−a⋅(∇×b)
- ∇ × ( φ a ) = φ ∇ × a + ∇ φ × a \nabla\times(\varphi\mathbf{a})=\varphi\nabla\times\mathbf{a}+\nabla\varphi\times\mathbf{a} ∇×(φa)=φ∇×a+∇φ×a
- ∇ ( a ⋅ b ) = ( b ⋅ ∇ ) a + ( a ⋅ ∇ ) b + b × ( ∇ × a ) + a × ( ∇ × b ) \nabla(\mathbf{a}\cdot\mathbf{b})=(\mathbf{b}\cdot\nabla)\mathbf{a}+(\mathbf{a}\cdot\nabla)\mathbf{b}+\mathbf{b}\times(\nabla\times\mathbf{a})+\mathbf{a}\times(\nabla\times\mathbf{b}) ∇(a⋅b)=(b⋅∇)a+(a⋅∇)b+b×(∇×a)+a×(∇×b)
- a × ( b × c ) = b ( a ⋅ c ) − c ( a ⋅ b ) \mathbf{a}\times(\mathbf{b}\times\mathbf{c})=\mathbf{b}(\mathbf{a}\cdot\mathbf{c})-\mathbf{c}(\mathbf{a}\cdot\mathbf{b}) a×(b×c)=b(a⋅c)−c(a⋅b)
1.9 哈密顿算子
∇ = i ∂ ∂ x + j ∂ ∂ y + k ∂ ∂ z \nabla=\mathbf{i}\dfrac{\partial}{\partial x}+\mathbf{j}\dfrac{\partial}{\partial y}+\mathbf{k}\dfrac{\partial}{\partial z} ∇=i∂x∂+j∂y∂+k∂z∂
1.10张量表示法
- 约定求和法则
a i b i = a 1 b 1 + a 2 b 2 + a 3 b 3 a_ib_i=a_1b_1+a_2b_2+a_3b_3 aibi=a1b1+a2b2+a3b3
∂ a i ∂ x i = ∂ a 1 ∂ x 1 + ∂ a 2 ∂ x 2 + ∂ a 3 ∂ x 3 = ∇ ⋅ a \dfrac{\partial a_i}{\partial x_i}=\dfrac{\partial a_1}{\partial x_1}+\dfrac{\partial a_2}{\partial x_2}+\dfrac{\partial a_3}{\partial x_3}=\nabla\cdot\mathbf{a} ∂xi∂ai=∂x1∂a1+∂x2∂a2+∂x3∂a3=∇⋅a
( a ⋅ ∇ ) b = a j ∂ b j ∂ x j (\mathbf{a}\cdot\nabla)\mathbf{b}=a_j\dfrac{\partial b_j}{\partial x_j} (a⋅∇)b=aj∂xj∂bj
Δ a = ∇ 2 a = ∇ ⋅ ∇ a = ∂ ∂ x i ( ∂ a j ∂ x i ) = ∂ 2 a j ∂ x i ∂ x i \Delta\mathbf{a}=\nabla^2\mathbf{a}=\nabla\cdot\nabla\mathbf{a}=\dfrac{\partial}{\partial x_i}\Big(\dfrac{\partial a_j}{\partial x_i}\Big)=\dfrac{\partial^2a_j}{\partial x_i\partial x_i} Δa=∇2a=∇⋅∇a=∂xi∂(∂xi∂aj)=∂xi∂xi∂2aj - Kronecker
δ
\delta
δ: a replacement operator
δ i j = { 0 , i ≠ j 1 , i = j \delta_{ij}=\begin{cases} 0, \quad i\neq j\\ 1,\quad i=j \end{cases} δij={0,i=j1,i=j - 置换符号
ε
i
j
k
\varepsilon_{ijk}
εijk
ε = { 0 , more than two indicators of i, j, k are the same 1 , i, j, k are even permutations − 1 , i, j, k are odd permutations \varepsilon=\begin{cases} 0,\quad\text{more than two indicators of i, j, k are the same}\\ 1,\quad\text{i, j, k are even permutations}\\ -1,\quad\text{i, j, k are odd permutations} \end{cases} ε=⎩⎪⎨⎪⎧0,more than two indicators of i, j, k are the same1,i, j, k are even permutations−1,i, j, k are odd permutations
a × b = ε i j k a j b k ∇ × a = ε i j k ∂ a k ∂ x j \mathbf{a}\times\mathbf{b}=\varepsilon_{ijk}a_jb_k\qquad\nabla\times\mathbf{a}=\varepsilon_{ijk}\dfrac{\partial a_k}{\partial x_j} a×b=εijkajbk∇×a=εijk∂xj∂ak
Δ = ∣ a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ∣ = ε i j k a i 1 a j 2 a k 3 \Delta=\begin{vmatrix} a_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23}\\ a_{31}&a_{32}&a_{33}\\ \end{vmatrix}=\varepsilon_{ijk}a_{i1}a_{j2}a_{k3} Δ=∣∣∣∣∣∣a11a21a31a12a22a32a13a23a33∣∣∣∣∣∣=εijkai1aj2ak3 -
ε
−
δ
\varepsilon - \delta
ε−δ恒等式
ε i j k ε i s t = δ j s δ k t − δ j t δ k s \varepsilon_{ijk}\varepsilon_{ist}=\delta_{js}\delta_{kt}-\delta_{jt}\delta_{ks} εijkεist=δjsδkt−δjtδks - 证明
- ∇ ⋅ ( φ a ) = ∂ φ a i ∂ x i = φ ∂ a i ∂ x i + a i ∂ φ ∂ x i = φ ∇ ⋅ a + ∇ φ ⋅ a \nabla\cdot(\varphi\mathbf{a})=\dfrac{\partial\varphi a_i}{\partial x_i}=\varphi\dfrac{\partial a_i}{\partial x_i}+a_i\dfrac{\partial\varphi}{\partial x_i}=\varphi\nabla\cdot\mathbf{a}+\nabla\varphi\cdot\mathbf{a} ∇⋅(φa)=∂xi∂φai=φ∂xi∂ai+ai∂xi∂φ=φ∇⋅a+∇φ⋅a
- ∇ ⋅ ( a × b ) = ∂ ∂ x i ( ε i j k a j b k ) = ε i j k ∂ a j ∂ x i b k + ε i j k a j ∂ b k ∂ x i = ε i j k ∂ a j ∂ x i b k − ε j i k a j ∂ b k ∂ x i = b ⋅ ( ∇ × a ) − a ⋅ ( ∇ × b ) \nabla\cdot(\mathbf{a}\times\mathbf{b})=\dfrac{\partial}{\partial x_i}(\varepsilon_{ijk}a_jb_k)=\varepsilon_{ijk}\dfrac{\partial a_j}{\partial x_i}b_k+\varepsilon_{ijk}a_j\dfrac{\partial b_k}{\partial x_i}\\=\varepsilon_{ijk}\dfrac{\partial a_j}{\partial x_i}b_k-\varepsilon_{jik}a_j\dfrac{\partial b_k}{\partial x_i}=\mathbf{b}\cdot(\nabla\times\mathbf{a})-\mathbf{a}\cdot(\nabla\times\mathbf{b}) ∇⋅(a×b)=∂xi∂(εijkajbk)=εijk∂xi∂ajbk+εijkaj∂xi∂bk=εijk∂xi∂ajbk−εjikaj∂xi∂bk=b⋅(∇×a)−a⋅(∇×b)
- ∇ × ( φ a ) = ε i j k ∂ ( φ a k ) ∂ x j = ε i j k φ ∂ a k ∂ x j + ε i j k ∂ φ ∂ x j a k = φ ∇ × a + ( ∇ φ ) × a \nabla\times(\varphi\mathbf{a})=\varepsilon_{ijk}\dfrac{\partial(\varphi a_k)}{\partial x_j}=\varepsilon_{ijk}\varphi\dfrac{\partial a_k}{\partial x_j}+\varepsilon_{ijk}\dfrac{\partial\varphi}{\partial x_j}a_k=\\\varphi\nabla\times\mathbf{a}+(\nabla\varphi)\times\mathbf{a} ∇×(φa)=εijk∂xj∂(φak)=εijkφ∂xj∂ak+εijk∂xj∂φak=φ∇×a+(∇φ)×a
- ∇ × ( a × b ) = ε i j k ∂ ( ε k l m a l b m ) ∂ x j = ε i j k ε k l m ∂ a l b m ∂ x j = ( δ i l δ j m − δ i m δ j l ) ( a l ∂ b m ∂ x j + b m ∂ a l ∂ x j ) = a i ∂ b j ∂ x j − a j ∂ b i ∂ x j + b j ∂ a i ∂ x j − b i ∂ a j ∂ x j = a ( ∇ ⋅ b ) − ( a ⋅ ∇ ) b + ( b ⋅ ∇ ) a − b ( ∇ ⋅ a ) \nabla\times(\mathbf{a}\times\mathbf{b})=\varepsilon_{ijk}\dfrac{\partial(\varepsilon_{klm}a_lb_m)}{\partial x_j}=\varepsilon_{ijk}\varepsilon_{klm}\dfrac{\partial a_lb_m}{\partial x_j}\\=(\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl})\Big(a_l\dfrac{\partial b_m}{\partial x_j}+b_m\dfrac{\partial a_l}{\partial x_j}\Big)\\=a_i\dfrac{\partial b_j}{\partial x_j}-a_j\dfrac{\partial b_i}{\partial x_j}+b_j\dfrac{\partial a_i}{\partial x_j}-b_i\dfrac{\partial a_j}{\partial x_j}\\=\mathbf{a}(\nabla\cdot\mathbf{b})-(\mathbf{a}\cdot\nabla)\mathbf{b}+(\mathbf{b}\cdot\nabla)\mathbf{a}-\mathbf{b}(\nabla\cdot\mathbf{a}) ∇×(a×b)=εijk∂xj∂(εklmalbm)=εijkεklm∂xj∂albm=(δilδjm−δimδjl)(al∂xj∂bm+bm∂xj∂al)=ai∂xj∂bj−aj∂xj∂bi+bj∂xj∂ai−bi∂xj∂aj=a(∇⋅b)−(a⋅∇)b+(b⋅∇)a−b(∇⋅a)
- ( b ⋅ ∇ ) a + ( a ⋅ ∇ ) b + b × ( ∇ × a ) + a × ( ∇ × b ) = b j ∂ a i ∂ x j + a j ∂ b i ∂ x j + ε i j k b j ε k l m ∂ a m ∂ x l + ε i j k a j ε k l m ∂ b m ∂ x l = b j ∂ a i ∂ x j + a j ∂ b i ∂ x j + ( δ i l δ j m − δ i m δ j l ) ( b j ∂ a m ∂ x l + a j ∂ b m ∂ x l ) = b j ∂ a i ∂ x j + a j ∂ b i ∂ x j + b j ∂ a j ∂ x i − b j ∂ a i ∂ x j + a j ∂ b j ∂ x i − a j ∂ b i ∂ x j = b j a j x i + a j ∂ b j ∂ x i = ∂ a j b j ∂ x i = ∇ ( a ⋅ b ) (\mathbf{b}\cdot\nabla)\mathbf{a}+(\mathbf{a}\cdot\nabla)\mathbf{b}+\mathbf{b}\times(\nabla\times\mathbf{a})+\mathbf{a}\times(\nabla\times\mathbf{b})\\=b_j\dfrac{\partial a_i}{\partial x_j}+a_j\dfrac{\partial b_i}{\partial x_j}+\varepsilon_{ijk}b_j\varepsilon_{klm}\dfrac{\partial a_m}{\partial x_l}+\varepsilon_{ijk}a_j\varepsilon_{klm}\dfrac{\partial b_m}{\partial x_l}\\=b_j\dfrac{\partial a_i}{\partial x_j}+a_j\dfrac{\partial b_i}{\partial x_j}+(\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl})\Big(b_j\dfrac{\partial a_m}{\partial x_l}+a_j\dfrac{\partial b_m}{\partial x_l}\Big)\\=b_j\dfrac{\partial a_i}{\partial x_j}+a_j\dfrac{\partial b_i}{\partial x_j}+b_j\dfrac{\partial a_j}{\partial x_i}-b_j\dfrac{\partial a_i}{\partial x_j}+a_j\dfrac{\partial b_j}{\partial x_i}-a_j\dfrac{\partial b_i}{\partial x_j}\\=b_j\dfrac{a_j}{x_i}+a_j\dfrac{\partial b_j}{\partial x_i}=\dfrac{\partial a_jb_j}{\partial x_i}=\nabla(\mathbf{a}\cdot\mathbf{b}) (b⋅∇)a+(a⋅∇)b+b×(∇×a)+a×(∇×b)=bj∂xj∂ai+aj∂xj∂bi+εijkbjεklm∂xl∂am+εijkajεklm∂xl∂bm=bj∂xj∂ai+aj∂xj∂bi+(δilδjm−δimδjl)(bj∂xl∂am+aj∂xl∂bm)=bj∂xj∂ai+aj∂xj∂bi+bj∂xi∂aj−bj∂xj∂ai+aj∂xi∂bj−aj∂xj∂bi=bjxiaj+aj∂xi∂bj=∂xi∂ajbj=∇(a⋅b)
- ∇ × ( ∇ × a ) = ε i j k ∂ ∂ x j ( ε k l m ∂ a m ∂ x l ) = ( δ i l δ j m − δ i m δ j l ) ∂ 2 a m ∂ x j ∂ x l = ∂ 2 a j ∂ x j ∂ x i − ∂ 2 a i ∂ x j ∂ x j = ∇ ( ∇ ⋅ a ) − Δ a \nabla\times(\nabla\times\mathbf{a})=\varepsilon_{ijk}\dfrac{\partial}{\partial x_j}\Big(\varepsilon_{klm}\dfrac{\partial a_m}{\partial x_l}\Big)\\=(\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl})\dfrac{\partial^2 a_m}{\partial x_j\partial x_l}\\=\dfrac{\partial^2 a_j}{\partial x_j\partial x_i}-\dfrac{\partial^2 a_i}{\partial x_j\partial x_j}=\nabla(\nabla\cdot\mathbf{a})-\Delta\mathbf{a} ∇×(∇×a)=εijk∂xj∂(εklm∂xl∂am)=(δilδjm−δimδjl)∂xj∂xl∂2am=∂xj∂xi∂2aj−∂xj∂xj∂2ai=∇(∇⋅a)−Δa
1.11 梯度、散度、旋度、拉普拉斯算子在曲线坐标系中的表达式
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推导：拉梅系数
- { ∣ ∂ r ∂ q 1 ∣ = ( ∂ x ∂ q 1 ) 2 + ( ∂ y ∂ q 1 ) 2 + ( ∂ z ∂ q 1 ) 2 = H 1 ∣ ∂ r ∂ q 2 ∣ = ( ∂ x ∂ q 2 ) 2 + ( ∂ y ∂ q 2 ) 2 + ( ∂ z ∂ q 2 ) 2 = H 2 ∣ ∂ r ∂ q 3 ∣ = ( ∂ x ∂ q 3 ) 2 + ( ∂ y ∂ q 3 ) 2 + ( ∂ z ∂ q 3 ) 2 = H 3 \begin{cases} \Big|\dfrac{\partial r}{\partial q_1}\Big|=\sqrt{\Big(\dfrac{\partial x}{\partial q_1}\Big)^2+\Big(\dfrac{\partial y}{\partial q_1}\Big)^2+\Big(\dfrac{\partial z}{\partial q_1}\Big)^2}=H_1 \\ \Big|\dfrac{\partial r}{\partial q_2}\Big|=\sqrt{\Big(\dfrac{\partial x}{\partial q_2}\Big)^2+\Big(\dfrac{\partial y}{\partial q_2}\Big)^2+\Big(\dfrac{\partial z}{\partial q_2}\Big)^2}=H_2 \\ \Big|\dfrac{\partial r}{\partial q_3}\Big|=\sqrt{\Big(\dfrac{\partial x}{\partial q_3}\Big)^2+\Big(\dfrac{\partial y}{\partial q_3}\Big)^2+\Big(\dfrac{\partial z}{\partial q_3}\Big)^2}=H_3 \\ \end{cases} ⎩⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎧∣∣∣∂q1∂r∣∣∣=(∂q1∂x)2+(∂q1∂y)2+(∂q1∂z)2=H1∣∣∣∂q2∂r∣∣∣=(∂q2∂x)2+(∂q2∂y)2+(∂q2∂z)2=H2∣∣∣∂q3∂r∣∣∣=(∂q3∂x)2+(∂q3∂y)2+(∂q3∂z)2=H3
- 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 dV=H1H2H3dq1dq2dq3
- 柱坐标： H 1 = 1 H 2 = r H 3 = 1 H_1=1\quad H_2=r\quad H_3=1 H1=1H2=rH3=1
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结论
- 梯度
- 柱坐标
- ∇ ϕ = ∂ ϕ ∂ r e r + 1 r ∂ ϕ ∂ θ e θ + ∂ ϕ ∂ z e z \nabla\phi=\dfrac{\partial\phi}{\partial r}\mathbf{e}_r+\dfrac{1}{r}\dfrac{\partial\phi}{\partial\theta}\mathbf{e}_{\theta}+\dfrac{\partial\phi}{\partial z}\mathbf{e}_z ∇ϕ=∂r∂ϕer+r1∂θ∂ϕeθ+∂z∂ϕez
- 柱坐标
- 散度
- 柱坐标
- ∇ ⋅ u = 1 r ∂ ( r u r ) ∂ r + 1 r ∂ u θ ∂ θ + ∂ u z ∂ z \nabla\cdot\mathbf{u}=\dfrac{1}{r}\dfrac{\partial(ru_r)}{\partial r}+\dfrac{1}{r}\dfrac{\partial u_{\theta}}{\partial\theta}+\dfrac{\partial u_z}{\partial z} ∇⋅u=r1∂r∂(rur)+r1∂θ∂uθ+∂z∂uz
- 柱坐标
- 旋度
- ∇ × u = 1 H 1 H 2 H 3 ∣ H 1 e 1 H 2 e 2 H 3 e 3 ∂ ∂ q 1 ∂ ∂ q 2 ∂ ∂ q 3 H 1 u 1 H 2 u 2 H 3 u 3 ∣ \nabla\times\mathbf{u}=\dfrac{1}{H_1H_2H_3}\begin{vmatrix} H_1\mathbf{e}_1& H_2\mathbf{e}_2& H_3\mathbf{e}_3\\ \dfrac{\partial}{\partial q_1}& \dfrac{\partial}{\partial q_2}& \dfrac{\partial}{\partial q_3}&\\ H_1u_1& H_2u_2& H_3u_3\\ \end{vmatrix} ∇×u=H1H2H31∣∣∣∣∣∣∣H1e1∂q1∂H1u1H2e2∂q2∂H2u2H3e3∂q3∂H3u3∣∣∣∣∣∣∣
- Laplace
- 柱坐标
- Δ φ = 1 r ∂ ∂ r ( r ∂ φ ∂ r ) + 1 r 2 ∂ 2 φ ∂ θ 2 + ∂ 2 φ ∂ z 2 \Delta\varphi=\dfrac{1}{r}\dfrac{\partial}{\partial r}\Big(r\dfrac{\partial\varphi}{\partial r}\Big)+\dfrac{1}{r^2}\dfrac{\partial^2\varphi}{\partial\theta^2}+\dfrac{\partial^2\varphi}{\partial z^2} Δφ=r1∂r∂(r∂r∂φ)+r21∂θ2∂2φ+∂z2∂2φ
- 柱坐标
- 梯度
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