张量分析Ⅱ

张量分析——要命的数学知识增加了！

张量分析Ⅱ张量分析

A-6张量分析

• 哈密顿算子(梯度算子)
  $$\begin{aligned}
  \mathrm{d\varphi}&=\frac{\partial \varphi}{\partial x}\mathrm{dx}+\frac{\partial \varphi}{\partial y}\mathrm{dy}+\frac{\partial \varphi}{\partial z}\mathrm{dz}\\
  &=\partial_{i}\varphi\mathrm{dx_{i}}=\partial_{i}\varphi e_{i}\cdot e_{j}\mathrm{dx_{j}}\\
  &=\nabla\varphi\cdot\mathrm{d\vec{r}}
  \end{aligned}$$

  • 标量场的梯度
    $$\begin{aligned}
    grad\varphi=\frac{\partial \varphi}{\partial x}e_{1}+\frac{\partial \varphi}{\partial y}e_{2}+\frac{\partial \varphi}{\partial z}e_{3}=\nabla\varphi
    \end{aligned}$$
  • 矢量场u的散度
    $$\begin{aligned}
    div\varphi&=\frac{\partial u_{x}}{\partial x}+\frac{\partial u_{y}}{\partial y}+\frac{\partial u_{z}}{\partial z}=u_{j,j}\\
    &=e_{i}\partial_{i}\cdot u_{j}e_{j}=\nabla\cdot \vec{u}
    \end{aligned}$$
  • 矢量的旋度
    $$\begin{aligned}
    curl\vec{u}&=\begin{vmatrix}
    e_{1} & e_{2} & e_{3}\\
    \frac{\partial }{\partial x} & \frac{\partial }{\partial y} & \frac{\partial }{\partial z}\\
    u_{1} & u_{2} & u_{3}
    \end{vmatrix}\\
    &=e_{ijk}\partial_{i}u_{j}e_{k}=e_{i}\times e_{j}\partial_{i}u_{j}\\
    &=e_{i}\partial_{i}\times u_{j}e_{j}=\nabla\times\vec{u}
    \end{aligned}$$

• 张量场的微分

  • 张量A的梯度
    左梯度：
    $$\begin{aligned}
    \nabla A=e_{i}\partial_{i}A_{jk}e_{j}e_{k}=A_{jk,j}e_{i}e_{j}e_{k}
    \end{aligned}$$
    右梯度：
    $$\begin{aligned}
    A\nabla=\partial_{i}A_{jk}e_{j}e_{k}e_{i}=A_{jk,i}e_{j}e_{j}e_{i}
    \end{aligned}$$张量的梯度为比原张量高一阶的新张量
  • 张量A的散度
    左散度：
    $$\begin{aligned}
    \nabla\cdot A=e_{i}\partial_{i}\cdot A_{jk}e_{j}e_{k}=A_{jk,i}\delta_{ij}e_{k}=A_{jk,j}e_{k}
    \end{aligned}$$
    右散度：
    $$\begin{aligned}
    A\cdot\nabla=A_{jk}e_{j}e_{j}\cdot e_{i}\partial_{i}=A_{jk,i}e_{j}\delta_{ki}=A_{jk,k}e_{j}=A_{kj,j}e_{k}
    \end{aligned}$$张量的散度为比原张量低一阶的新张量
  • 张量A的旋度
    左旋度：
    $$\begin{aligned}
    \nabla\times A&=e_{i}\partial_{i}\times A_{jk}e{j}e_{k}=A_{jk,i}e_{ijr}e_{r}e_{k}\\
    &=e_{ijr}A_{jk,r}e_{r}e_{k}\\
    &=e_{rki}A_{kj,r}e_{i}e_{k}
    \end{aligned}$$
    右旋度：
    $$\begin{aligned}
    A\times\nabla&=A{jk}e_{j}e_{k}\times partial_{i}e_{i}=e_{kir}A_{ij,i}e_{j}e_{r}\\
    &=e_{kri}A_{jk,r}e_{j}e_{i}\\
    &=e_{krj}A_{ik,r}e_{i}e_{j}
    \end{aligned}$$

• 散度定理
  高斯积分公式为：
  $$\begin{aligned}
  &\int_{V}\big(\frac{\partial V_{x}}{\partial x}+\frac{\partial V_{y}}{\partial y}+\frac{\partial V_{z}}{\partial z}\big)\mathrm{dv}=\oint_{S}(V_{x}\cos\alpha+V_{y}\cos\beta+V_{z}\cos\gamma)\mathrm{ds}\\
  \Rightarrow &\int_{V}V_{i,i}\mathrm{dv}=\oint_{S}V_{i}n_{i}\mathrm{ds}
  \end{aligned}$$
  对于任意阶张量的高斯积分公式为：
  $$\begin{aligned}
  &\int_{V}A_{ijk,k}\mathrm{dv}=\oint_{S}A_{ijk}n_{k}\mathrm{ds}\\
  &\int_{V} A\cdot\nabla\mathrm{dv}=\oint_{S}A\cdot n\mathrm{ds}\\
  &\int_{V}\nabla\cdot A\mathrm{dv}=\oint_{S}n\cdot A\mathrm{ds}
  \end{aligned}$$

  A-7曲线坐标系下的张量分析

  尽管一般在讨论张量时都是在笛卡尔坐标系下进行，但在解决具体问题时，可能会用到更复杂的坐标系

• 曲线坐标
  在笛卡尔坐标下，空间中任一点的位矢为$\vec{r}=x_{i}e_{i}$
  在三维空间某连通区域内，给定笛卡尔坐标的三个连续可微的单值函数
  $$\begin{aligned}
  x_{i\prime}=x_{i\prime}(x_{i})\rightarrow(反函数) x_{i}=x_{i}(x_{i\prime})
  \end{aligned}$$
  若函数不是线性函数，则称其为曲线坐标系
  $$\begin{aligned}
  &J=|\frac{\partial x_{i\prime}}{\partial x_{i}}|>0\\
  &J^{-1}=|\frac{\partial x_{i}}{\partial x_{i\prime}}|>0
  \end{aligned}$$
  用于编排指标i’的次序

• 局部基矢量

  • 自然基:取一点出坐标曲线的切向量
    $$\begin{aligned}
    g_{i}=\frac{\partial \vec{r}}{\partial x_{i\prime}}=\frac{\partial }{\partial x_{i\prime}}(x_{i}e_{i})=\frac{\partial x_{i}}{\partial x_{i\prime}}e_{i}
    \end{aligned}$$
  • 度量张量
    $$\begin{aligned}
    \boldsymbol{g_{i}}\cdot \boldsymbol{g_{j}}=g_{ij}
    \end{aligned}$$
    例如对于圆柱坐标系
    $$\begin{aligned}
    &x=r\cos\theta,y=r\sin\theta,z=z\\
    &\vec{r}=r\cos\theta e_{1}+r\sin\theta e_{2}+ze_{3}\\
    &g_{1}=\frac{\partial \vec{r}}{\partial r}=\cos\theta e_{1}+\sin\theta_{2}\\
    &g_{2}=\frac{\partial \vec{r}}{\partial \theta}=-r\sin\theta e_{1}+r\cos\theta e_{2}\\
    &g_{3}=\frac{\partial \vec{r}}{\partial z}=e_{3}\\
    &g_{ij}=\begin{vmatrix}
    g_{11} & g_{12} & g_{13}\\
    g_{21} & g_{22} & g_{23}\\
    g_{31} & g_{32} & g_{33}
    \end{vmatrix}=\begin{vmatrix}
    1 & 0 & 0 \\
    0 & r^2 & 0 \\
    0 & 0 & 1
    \end{vmatrix}
    \end{aligned}$$
    例如对于球坐标系
    $$\begin{aligned}
    &x=r\cos\theta\sin\varphi,y=r\sin\theta\sin\varphi,z=r\cos\varphi\\
    &\vec{r}=xe_{1}+ye_{2}+ze_{3}\\
    &g_{1}=\cos\theta\sin\varphi e_{1}+\sin\theta\sin\varphi e_{2} + \cos\varphi e_{3}\\
    &g_{2}=-r\sin\theta\sin\varphi e_{1}+r\cos\theta\sin\varphi e_{2}\\
    &g_{3}=r\cos\theta\cos\varphi e_{1}+r\sin\theta\cos\varphi e_{2}-r\sin\varphi e_{3}\\
    &g_{ij}=\begin{vmatrix}
    1 & 0 & 0\\
    0 & r^2\sin^2\varphi & 0\\
    0 & 0 & r^2
    \end{vmatrix}
    \end{aligned}$$
    笛卡尔坐标系中关于张量的定义与运算可以推广到曲线坐标系中，区别只在于这时的基矢量$g_{i}$及变换系数$a_{i,i}$是空间点位置的函数，如张量A在曲线坐标系中可以写成：
    $$\begin{aligned}
    A=A_{ijk}g_{i}g_{j}g_{k}
    \end{aligned}$$在曲线坐标中并非所有坐标都具有长度量纲，因此相对应的自然基矢量就不是无量纲的单位矢量。具有一定物理意义的向量或张量在这样的基上各分量不具有物理量纲，会在物理解释上带来不便。为了使张量在每个具体的坐标系里能起的具有物理量纲的分量，在正交曲线坐标系中，取切于坐标曲线的无量纲单位矢量作为基矢量，即： $$\begin{aligned}
    \hat{e_{i}}=\frac{1}{|\boldsymbol{g_{i}}|}\boldsymbol{g_{i}}=\frac{1}{\sqrt{g_{ii}}}\boldsymbol{g_{i}}
    \end{aligned}$$正交单位标架为物理标架，或称物理基

• 圆柱坐标下的物理基
  $$\begin{aligned}
  \begin{pmatrix}
  \hat{e_{1}}\\
  \hat{e_{2}}\\
  \hat{e_{3}}
  \end{pmatrix}=\begin{pmatrix}
  g_{1}\\
  \frac{1}{r}g_{2}\\
  g_{3}
  \end{pmatrix}=\begin{bmatrix}
  \cos\theta & \sin\theta & 0\\
  -\sin\theta & \cos\theta & 0\\
  0 & 0 & 1
  \end{bmatrix}\begin{pmatrix}
  e_{1}\\
  e_{2}\\
  e_{3}
  \end{pmatrix}
  \end{aligned}$$

• 球坐标系下的物理基
  $$\begin{aligned}
  \begin{pmatrix}
  \hat{e_{1}}\\
  \hat{e_{2}}\\
  \hat{e_{3}}
  \end{pmatrix}=\begin{pmatrix}
  g_{1}\\
  \frac{1}{r\sin\varphi}g_{2}\\
  \frac{1}{r}g_{3}
  \end{pmatrix}=\begin{bmatrix}
  \cos\theta\sin\varphi & \sin\theta\sin\varphi & \sin\varphi\\
  -\sin\theta & \cos\theta & 0\\
  \cos\theta\cos\varphi & \sin\theta\cos\varphi & -\sin\varphi
  \end{bmatrix}\begin{pmatrix}
  e_{1} \\
  e_{2} \\
  e_{3}
  \end{pmatrix}
  \end{aligned}$$

• 张量对曲线坐标的导数

  • 标量场φ沿s方向的方向导数：
    $$\begin{aligned}
    &\frac{\partial \varphi}{\partial S}=\nabla \varphi\cdot \vec{s}\\
    &\vec{s}=\frac{\partial \vec{r}}{\partial S}=\frac{\partial \vec{r}}{\partial x_{i}}\cdot\frac{\partial x_{i}}{\partial S}=\frac{\partial x_{i}}{\partial S}\boldsymbol{g_{i}}=\frac{\partial x_{i}}{\partial S}\sqrt{g_{ii}}\hat{e_{i}}\\
    \because &\frac{\partial \varphi}{\partial S}=\frac{\partial \varphi}{\partial x_{i}}\cdot\frac{\partial x_{i}}{\partial S}\\
    \because&\frac{\partial x_{i}}{\partial S}=\frac{1}{\sqrt{g_{ii}}}\hat{e_{i}}\cdot \vec{s}\\
    \therefore &\frac{\partial \varphi}{\partial S}=\frac{1}{\sqrt{g_{ii}}}\frac{\partial \varphi}{\partial x_{i}}\hat{e_{i}}\cdot \vec{s}\\
    \therefore & \nabla\varphi=\frac{1}{\sqrt{g_{ii}}}\frac{\partial \varphi}{\partial x_{i}}\hat{e_{i}}=\hat{e_{i}}\partial_{i}\varphi
    \end{aligned}$$

  • 克里斯多弗符号
    $$\begin{aligned}
    &\partial_{i}\hat{e_{j}}=\Gamma_{ijk}\hat{e_{k}}\quad 微分等于\\
    &\Gamma_{ijk}=\frac{1}{\sqrt{g_{ii}g_{jj}g_{kk}}}\big[\frac{1}{2}(g_{jk,i}+g_{ki,j}-g_{ij,k})+\sqrt{g_{jj}}\frac{\partial }{\partial x_{i}}(\frac{1}{\sqrt{g_{jj}}})g_{jk}\big]
    \end{aligned}$$
    $$\begin{aligned}
    \Gamma_{ijk}&=\partial_{i}\hat{e_{j}}\cdot\hat{e_{k}}=\frac{1}{\sqrt{g_{ii}}}\frac{\partial }{\partial x_{i}}(\frac{1}{\sqrt{g_{jj}}}g_{j})\cdot\frac{g_{k}}{\sqrt{g_{kk}}}
    \end{aligned}$$

• 张量的梯度
  $$\begin{aligned}
  \nabla A&=e_{i}\partial_{i}(A_{jk}e_{j}e_{k})\\
  &=e_{i}(\partial_{i}A_{jk}e_{j}e_{k}+A_{jk}\partial_{i}e_{j}e_{k}+A_{jk}e_{j}\partial_{i}e_{k})\\
  &=e_{i}(\partial_{i}A_{jk}e_{j}e_{k}+A_{jk}\Gamma_{ijr}e_{r}e_{k}+A_{jk}e_{j}\Gamma_{ikr}e_{r})\\
  &=(\partial_{i}A_{jk}+\Gamma_{irj}A_{rk}+\Gamma_{irk}A_{jr})e_{i}e_{j}e_{j}\\
  &=\nabla_{i}A_{jk}e_{i}e_{j}e_{k}
  \end{aligned}$$

• 圆柱坐标系下张量的导数公式

  • 梯度
    $$\begin{aligned}
    &grad \varphi=\nabla\varphi=e_{i}\nabla_{i}\varphi=e_{i}\partial_{i}\varphi\\
    &\partial_{i}=\frac{1}{\sqrt{g_{ii}}}\frac{\partial }{\partial x_{i}}\\
    &g_{11}=1,g_{22}=r^2,g_{33}=1\\
    &grad\varphi=\big(e_{r}\frac{\partial }{\partial r}+\frac{1}{r}e_{\theta}\frac{\partial }{\partial \theta}+e_{z}\frac{\partial }{\partial z}\big)\varphi
    \end{aligned}$$
  • 散度(矢量)
    $$\begin{aligned}
    div u&=\nabla\cdot u=e_{i}\nabla_{i}\cdot u_{j}e_{j}=\partial_{i}u_{i}+\Gamma_{iki}u_{k}\\
    &=\frac{\partial u_{r}}{\partial r}+\frac{1}{r}\frac{\partial u_{\theta}}{\partial \theta}+\frac{\partial u_{z}}{\partial z}+\frac{u_{r}}{r}
    \end{aligned}$$
  • 拉普拉斯算子
    $$\begin{aligned}
    &\nabla^2\varphi=\nabla\cdot\nabla\varphi(\partial_{i}\partial_{i}+\Gamma_{iji}\partial_{j})\varphi\\
    &=\big(\frac{\partial^2 }{\partial r^2}+\frac{1}{r}\frac{\partial }{\partial \theta}(\frac{1}{r}\frac{\partial }{\partial \theta})+\frac{\partial^2 }{\partial z^2}+\Gamma_{212\frac{\partial }{\partial r}}\big)\varphi\\
    &=\big(\frac{\partial^2 }{\partial r^2}+\frac{1}{r}\frac{\partial }{\partial r}+\frac{1}{r^2}\frac{\partial^2 }{\partial \theta^2}+\frac{\partial^2 }{\partial z^2}\big)\varphi
    \end{aligned}$$
  • 散度(张量)
    $$\begin{aligned}
    \nabla\cdot A&=e_{i}\partial_{i}\cdot A_{jk}e_{j}e_{k}\\
    &=e_{r}\big(\frac{\partial A_{rr}}{\partial r}+\frac{1}{r}\frac{\partial A_{\theta r}}{\partial \theta}+\frac{\partial A_{zr}}{\partial z}+\frac{A_{rr}-A_{\theta\theta}}{r}\big)\\
    &+e_{\theta}\big(\frac{\partial A_{r\theta}}{\partial r}+\frac{1}{r}\frac{\partial A_{\theta \theta}}{\partial \theta}+\frac{\partial A_{z\theta}}{\partial z}+\frac{A_{r\theta}-A_{\theta r}}{r}\big)\\
    &+e_{z}\big(\frac{\partial A_{rz}}{\partial r}+\frac{1}{r}\frac{\partial A_{\theta z}}{\partial \theta}+\frac{\partial A_{zz}}{\partial z}+\frac{A_{rz}}{r}\big)
    \end{aligned}$$$

特别鸣谢

• 特别鸣谢上海交通大学(SJTU)对本文的资源支持

The End