曲线坐标张量分析
$\Gamma$
  记$\boldsymbol{r}$为矢径，$x^i$为曲线坐标，$\partial_i=\displaystyle\frac{\partial}{\partial x^i}$，则协变基矢量可表示为$\boldsymbol{g_i}=\partial_i\boldsymbol{r}$，用笛卡尔坐标$\boldsymbol{r}=\overline{x}^p\boldsymbol{i_p}$表示协变基矢量、逆变基矢量和度量张量为
$$\boldsymbol{g_k}=\frac{\partial\overline{x}^p}{\partial x^k}\boldsymbol{i_p}\qquad
\boldsymbol{g^k}=\frac{\partial x^k}{\partial \overline{x}^p}\boldsymbol{i^p}\\
g_{kl}=\frac{\partial\overline{x}^p}{\partial x^k}\frac{\partial\overline{x}^p}{\partial x^l}\qquad
g^{kl}=\frac{\partial x^k}{\partial \overline{x}^p}\frac{\partial x^l}{\partial \overline{x}^p}$$
  引入Christoffel符号$\Gamma$表示协变基矢量的导数
$$\partial_i\boldsymbol{g_j}=\Gamma_{ij}^k\boldsymbol{g_k}=\Gamma_{ij,k}\boldsymbol{g^k}$$
  逆变基矢量的导数，可对$\boldsymbol{g^j}\cdot\boldsymbol{g_k}=\delta_k^j$求导，得$\partial_i\boldsymbol{g^j}\cdot\boldsymbol{g_k}=-\boldsymbol{g^j}\cdot\partial_i\boldsymbol{g_k}$，即
$$\partial_i\boldsymbol{g^j}=-\Gamma_{ik}^j\boldsymbol{g^k}$$
  由于$\partial_i\boldsymbol{g_j}=\partial_i\partial_j\boldsymbol{r}=\partial_j\partial_i\boldsymbol{r}=\partial_j\boldsymbol{g_i}$，故$\Gamma_{ij}^k$和$\Gamma_{ij,k}$均关于指标$ij$对称，由于在直线坐标系下基矢量不变，$\Gamma=0$，而曲线坐标系下$\Gamma$不一定为零，故$\Gamma$不是张量的分量，用笛卡尔坐标表示为
$$\Gamma_{ij}^k=\frac{\partial^2\overline{x}^p}{\partial x^i\partial x^j}\frac{\partial x^k}{\partial \overline{x}^p}\qquad \Gamma_{ij,k}=\frac{\partial^2\overline{x}^p}{\partial x^i\partial x^j}\frac{\partial \overline{x}^p}{\partial x^k}$$
  借助于笛卡尔坐标对度量张量协变分量求导可知
$$\partial_kg_{ij}=\Gamma_{ki,j}+\Gamma_{kj,i}\\
\partial_ig_{jk}=\Gamma_{ij,k}+\Gamma_{ik,j}\\
\partial_jg_{ki}=\Gamma_{jk,i}+\Gamma_{ji,k}$$
  由此三式可知$\Gamma$用度量张量协变分量表示为
$$\Gamma_{ij,k}=\frac{1}{2}(\partial_ig_{jk}+\partial_{j}g_{ki}-\partial_{k}g_{ij})$$
  三维空间中三个协变基矢量构成的平行六面体体积为
$$\sqrt{g}=[\boldsymbol{g_1}\quad\boldsymbol{g_2}\quad\boldsymbol{g_3}]=(\boldsymbol{g_1}\times\boldsymbol{g_2})\cdot\boldsymbol{g_3}$$
  对其求导
$$\partial_i\sqrt{g}=(\Gamma_{i1}^l\boldsymbol{g_l}\times\boldsymbol{g_2})\cdot\boldsymbol{g_3}+(\boldsymbol{g_1}\times\Gamma_{i2}^l\boldsymbol{g_l})\cdot\boldsymbol{g_3}+(\boldsymbol{g_1}\times\boldsymbol{g_2})\cdot\Gamma_{i3}^l\boldsymbol{g_l}\\
=(\Gamma_{i1}^1\boldsymbol{g_1}\times\boldsymbol{g_2})\cdot\boldsymbol{g_3}+(\boldsymbol{g_1}\times\Gamma_{i2}^2\boldsymbol{g_2})\cdot\boldsymbol{g_3}+(\boldsymbol{g_1}\times\boldsymbol{g_2})\cdot\Gamma_{i3}^3\boldsymbol{g_3}\\
=\Gamma_{ij}^j\sqrt{g}$$
  故$\Gamma$也可用$g$表示为
$$\Gamma_{ij}^j=\frac{1}{\sqrt{g}}\partial_i\sqrt{g}=\frac{1}{2}\partial_i\ln{g}$$
  当坐标由$x^i$转变为$x^{i^\prime}$时，$\Gamma$的转变如下
$$\Gamma_{i’j’}^{k’}=\partial_{i’}\boldsymbol{g_{j’}}\cdot\boldsymbol{g^{k’}}\\
=\partial_{i’}(\partial_{j’}x^l\boldsymbol{g_l})\cdot\partial_{m}x^{k’}\boldsymbol{g^m}\\
=(\partial_{i’}\partial_{j’}x^l\boldsymbol{g_l}+\partial_{j’}x^l\partial_{i’}x^n\partial_n\boldsymbol{g_l})\cdot\partial_{m}x^{k’}\boldsymbol{g^m}\\
=(\partial_{i’}\partial_{j’}x^l\boldsymbol{g_l}+\partial_{j’}x^l\partial_{i’}x^n\Gamma_{nl}^o\boldsymbol{g_o})\cdot\partial_{m}x^{k’}\boldsymbol{g^m}\\
=\partial_{i’}\partial_{j’}x^m\partial_mx^{k’}+\beta_{i’}^n\beta_{j’}^l\beta_m^{k’}\Gamma_{nl}^m$$
  在曲线坐标系下$\partial_{i’}\partial_{j’}x^m$一般不为零，即$\Gamma$不服从张量分量的转换关系，而在直线坐标系下$\partial_{i’}\partial_{j’}x^m=0$，但此时$\Gamma=0$，转换没有实际意义
$\boldsymbol{\nabla}$
  记张量场函数$\boldsymbol{T(r)}$对于$\boldsymbol{r}$的增量$\boldsymbol{u}$的有限微分为
$$\boldsymbol{T}’(\boldsymbol{r};\boldsymbol{u})=\lim\limits_{h\to 0}\frac{1}{h}[\boldsymbol{T}(\boldsymbol{r}+h\boldsymbol{u})-\boldsymbol{T}(\boldsymbol{r})]$$
  有限微分对于$\boldsymbol{u}$具有线性性质，证明如下
$$\boldsymbol{T}’(\boldsymbol{r};a\boldsymbol{u})=\lim\limits_{h\to 0}\frac{1}{h}[\boldsymbol{T}(\boldsymbol{r}+ha\boldsymbol{u})-\boldsymbol{T}(\boldsymbol{r})]\\=a\lim\limits_{ha\to 0}\frac{1}{ha}[\boldsymbol{T}(\boldsymbol{r}+ha\boldsymbol{u})-\boldsymbol{T}(\boldsymbol{r})]=a\boldsymbol{T}’(\boldsymbol{r};\boldsymbol{u})\\
\quad\\
\boldsymbol{T}’(\boldsymbol{r};\boldsymbol{u}+\boldsymbol{v})=\lim\limits_{h\to 0}\frac{1}{h}[\boldsymbol{T}(\boldsymbol{r}+h\boldsymbol{u}+h\boldsymbol{v})-\boldsymbol{T}(\boldsymbol{r})]\\
=\lim\limits_{h\to 0}\frac{1}{h}[\boldsymbol{T}(\boldsymbol{r}+h\boldsymbol{u}+h\boldsymbol{v})-\boldsymbol{T}(\boldsymbol{r}+h\boldsymbol{u})+\boldsymbol{T}(\boldsymbol{r}+h\boldsymbol{u})-\boldsymbol{T}(\boldsymbol{r})]\\
=\boldsymbol{T}’(\boldsymbol{r};\boldsymbol{u})+\boldsymbol{T}’(\boldsymbol{r};\boldsymbol{v})$$
  即$\boldsymbol{u}$可通过线性变换映射为张量$\boldsymbol{T}’(\boldsymbol{r};\boldsymbol{u})$，根据商法则，此线性变换通过一个$n+1$阶张量实现，记为
$$\boldsymbol{T}’(\boldsymbol{r};\boldsymbol{u})=\boldsymbol{T}’(\boldsymbol{r})\cdot \boldsymbol{u}$$
  称$\boldsymbol{T}’(\boldsymbol{r})=\displaystyle\frac{\dd \boldsymbol{T}}{\dd \boldsymbol{r}}$为$n$阶张量场$\boldsymbol{T}(\boldsymbol{r})$对矢径$\boldsymbol{r}$的导数，当增量$\boldsymbol{u}=\boldsymbol{g_i}$时
$$\boldsymbol{T}’(\boldsymbol{r};\boldsymbol{g_i})=\lim\limits_{h\to 0}\frac{1}{h}[\boldsymbol{T}(\boldsymbol{r}+h\boldsymbol{g_i})-\boldsymbol{T}(\boldsymbol{r})]$$
  又根据多元函数偏导数的定义
$$\partial_i\boldsymbol{T}=\lim\limits_{h\to 0}\frac{1}{h}[\boldsymbol{T}(\boldsymbol{r}(x^l+h\delta_i^l))-\boldsymbol{T}(\boldsymbol{r})]\\
=\lim\limits_{h\to 0}\frac{1}{h}[\boldsymbol{T}(\boldsymbol{r}+h\boldsymbol{g_i}+o(h))-\boldsymbol{T}(\boldsymbol{r})]$$
  可知$\boldsymbol{T}’(\boldsymbol{r};\boldsymbol{g_i})=\partial_i\boldsymbol{T}$，故
$$\boldsymbol{T}’(\boldsymbol{r};\boldsymbol{u})=\boldsymbol{T}’(\boldsymbol{r};u^i\boldsymbol{g_i})\\
=u^i\partial_i\boldsymbol{T}\\
=(\partial_i\boldsymbol{T}\boldsymbol{g^i})\cdot \boldsymbol{u}$$
  观察形式即得$\boldsymbol{T}(\boldsymbol{r})$对$\boldsymbol{r}$的导数为
$$\boldsymbol{T}’(\boldsymbol{r})=\partial_i\boldsymbol{T}\boldsymbol{g^i}$$
  记$\varphi$为标量场函数，$\boldsymbol{u,v}$为矢量场函数，$\boldsymbol{T}$为张量场函数，定义其梯度、散度、旋度如下
$$\boldsymbol{\nabla}\boldsymbol{T}=\boldsymbol{g^i}\partial_i\boldsymbol{T}\qquad\boldsymbol{T}\boldsymbol{\nabla}=\partial_i\boldsymbol{T}\boldsymbol{g^i}\\
\boldsymbol{\nabla}\cdot\boldsymbol{T}=\boldsymbol{g^i}\cdot\partial_i\boldsymbol{T}\qquad\boldsymbol{T}\cdot\boldsymbol{\nabla}=\partial_i\boldsymbol{T}\cdot\boldsymbol{g^i}\\
\boldsymbol{\nabla}\times\boldsymbol{T}=\boldsymbol{g^i}\times\partial_i\boldsymbol{T}\qquad\boldsymbol{T}\times\boldsymbol{\nabla}=\partial_i\boldsymbol{T}\times\boldsymbol{g^i}$$
  易知张量场的梯度、散度、旋度仍为张量，为用基矢量表示$\partial_i \boldsymbol{T}$，可引入协变导数，以三阶张量$\boldsymbol{T}=T_{\cdot\cdot k}^{ij}\boldsymbol{g_i g_j g^k}$为例，对其求导得
$$\partial_l\boldsymbol{T}=\partial_lT_{\cdot\cdot k}^{ij}\boldsymbol{g_i g_j g^k}+T_{\cdot\cdot k}^{ij}\Gamma_{il}^m\boldsymbol{g_mg_jg^k}+T_{\cdot\cdot k}^{ij}\boldsymbol{g_i}\Gamma_{jl}^m\boldsymbol{g_mg^k}-T_{\cdot\cdot k}^{ij}\boldsymbol{g_ig_j}\Gamma_{ml}^k\boldsymbol{g^m}\\
=(\partial_lT_{\cdot\cdot k}^{ij}+T_{\cdot\cdot k}^{mj}\Gamma_{ml}^i+T_{\cdot\cdot k}^{im}\Gamma_{ml}^j-T_{\cdot\cdot m}^{ij}\Gamma_{kl}^m)\boldsymbol{g_i g_j g^k}$$
  定义张量分量$T_{\cdot\cdot k}^{ij}$对坐标的协变导数
$$T_{\cdot\cdot k;l}^{ij}=\partial_lT_{\cdot\cdot k}^{ij}+T_{\cdot\cdot k}^{mj}\Gamma_{ml}^i+T_{\cdot\cdot k}^{im}\Gamma_{ml}^j-T_{\cdot\cdot m}^{ij}\Gamma_{kl}^m$$
  则$\partial_l \boldsymbol{T}$可简记为
$$\partial_l \boldsymbol{T}=T_{\cdot\cdot k;l}^{ij}\boldsymbol{g_i g_j g^k}$$
  引入符号$\nabla_i()=()_{;i}$作为张量分量协变导数的另一种符号，以右梯度与左梯度为例，其借助这两种协变导数符号可按指标顺序分别记为
$$\boldsymbol{T}\boldsymbol{\nabla}=\partial_l\boldsymbol{T}\boldsymbol{g^l}=T_{\cdot\cdot k;l}^{ij}\boldsymbol{g_i g_j g^k g^l}\\
\boldsymbol{\nabla}\boldsymbol{T}=\boldsymbol{g^l}\partial_l\boldsymbol{T}=\nabla_lT_{\cdot\cdot k}^{ij}\boldsymbol{g^l g_i g_j g^k}$$
  矢量场函数$\boldsymbol{F}$的散度和旋度借助协变导数可写为
$$\boldsymbol{\nabla}\cdot\boldsymbol{F}=\nabla_iF^i=\partial_iF^i+F^m\Gamma_{im}^{i}\\
=\partial_iF^i+F^m\frac{1}{\sqrt{g}}\partial_m\sqrt{g}\\
=\frac{1}{\sqrt{g}}\partial_i(\sqrt{g}F^i)\\
\quad\\
\boldsymbol{\nabla}\times\boldsymbol{F}=\boldsymbol{g^i}\times\nabla_iF_j\boldsymbol{g^j}=\epsilon^{ijk}\nabla_iF_j\boldsymbol{g_k}\\
=\epsilon^{ijk}(\partial_iF_j-F_l\Gamma_{ij}^l)\boldsymbol{g_k}=\epsilon^{ijk}\partial_iF_j\boldsymbol{g_k}\\
=\frac{1}{\sqrt{g}}\begin{vmatrix}\boldsymbol{g_1}&\boldsymbol{g_2}&\boldsymbol{g_3}\\
\partial_1&\partial_2&\partial_3\\ F_1&F_2&F_3\end{vmatrix}$$
  协变导数满足Leibnitz法则，即张量分量缩并与求协变导数可交换次序；张量分量乘积的协变导数服从函数乘积的普通偏导数的运算规则，例如
$$\nabla_s(A_{\cdot\cdot k}^{ij}B_{\cdot m}^l)=(\nabla_sA_{\cdot\cdot k}^{ij})B_{\cdot m}^l+A_{\cdot\cdot k}^{ij}(\nabla_sB_{\cdot m}^l)$$
  协变导数还具有如下性质
$$\nabla_ig_{jk}=\partial_ig_{jk}-g_{lk}\Gamma_{ij}^l-g_{jl}\Gamma_{ik}^l\\
=\partial_ig_{jk}-\Gamma_{ij;k}-\Gamma_{ik;j}=0\\
\quad\\
\nabla_i\delta_j^k=\partial_i\delta_j^k+\delta_j^l\Gamma_{il}^k-\delta_l^k\Gamma_{ij}^l\\
=0+\Gamma_{ij}^k-\Gamma_{ij}^k=0\\
\quad\\
\boldsymbol{\nabla}\boldsymbol{\epsilon}=\boldsymbol{g^l}\partial_l(\epsilon^{ijk}\boldsymbol{g_i g_j g_k})\\=\boldsymbol{g^l}\partial_l([\boldsymbol{g^i\quad g^j\quad g^k}]\boldsymbol{g_i g_j g_k})\\
=\boldsymbol{g^l}(([-\Gamma_{lm}^i\boldsymbol{g^m\quad g^j\quad g^k}]+[\boldsymbol{g^i}\quad -\Gamma_{lm}^j\boldsymbol{g^m\quad g^k}]
\\+[\boldsymbol{g^i\quad g^j}\quad -\Gamma_{lm}^k\boldsymbol{g^m}])\boldsymbol{g_i g_j g_k}+[\boldsymbol{g^i\quad g^j\quad g^k}](\Gamma_{li}^m\boldsymbol{g_m g_j g_k}\\
+\boldsymbol{g_i}\Gamma_{lj}^m\boldsymbol{g_m g_k}+\boldsymbol{g_i g_j}\Gamma_{lk}^m\boldsymbol{g_m}))=\boldsymbol{O}$$
  梯度、散度、旋度有如下常用公式
$$\boldsymbol{\nabla}\times\boldsymbol{T}=\boldsymbol{\epsilon:}(\boldsymbol{\nabla}\boldsymbol{T})\qquad\boldsymbol{T}\times\boldsymbol{\nabla}=(\boldsymbol{T}\boldsymbol{\nabla})\boldsymbol{:\epsilon}\\
\boldsymbol{\nabla}\boldsymbol{r}=\boldsymbol{r}\boldsymbol{\nabla}=\boldsymbol{G}\\
\boldsymbol{\nabla}(\varphi\boldsymbol{T})=\boldsymbol{\nabla}\varphi\boldsymbol{T}+\varphi\boldsymbol{\nabla}\boldsymbol{T}\\
\boldsymbol{\nabla}\cdot(\varphi\boldsymbol{T})=\boldsymbol{\nabla}\varphi\cdot\boldsymbol{T}+\varphi\boldsymbol{\nabla}\cdot\boldsymbol{T}\\
\boldsymbol{\nabla}\times(\varphi\boldsymbol{T})=\boldsymbol{\nabla}\varphi\times\boldsymbol{T}+\varphi\boldsymbol{\nabla}\times\boldsymbol{T}\\
\boldsymbol{\nabla}(\boldsymbol{u}\cdot\boldsymbol{v})=\boldsymbol{\nabla}\boldsymbol{u}\cdot\boldsymbol{v}+\boldsymbol{\nabla}\boldsymbol{v}\cdot\boldsymbol{u}\\
\boldsymbol{\nabla}(\boldsymbol{T}\cdot\boldsymbol{v})=\boldsymbol{\nabla}\boldsymbol{T}\cdot\boldsymbol{v}+\boldsymbol{\nabla}\boldsymbol{v}\cdot\boldsymbol{T}^T\\
\boldsymbol{\nabla}(\boldsymbol{u}\cdot\boldsymbol{v})=\boldsymbol{u}\times(\boldsymbol{\nabla}\times\boldsymbol{v})+\boldsymbol{v}\times(\boldsymbol{\nabla}\times\boldsymbol{u})+\boldsymbol{u}\cdot(\boldsymbol{\nabla}\boldsymbol{v})+\boldsymbol{v}\cdot(\boldsymbol{\nabla}\boldsymbol{u})\\
\boldsymbol{\nabla}\times\boldsymbol{u}\times \boldsymbol{v}=(\boldsymbol{u}\boldsymbol{\nabla}-\boldsymbol{\nabla}\boldsymbol{u})\cdot\boldsymbol{v}\\
\boldsymbol{\nabla}\times(\boldsymbol{u}\times \boldsymbol{v})=\boldsymbol{u}(\boldsymbol{\nabla}\cdot\boldsymbol{v})-\boldsymbol{v}(\boldsymbol{\nabla}\cdot\boldsymbol{u})-\boldsymbol{u}\cdot(\boldsymbol{\nabla}\boldsymbol{v})+\boldsymbol{v}\cdot(\boldsymbol{\nabla}\boldsymbol{u})\\
\boldsymbol{\nabla}\times\boldsymbol{\nabla}\varphi=0\\
\boldsymbol{\nabla}\cdot(\boldsymbol{\nabla}\times\boldsymbol{u})=0\\
\small{欧氏空间中有}
\normalsize\qquad\boldsymbol{\nabla}\times(\boldsymbol{\nabla}\times\boldsymbol{u})=\boldsymbol{\nabla}(\boldsymbol{\nabla}\cdot\boldsymbol{u})-\boldsymbol{\nabla}\cdot(\boldsymbol{\nabla}\boldsymbol{u})$$
$\displaystyle\int$
  若$S$为封闭曲面，$\boldsymbol{a}$为常矢量，易知$\displaystyle\oint_S\boldsymbol{a}\cdot\dd\boldsymbol{S}=0$，由$\boldsymbol{a}$的任意性即得
$$\oint_S\dd\boldsymbol{S}=0$$
  该式说明封闭曲面的面素矢量之和为零，另外下式也能表示与其相同的几何意义
$$\partial_i(\sqrt{g}\boldsymbol{g^i})=\partial_i\sqrt{g}\boldsymbol{g^i}+\sqrt{g}\partial_i\boldsymbol{g^i}\\
=\partial_i\sqrt{g}\boldsymbol{g^i}-\sqrt{g}\Gamma_{il}^i\boldsymbol{g^l}\\
=\partial_i\sqrt{g}\boldsymbol{g^i}-\partial_l\sqrt{g}\boldsymbol{g^l}=0$$
  考虑坐标差异为$\dd x^i$的曲面微元体$\Delta V$，其六个面素之和为
$$\sum_{\Delta S}\dd\boldsymbol{S}=\dd\boldsymbol{S_左}+\dd\boldsymbol{S_右}+\dd\boldsymbol{S_前}+\dd\boldsymbol{S_后}+\dd\boldsymbol{S_下}+\dd\boldsymbol{S_上}\\
=-\boldsymbol{g_2}\times\boldsymbol{g_3}\dd x^2\dd x^3+(\boldsymbol{g_2}+\partial_1\boldsymbol{g_2}\dd x^1)\times(\boldsymbol{g_3}+\partial_1\boldsymbol{g_3}\dd x^1)\dd x^2\dd x^3\\
+-\boldsymbol{g_3}\times\boldsymbol{g_1}\dd x^3\dd x^1+(\boldsymbol{g_3}+\partial_2\boldsymbol{g_3}\dd x^2)\times(\boldsymbol{g_1}+\partial_2\boldsymbol{g_1}\dd x^2)\dd x^3\dd x^1\\
+-\boldsymbol{g_1}\times\boldsymbol{g_2}\dd x^1\dd x^2+(\boldsymbol{g_1}+\partial_3\boldsymbol{g_1}\dd x^3)\times(\boldsymbol{g_2}+\partial_3\boldsymbol{g_2}\dd x^3)\dd x^1\dd x^2\\
=(\partial_1(\boldsymbol{g_2}\times\boldsymbol{g_3})+\partial_2(\boldsymbol{g_3}\times\boldsymbol{g_1})+\partial_3(\boldsymbol{g_1}\times\boldsymbol{g_2}))\dd x^1\dd x^2\dd x^3\\
=\partial_i(\sqrt{g}\boldsymbol{g^i})\dd x^1\dd x^2\dd x^3$$
  可见$\partial_i(\sqrt{g}\boldsymbol{g^i})=0$从微观角度表明了封闭曲面的面素矢量之和为零，当然，这只是不严谨的分析，事实上大部分封闭曲面都不能仅仅由微元六面体构成，正确的做法是考虑任意形状的微元四面体。另外联想生活中的物理现象，任意形状的空腔，其腔壁受到内部气体的均匀压强$p$作用，然而却能保持平衡，说明所受内部压力的合力为零，亦即$\displaystyle\oint_S p\dd\boldsymbol{S}=0$
  以下不加证明地给出张量函数$\boldsymbol{T}$的Green和Stokes变换公式
$$\text{Green}\\
\int_V\dd V\;\boldsymbol{\nabla T}=\oint_S\dd\boldsymbol{S}\;\boldsymbol{T}\qquad\int_V\dd V\;\boldsymbol{T\nabla }=\oint_S\boldsymbol{T}\;\dd\boldsymbol{S}\\
\int_V\dd V\;\boldsymbol{\nabla} \cdot \boldsymbol{T}=\oint_S\dd\boldsymbol{S}\cdot\boldsymbol{T}\qquad\int_V\dd V\;\boldsymbol{T}\cdot\boldsymbol{\nabla }=\oint_S\boldsymbol{T}\cdot\dd\boldsymbol{S}\\
\int_V\dd V\;\boldsymbol{\nabla}\times \boldsymbol{T}=\oint_S\dd\boldsymbol{S}\times\boldsymbol{T}\qquad\int_V\dd V\;\boldsymbol{T}\times\boldsymbol{\nabla }=\oint_S\boldsymbol{T}\times\dd\boldsymbol{S}\\
\quad\\
\text{Stokes}\\
\int_S\dd\boldsymbol{S}\cdot(\boldsymbol{\nabla}\times\boldsymbol{T})=\oint_l\dd\boldsymbol{l}\cdot\boldsymbol{T}\\\int_S(\boldsymbol{T}\times\boldsymbol{\nabla})\cdot\dd\boldsymbol{S}=-\oint_l\boldsymbol{T}\cdot\dd\boldsymbol{l}$$
$\displaystyle\boldsymbol{R}$
  线元平方由正定二次型决定的空间，称为黎曼空间，在黎曼空间中一般找不到适用于全空间的笛卡尔坐标系，即找不到一个坐标使$\Gamma_{ij}^k\equiv 0$，除非此黎曼空间为欧氏空间，以下将寻找黎曼空间成为欧氏空间的数学表述
  给定一组曲线坐标$x^i$，得到其度量张量$g_{ij}$和$\Gamma_{ij}^k$，设存在新坐标$y^{i’}$，在新坐标下$\Gamma_{i’j’}^{k’}\equiv 0$，由坐标转换关系可得
$$\Gamma_{i’j’}^{k’}=\partial_{i’}\partial_{j’}x^m\partial_mx^{k’}+\beta_{i’}^n\beta_{j’}^l\beta_m^{k’}\Gamma_{nl}^m\\
=(\partial_{i’}\partial_{j’}x^m+\partial_{i’}x^n\partial_{j’}x^l\Gamma_{nl}^m)\partial_my^{k’}$$