曲线积分、曲面积分

第一类曲线积分（线质量模型）

设空间中有曲线 $\mathbf{\Gamma }(t):[\alpha ,\beta ]\ni t\mapsto \mathbf{\Gamma }(t)=\left[\begin{array}{c}x(t)\\ y(t)\\ z(t)\end{array}\right]$.

设 $f(x,y,z)$ 为定义在曲线 $\mathbf{\Gamma }$ 上的线密度，现要求曲线的质量。

• 整体细分

  在 $[\alpha ,\beta ]$ 上引入分割 $P_t:\alpha =t_0<t_1<...<t_{i-1}<t_i<...<t_N=\beta$.

  由此产生物理域上的分割

  $$P:\mathbf{\Gamma }(\alpha )=\mathbf{\Gamma }(t_0)<\mathbf{\Gamma }(t_1)<...<\mathbf{\Gamma }(t_{i-1})<\mathbf{\Gamma }(t_i)<...<\mathbf{\Gamma }(t_N)=\mathbf{\Gamma }(\beta ).$$

• 局部近似

  考虑用折线段 $\mathbf{\Gamma }(t_i)-\mathbf{\Gamma }(t_{i-1})$ 来近似 $\mathbf{\Gamma }(t_{i-1})$ 与 $\mathbf{\Gamma }(t_i)$ 之间的曲线段，并取 $f\circ \mathbf{\Gamma }({\xi }_i),\mathrm{\forall }{\xi }_i\in [t_{i-1},t_i]$ 作为等效线密度。

  则这段曲线段的质量可以近似为：

  $$f\circ \mathbf{\Gamma }({\xi }_i)|\mathbf{\Gamma }(t_i)-\mathbf{\Gamma }(t_{i-1}){|}_{{\mathbb{R}}^3}$$

  因此可有总质量的近似

  $$\sum _{i=1}^{N}f\circ \mathbf{\Gamma }({\xi }_i)|\mathbf{\Gamma }(t_i)-\mathbf{\Gamma }(t_{i-1}){|}_{{\mathbb{R}}^3},\mathrm{\forall }{\xi }_i\in [t_{i-1},t_i].$$

• 引入极限

  曲线的质量为

  $$m=\underset{|P_t|\to 0}{lim}\sum _{i=1}^{N}f\circ \mathbf{\Gamma }({\xi }_i)|\mathbf{\Gamma }(t_i)-\mathbf{\Gamma }(t_{i-1}){|}_{{\mathbb{R}}^3},\mathrm{\forall }{\xi }_i\in [t_{i-1},t_i].$$

  • 将非标准部分和转化为标准部分和

    先选取 ${\xi }_i\in [t_{i-1},t_i]$，然后根据可微性可有

    $$\begin{array}{rl}& \mathbf{\Gamma }(t_i)=\mathbf{\Gamma }({\xi }_i)+\frac{\text{d}\mathbf{\Gamma }}{\text{d}t}({\xi }_i)(t_i-{\xi }_i)(1+o(1))\\ & \mathbf{\Gamma }(t_{i-1})=\mathbf{\Gamma }({\xi }_i)+\frac{\text{d}\mathbf{\Gamma }}{\text{d}t}({\xi }_i)(t_{i-1}-{\xi }_i)(1+o(1)).\end{array}$$

    因此考虑切线段近似：

    $$\mathbf{\Gamma }(t_i)-\mathbf{\Gamma }(t_{i-1})=\frac{\text{d}\mathbf{\Gamma }}{\text{d}t}({\xi }_i)(t_i-t_{i-1})(1+o(1)),\mathrm{\forall }{\xi }_i\in [t_{i-1},t_i].$$

    因此求和形式可以化为

    $$m=\underset{|P_t|\to 0}{lim}\sum _{i=1}^{N}f\circ \mathbf{\Gamma }({\xi }_i)|\frac{\text{d}\mathbf{\Gamma }}{\text{d}t}({\xi }_i){|}_{{\mathbb{R}}^3}\mathrm{\Delta }{t}_i,\mathrm{\forall }{\xi }_i\in [t_{i-1},t_i].$$

    根据定积分的定义，可有

    $$m=\underset{|P_t|\to 0}{lim}\sum _{i=1}^{N}f\circ \mathbf{\Gamma }({\xi }_i)|\frac{\text{d}\mathbf{\Gamma }}{\text{d}t}({\xi }_i){|}_{{\mathbb{R}}^3}\mathrm{\Delta }{t}_i={\int }_{\alpha }^{\beta }f\circ \mathbf{\Gamma }(t)|\frac{\text{d}\mathbf{\Gamma }}{\text{d}t}(t){|}_{{\mathbb{R}}^3}\text{d}t.$$

第一类曲线积分的极限表示

如有 $f\circ \mathbf{\Gamma }(t)\in \mathrm{\Re }[\alpha ,\beta ]$ 并且 $\frac{\text{d}\mathbf{\Gamma }}{\text{d}t}\in \mathrm{\Re }[\alpha ,\beta ]$，则有

$$\mathrm{\exists }\underset{|P_t|\to 0}{lim}\sum _{i=1}^{N}f\circ \mathbf{\Gamma }({\xi }_i)|\mathbf{\Gamma }(t_i)-\mathbf{\Gamma }(t_{i-1}){|}_{{\mathbb{R}}^3}={\int }_{\alpha }^{\beta }f\circ \mathbf{\Gamma }(t)|\frac{\text{d}\mathbf{\Gamma }}{\text{d}t}(t){|}_{{\mathbb{R}}^3}\text{d}t.$$

证明

计算

$$\begin{array}{rl}& \sum _{i=1}^{N}f\circ \mathbf{\Gamma }({\xi }_i)|\mathbf{\Gamma }(t_i)-\mathbf{\Gamma }(t_{i-1}){|}_{{\mathbb{R}}^3}\\ & =\sum _{i=1}^{N}f\circ \mathbf{\Gamma }({\xi }_i)\sqrt{[x(t_i)-x(t_{i-1}){]}^2+[y(t_i)-y(t_{i-1}){]}^2+[z(t_i)-z(t_{i-1}){]}^2}\\ & =\sum _{i=1}^{N}f\circ \mathbf{\Gamma }({\xi }_i)\sqrt{{\dot{x}}^2({\alpha }_i)+{\dot{y}}^2({\beta }_i)+{\dot{z}}^2({\zeta }_i)}\mathrm{\Delta }{t}_i,{\alpha }_i,{\beta }_i,{\zeta }_i\in (t_{i-1},t_i)\end{array}$$

估计

$$\begin{array}{rl}& |\sum _{i=1}^{N}f\circ \mathbf{\Gamma }({\xi }_i)\sqrt{{\dot{x}}^2({\alpha }_i)+{\dot{y}}^2({\beta }_i)+{\dot{z}}^2({\zeta }_i)}\mathrm{\Delta }{t}_i-{\int }_{\alpha }^{\beta }f\circ \mathbf{\Gamma }(t)|\frac{\text{d}\mathbf{\Gamma }}{\text{d}t}{|}_{{\mathbb{R}}^3}\text{d}t|\\ & \le |\sum _{i=1}^{N}f\circ \mathbf{\Gamma }({\xi }_i)\sqrt{{\dot{x}}^2({\alpha }_i)+{\dot{y}}^2({\beta }_i)+{\dot{z}}^2({\zeta }_i)}\mathrm{\Delta }{t}_i-\sum _{i=1}^{N}f\circ \mathbf{\Gamma }({\xi }_i)\sqrt{{\dot{x}}^2({\xi }_i)+{\dot{y}}^2({\xi }_i)+{\dot{z}}^2({\xi }_i)}\mathrm{\Delta }{t}_i|\\ & +|\sum _{i=1}^{N}f\circ \mathbf{\Gamma }({\xi }_i)\sqrt{{\dot{x}}^2({\xi }_i)+{\dot{y}}^2({\xi }_i)+{\dot{z}}^2({\xi }_i)}\mathrm{\Delta }{t}_i-{\int }_{\alpha }^{\beta }f\circ \mathbf{\Gamma }(t)|\frac{\text{d}\mathbf{\Gamma }}{\text{d}t}{|}_{{\mathbb{R}}^3}\text{d}t|\end{array}$$

上式中

$$\begin{array}{rl}& RH{S}^2<ϵ,\mathrm{\forall }|P_t|<{\delta }_{ϵ}\\ & RH{S}^1\le \sum _{i=1}^N|f\circ \mathbf{\Gamma }({\xi }_i)||\sqrt{{\dot{x}}^2({\alpha }_i)+{\dot{y}}^2({\beta }_i)+{\dot{z}}^2({\zeta }_i)}-\sqrt{{\dot{x}}^2({\xi }_i)+{\dot{y}}^2({\xi }_i)+{\dot{z}}^2({\xi }_i)}|\mathrm{\Delta }{t}_i\\ & \le \sum _{i=1}^N|f\circ \mathbf{\Gamma }({\xi }_i)|[|\dot{x}({\alpha }_i)-\dot{x}({\xi }_i)|+|\dot{y}({\beta }_i)-\dot{y}({\xi }_i)|+|\dot{z}({\zeta }_i)-\dot{z}({\xi }_i)|]\mathrm{\Delta }{t}_i\\ & \le \underset{t\in [\alpha ,\beta ]}{sup}|f\circ \mathbf{\Gamma }(t)|\sum _{i=1}^N[|\dot{x}({\alpha }_i)-\dot{x}({\xi }_i)|+|\dot{y}({\beta }_i)-\dot{y}({\xi }_i)|+|\dot{z}({\zeta }_i)-\dot{z}({\xi }_i)|]\mathrm{\Delta }{t}_i\\ & =\underset{t\in [\alpha ,\beta ]}{sup}|f\circ \mathbf{\Gamma }(t)|[\mathrm{\Omega }(x^{\prime };P_t)+\mathrm{\Omega }(y^{\prime };P_t)+\mathrm{\Omega }(z^{\prime };P_t)]\\ & <3\underset{t\in [\alpha ,\beta ]}{sup}|f\circ \mathbf{\Gamma }(t)|ϵ,\mathrm{\forall }|P_t|<min\{{\delta }_{{ϵ}_x},{\delta }_{{ϵ}_y},{\delta }_{{ϵ}_z}\}\end{array}$$

因此有

$$\mathrm{\exists }\underset{|P_t|\to 0}{lim}\sum _{i=1}^{N}f\circ \mathbf{\Gamma }({\xi }_i)|\mathbf{\Gamma }(t_i)-\mathbf{\Gamma }(t_{i-1}){|}_{{\mathbb{R}}^3}={\int }_{\alpha }^{\beta }f\circ \mathbf{\Gamma }(t)|\frac{\text{d}\mathbf{\Gamma }}{\text{d}t}(t){|}_{{\mathbb{R}}^3}\text{d}t.◻$$

第二类曲线积分（做功模型）

设 $\mathit{F}(x,y,z)$ 为定义在曲线上的力，现要求力 $\mathit{F}$ 从曲线的一端到另一端所做的功。对局部作 “均匀化” 假设，即

• 以直线段 $\mathbf{\Gamma }(t_i)-\mathbf{\Gamma }(t_{i-1})$ 近似局部的路径；
• 取 $\mathit{F}\circ \mathbf{\Gamma }({\xi }_i)$ 作为整个局部路径的力。

则有

$$W_i=\mathit{F}\circ \mathbf{\Gamma }({\xi }_i)·[\mathbf{\Gamma }(t_i)-\mathbf{\Gamma }(t_{i-1})],i=1,2,...,N.$$

故有

$$\sum _{i=1}^N{W}_i=\sum _{i=1}^N\mathit{F}\circ \mathbf{\Gamma }({\xi }_i)[\mathbf{\Gamma }(t_i)-\mathbf{\Gamma }(t_{i-1})].$$

取其极限形式，有

$$\underset{|P_t|\to 0}{lim}\sum _{i=1}^N=\underset{|P_t|\to 0}{lim}\sum _{i=1}^N\mathit{F}\circ \mathbf{\Gamma }({\xi }_i)[\mathbf{\Gamma }(t_i)-\mathbf{\Gamma }(t_{i-1})].$$

需将上式标准化，即需处理 $\mathbf{\Gamma }(t_i)-\mathbf{\Gamma }(t_{i-1})$.

用切线段近似，则有

$$\mathbf{\Gamma }(t_i)-\mathbf{\Gamma }(t_{i-1})=\dot{\mathbf{\Gamma }}({\xi }_i)\mathrm{\Delta }{t}_i.$$

代入部分和表达式，有

$$\underset{|P_t|\to 0}{lim}\sum _{i=1}^N\mathit{F}\circ \mathbf{\Gamma }({\xi }_i)[\mathbf{\Gamma }(t_i)-\mathbf{\Gamma }(t_{i-1})]=\underset{|P_t|\to 0}{lim}\sum _{i=1}^N\mathit{F}\circ \mathbf{\Gamma }({\xi }_i)·\dot{\mathbf{\Gamma }}({\xi }_i)\mathrm{\Delta }{t}_i.$$

因此定义

$${\int }_{\alpha }^{\beta }(\mathit{F}\circ \mathbf{\Gamma }(t),\frac{\text{d}\mathbf{\Gamma }}{\text{d}t}(t){)}_{{\mathbb{R}}^3}\text{d}t=\underset{|P_t|\to 0}{lim}\sum _{i=1}^N\mathit{F}\circ \mathbf{\Gamma }({\xi }_i)·\dot{\mathbf{\Gamma }}({\xi }_i)\mathrm{\Delta }{t}_i.$$

第二类曲线积分的极限表示

如有 $\mathit{F}\circ \mathbf{\Gamma }(t)\in \mathrm{\Re }[\alpha ,\beta ]$ 并且 $\frac{\text{d}\mathbf{\Gamma }}{\text{d}t}\in \mathrm{\Re }[\alpha ,\beta ]$，则有

$$\underset{|P_t|\to 0}{lim}\sum _{i=1}^N\mathbf{(}F\circ \mathbf{\Gamma }({\xi }_i),[\mathbf{\Gamma }(t_i)-\mathbf{\Gamma }(t_{i-1})]{)}_{{\mathbb{R}}^3}={\int }_{\alpha }^{\beta }(\mathit{F}\circ \mathbf{\Gamma }(t),\frac{\text{d}\mathbf{\Gamma }}{\text{d}t}(t){)}_{{\mathbb{R}}^3}\text{d}t.$$

证明

$$\begin{array}{rl}& \sum _{i=1}^N(\mathit{F}\circ \mathbf{\Gamma }({\xi }_i),[\mathbf{\Gamma }(t_i)-\mathbf{\Gamma }(t_{i-1})]{)}_{{\mathbb{R}}^3}\\ & =\sum _{i=1}^N[F_1\circ \mathbf{\Gamma }({\xi }_i)·\dot{x}({\alpha }_i)+F_2\circ \mathbf{\Gamma }({\xi }_i)·\dot{y}({\beta }_i)+F_3\circ \mathbf{\Gamma }({\xi }_i)·\dot{(}{\zeta }_i)]\mathrm{\Delta }{t}_i.\end{array}$$

估计

$$\begin{array}{rl}& |\sum _{i=1}^N[F_1\circ \mathbf{\Gamma }({\xi }_i)·\dot{x}({\alpha }_i)+F_2\circ \mathbf{\Gamma }({\xi }_i)·\dot{y}({\beta }_i)+F_3\circ \mathbf{\Gamma }({\xi }_i)·\dot{(}{\zeta }_i)]\mathrm{\Delta }{t}_i-{\int }_{\alpha }^{\beta }\mathbf{(}F\circ \mathbf{\Gamma }(t),\frac{\text{d}\mathbf{\Gamma }}{\text{d}t}(t){)}_{{\mathbb{R}}^3}\text{d}t|\\ & \le |\sum _{i=1}^N[F_1\circ \mathbf{\Gamma }({\xi }_i)·\dot{x}({\alpha }_i)+F_2\circ \mathbf{\Gamma }({\xi }_i)·\dot{y}({\beta }_i)+F_3\circ \mathbf{\Gamma }({\xi }_i)·\dot{(}{\zeta }_i)]\mathrm{\Delta }{t}_i-\sum _{i=1}^N[F_1\circ \mathbf{\Gamma }({\xi }_i)·\dot{x}({\xi }_i)+F_2\circ \mathbf{\Gamma }({\xi }_i)·\dot{y}({\xi }_i)+F_3\circ \mathbf{\Gamma }({\xi }_i)·\dot{(}{\xi }_i)]\mathrm{\Delta }{t}_i|\\ & +|\sum _{i=1}^N[F_1\circ \mathbf{\Gamma }({\xi }_i)·\dot{x}({\xi }_i)+F_2\circ \mathbf{\Gamma }({\xi }_i)·\dot{y}({\xi }_i)+F_3\circ \mathbf{\Gamma }({\xi }_i)·\dot{(}{\xi }_i)]\mathrm{\Delta }{t}_i-{\int }_{\alpha }^{\beta }\mathbf{(}F\circ \mathbf{\Gamma }(t),\frac{\text{d}\mathbf{\Gamma }}{\text{d}t}(t){)}_{{\mathbb{R}}^3}\text{d}t|.\end{array}$$

右端第二项满足

$$RH{S}^2\le ϵ,\mathrm{\exists }|P_t|<{\delta }_{ϵ}.$$

采取和第一类曲线积分同样的手法，可估计得

$$RH{S}^1\le \underset{[\alpha ,\beta ]}{sup}|F_1\circ \mathbf{\Gamma }(t)|\mathrm{\Omega }(x^{\prime }(t);P_t)+\underset{[\alpha ,\beta ]}{sup}|F_2\circ \mathbf{\Gamma }(t)|\mathrm{\Omega }(y^{\prime }(t);P_t)+\underset{[\alpha ,\beta ]}{sup}|F_3\circ \mathbf{\Gamma }(t)|\mathrm{\Omega }(z^{\prime }(t);P_t).$$

故有

$$RH{S}^1<3ϵ,\mathrm{\forall }|P_t|<{\delta }_{ϵ}.$$

综上即可得到结论。$◻$

曲面上的积分

设 ${\mathbb{R}}^3$ 空间中有曲面，其向量值映照表示为

$$\mathbf{\Sigma }=E_{\lambda \mu }\ni \left[\begin{array}{c}\lambda \\ \mu \end{array}\right]\mapsto \mathbf{\Sigma }(\lambda ,\mu )=\left[\begin{array}{c}x(\lambda ,\mu )\\ y(\lambda ,\mu )\\ z(\lambda ,\mu )\end{array}\right]\in {\mathbb{R}}^3.$$

第一类曲面积分（面质量模型）

设 $f(x,y,z)$ 为定义在曲面上的面密度（单位面积上的质量），现要求曲面的质量。

• 整体细分

  在 $\mathbf{\Sigma }$ 的定义域（不妨设为矩形） $E_{\lambda \mu }=[a,b]\times [c,d]$ 上引入分割

  $$\begin{array}{rl}& P_{\lambda }:a={\lambda }_0<{\lambda }_1<...<{\lambda }_{i-1}<{\lambda }_i<...<{\lambda }_M=b\\ & P_{\mu }:c={\mu }_0<{\mu }_1<...<{\mu }_{i-1}<{\mu }_i<...<{\mu }_N=d.\end{array}$$

  由此形成参数域 $E_{\lambda \mu }$ 上的规则分割

  $$E_{\lambda \mu }=\bigcup _{i=1}^M\bigcup _{j=1}^N{I}_{ij}.$$

  由曲面 $\mathbf{\Sigma }$ 的向量值映照， $P_{\lambda },P_{\mu }$ 分割即对应物理区域（曲面）上的分割。

• 局部近似

  考虑以物理域上的 $\mathit{A}\mathit{B}\times \mathit{C}\mathit{D}$ 近似替代曲面的某一部分的面积。

  设 $A,B,C,D$ 在参数域上分别对应 $({\lambda }_{i-1},{\mu }_{j-1}),({\lambda }_i,{\mu }_{j-1}),({\lambda }_i,{\mu }_j),({\lambda }_{i-1},{\mu }_j)$.

  则有

  $$\begin{array}{rl}& \mathit{A}\mathit{B}={\mathit{r}}_B-{\mathit{r}}_A=\mathbf{\Sigma }({\lambda }_i,{\mu }_{j-1})-\mathbf{\Sigma }({\lambda }_{i-1},{\mu }_{j-1})=\frac{\partial \mathbf{\Sigma }}{\partial \lambda }({\lambda }_{i-1},{\mu }_{j-1})\mathrm{\Delta }{\lambda }_i(1+o(1)).\\ & \mathit{A}\mathit{D}={\mathit{r}}_D-{\mathit{r}}_A=\mathbf{\Sigma }({\lambda }_{i-1},{\mu }_j)-\mathbf{\Sigma }({\lambda }_{i-1},{\mu }_{j-1})=\frac{\partial \mathbf{\Sigma }}{\partial \mu }({\lambda }_{i-1},{\mu }_{j-1})\mathrm{\Delta }{\mu }_i(1+o(1)).\end{array}$$

  故有

  $$\mathit{A}\mathit{B}\times \mathit{C}\mathit{D}=(\frac{\partial \mathbf{\Sigma }}{\partial \lambda }\times \frac{\partial \mathbf{\Sigma }}{\partial \mu })({\lambda }_{i-1},{\mu }_{j-1})\mathrm{\Delta }{\lambda }_i\mathrm{\Delta }{\lambda }_j.$$

  另特取 $({\xi }_i,{\eta }_j)\in I_{ij}=[{\lambda }_{i-1},{\lambda }_i]\times [{\mu }_{j-1},{\mu }_j]$，以 $f\circ \mathbf{\Sigma }({\xi }_i,{\eta }_i)$ 来表示曲面 $\mathit{A}\mathit{B}\times \mathit{C}\mathit{D}$ 上的等效面密度。则此部分的质量为

  $$m_{ij}=f\circ \mathbf{\Sigma }({\xi }_i,{\eta }_i){|\frac{\partial \mathbf{\Sigma }}{\partial \lambda }\times \frac{\partial \mathbf{\Sigma }}{\partial \mu }|}_{{\mathbb{R}}^3}({\lambda }_{i-1},{\mu }_{j-1})\mathrm{\Delta }{\lambda }_i\mathrm{\Delta }{\lambda }_j.$$

  求和可得总质量

  $$m=\sum _{i=1}^M\sum _{j=1}^{N}f\circ \mathbf{\Sigma }({\xi }_i,{\eta }_i){|\frac{\partial \mathbf{\Sigma }}{\partial \lambda }\times \frac{\partial \mathbf{\Sigma }}{\partial \mu }|}_{{\mathbb{R}}^3}({\lambda }_{i-1},{\mu }_{j-1})\mathrm{\Delta }{\lambda }_i\mathrm{\Delta }{\lambda }_j.$$

• 引入极限，即

  $$\underset{|P_{\lambda }\times P_{\mu }|\to 0}{lim}\sum _{i=1}^M\sum _{j=1}^N{m}_{ij}=\underset{|P_{\lambda }\times P_{\mu }|\to 0}{lim}\sum _{i=1}^M\sum _{j=1}^{N}f\circ \mathbf{\Sigma }({\xi }_i,{\eta }_i){|\frac{\partial \mathbf{\Sigma }}{\partial \lambda }\times \frac{\partial \mathbf{\Sigma }}{\partial \mu }|}_{{\mathbb{R}}^3}({\lambda }_{i-1},{\mu }_{j-1})\mathrm{\Delta }{\lambda }_i\mathrm{\Delta }{\lambda }_j.$$

• 将非标准部分和转化为标准部分和

  另有方案：考虑先选取 $({\xi }_i,{\eta }_j)\in I_{ij}=[{\lambda }_{i-1},{\lambda }_i]\times [{\mu }_{j-1},{\mu }_j]$，并用该点的切平面来近似曲面。

  $$m_{ij}=f\circ \mathbf{\Sigma }({\xi }_i,{\eta }_j){|\frac{\partial \mathbf{\Sigma }}{\partial \lambda }\times \frac{\partial \mathbf{\Sigma }}{\partial \mu }|}_{{\mathbb{R}}^3}({\xi }_i,{\eta }_j)\mathrm{\Delta }{\lambda }_i\mathrm{\Delta }{\mu }_j.$$

  求和，取极限，根据重积分的定义，有

  $$m=\underset{|P_{\lambda }\times P_{\mu }|\to 0}{lim}f\circ \mathbf{\Sigma }({\xi }_i,{\eta }_j){|\frac{\partial \mathbf{\Sigma }}{\partial \lambda }\times \frac{\partial \mathbf{\Sigma }}{\partial \mu }|}_{{\mathbb{R}}^3}({\xi }_i,{\eta }_j)\mathrm{\Delta }{\lambda }_i\mathrm{\Delta }{\mu }_j={\int }_{E_{\lambda \mu }}f\circ \mathbf{\Sigma }(\lambda ,\mu ){|\frac{\partial \mathbf{\Sigma }}{\partial \lambda }\times \frac{\partial \mathbf{\Sigma }}{\partial \mu }|}_{{\mathbb{R}}^3}(\lambda ,\mu )\text{d}\lambda \text{d}\mu .$$

第一类曲面积分的极限表示 —— 基于切平面近似

如有 $f\circ \mathbf{\Sigma }(\lambda ,\mu )\in \mathrm{\Re }(E_{\lambda \mu }),\frac{\partial \mathbf{\Sigma }}{\partial \lambda }(\lambda ,\mu ),\frac{\partial \mathbf{\Sigma }}{\partial \mu }(\lambda ,\mu )\in \mathrm{\Re }(E_{\lambda \mu })$，则有

$$\begin{array}{rl}& \mathrm{\exists }\underset{|P_{\lambda }\times P_{\mu }|\to 0}{lim}\sum _{i=1}^M\sum _{j=1}^{N}f\circ \mathbf{\Sigma }({\xi }_i,{\eta }_j){|\frac{\partial \mathbf{\Sigma }}{\partial \lambda }\times \frac{\partial \mathbf{\Sigma }}{\partial \mu }|}_{{\mathbb{R}}^3}({\lambda }_{i-1},{\mu }_{j-1})\mathrm{\Delta }{\lambda }_i\mathrm{\Delta }{\mu }_j\\ & ={\int }_{E_{\lambda \mu }}\mathbf{\Sigma }(\lambda ,\mu ){|\frac{\partial \mathbf{\Sigma }}{\partial \lambda }\times \frac{\partial \mathbf{\Sigma }}{\partial \mu }|}_{{\mathbb{R}}^3}(\lambda ,\mu )\text{d}\lambda \text{d}\mu .\end{array}$$

证明从略。

第一类曲面积分的极限表示 —— 基于平面近似

如有 $f\circ \mathbf{\Sigma }(\lambda ,\mu )\in \mathrm{\Re }(E_{\lambda \mu }),\frac{\partial \mathbf{\Sigma }}{\partial \lambda }(\lambda ,\mu ),\frac{\partial \mathbf{\Sigma }}{\partial \mu }(\lambda ,\mu )\in \mathrm{\Re }(E_{\lambda \mu })$，则有

$$\begin{array}{rl}& \mathrm{\exists }\underset{|P_{\lambda }\times P_{\mu }|\to 0}{lim}\sum _{i=1}^M\sum _{j=1}^{N}f\circ \mathbf{\Sigma }({\xi }_i,{\eta }_i)|[\mathbf{\Sigma }({\lambda }_i,{\mu }_{j-1})-\mathbf{\Sigma }({\lambda }_{i-1},{\mu }_{j-1})]\times [\mathbf{\Sigma }({\lambda }_{i-1},{\mu }_j)-\mathbf{\Sigma }({\lambda }_{i-1},{\mu }_{j-1})]|\\ & ={\int }_{E_{\lambda \mu }}\mathbf{\Sigma }(\lambda ,\mu ){|\frac{\partial \mathbf{\Sigma }}{\partial \lambda }\times \frac{\partial \mathbf{\Sigma }}{\partial \mu }|}_{{\mathbb{R}}^3}(\lambda ,\mu )\text{d}\lambda \text{d}\mu .\end{array}$$

证明从略。

第二类曲面积分（通量模型）

设有曲面 $\mathbf{\Sigma }\subset {\mathbb{R}}^3$ 上的向量场，其向量值映照表示为

$$\mathit{V}:\mathbf{\Sigma }\ni \left[\begin{array}{c}x\\ y\\ z\end{array}\right]\mapsto \mathit{V}(x,y,z)=\left[\begin{array}{c}P\\ Q\\ R\end{array}\right](x,y,z)\in {\mathbb{R}}^3.$$

第二类曲线积分的极限表示 —— 基于切平面近似

如有 $\mathit{V}\circ \mathbf{\Sigma }(\lambda ,\mu )\in \mathrm{\Re }(E_{\lambda \mu })$ 并且有 $\frac{\partial \mathbf{\Sigma }}{\partial \lambda }(\lambda ,\mu ),\frac{\partial \mathbf{\Sigma }}{\partial \mu }(\lambda ,\mu )\in \mathrm{\Re }(E_{\lambda \mu })$，则有

$$\begin{array}{rl}& \mathrm{\exists }\underset{|P_{\lambda }\times P_{\mu }|\to 0}{lim}\sum _{i=1}^M\sum _{j=1}^N\mathit{V}\circ \mathbf{\Sigma }({\xi }_i,{\eta }_j)·(\frac{\partial \mathbf{\Sigma }}{\partial \lambda }\times \frac{\partial \mathbf{\Sigma }}{\partial \mu })({\lambda }_{i-1},{\mu }_{j-1})\mathrm{\Delta }{\lambda }_i\mathrm{\Delta }{\mu }_j\\ & ={\int }_{E_{\lambda \mu }}\mathit{V}\circ \mathbf{\Sigma }(\lambda ,\mu )·(\frac{\partial \mathbf{\Sigma }}{\partial \lambda }\times \frac{\partial \mathbf{\Sigma }}{\partial \mu })(\lambda ,\mu )\text{d}\lambda \text{d}\mu .\end{array}$$

第二类曲线积分的极限表示 —— 基于平面近似

如有 $\mathit{V}\circ \mathbf{\Sigma }(\lambda ,\mu )\in \mathrm{\Re }(E_{\lambda \mu })$，并且有 $\frac{\partial \mathbf{\Sigma }}{\partial \lambda }(\lambda ,\mu ),\frac{\partial \mathbf{\Sigma }}{\partial \mu }(\lambda ,\mu )\in \mathrm{\Re }(E_{\lambda \mu })$，则有

$$\begin{array}{rl}& \mathrm{\exists }\underset{|P_{\lambda }\times P_{\mu }|\to 0}{lim}\sum _{i=1}^M\sum _{j=1}^N\mathit{V}\circ \mathbf{\Sigma }({\xi }_i,{\eta }_i)·[(\mathbf{\Sigma }({\lambda }_i,{\mu }_{j-1})-\mathbf{\Sigma }({\lambda }_{i-1},{\mu }_{j-1}))\times (\mathbf{\Sigma }({\lambda }_{i-1},{\mu }_j)-\mathbf{\Sigma }({\lambda }_{i-1},{\mu }_{j-1}))]\\ & ={\int }_{E_{\lambda \mu }}\mathit{V}\circ \mathbf{\Sigma }(\lambda ,\mu )·(\frac{\partial \mathbf{\Sigma }}{\partial \lambda }\times \frac{\partial \mathbf{\Sigma }}{\partial \mu })(\lambda ,\mu )\text{d}\lambda \text{d}\mu .\end{array}$$

上面这两个定理的证明与前面的证明类似，只需注意到以下表达式即可：

$$|(\mathit{a},\mathit{b}{)}_{{\mathbb{R}}^3}-(\mathit{a},\mathit{c}{)}_{{\mathbb{R}}^3}|\le {|\mathit{a}-\mathit{b}|}_{{\mathbb{R}}^3}|\mathit{c}{|}_{{\mathbb{R}}^3}.$$

标量场数乘单位法向量的积分

公式类似第二类曲面积分，只不过要把前面的矢量 $\mathit{V}\circ \mathbf{\Sigma }$ 变为标量 $p\circ \mathbf{\Sigma }$，这对应的是压力作用模型（压力 $=$ 压强 $\times$ 面积）。