# 曲面積分

## 純量場的面積分

${\displaystyle \iint _{S}f\,dS=\iint _{T}f(\mathbf {x} (s,t))\left|{\frac {\partial \mathbf {x} }{\partial s}}\times {\frac {\partial \mathbf {x} }{\partial t}}\right|ds\,dt}$

${\displaystyle A=\iint _{S}\,dS=\iint _{T}\left|{\frac {\partial \mathbf {r} }{\partial x}}\times {\frac {\partial \mathbf {r} }{\partial y}}\right|dx\,dy}$

{\displaystyle {\begin{aligned}A&{}=\iint _{T}\left\|\left(1,0,{\partial f \over \partial x}\right)\times \left(0,1,{\partial f \over \partial y}\right)\right\|dx\,dy\\&{}=\iint _{T}\left\|\left(-{\partial f \over \partial x},-{\partial f \over \partial y},1\right)\right\|dx\,dy\\&{}=\iint _{T}{\sqrt {\left({\partial f \over \partial x}\right)^{2}+\left({\partial f \over \partial y}\right)^{2}+1}}\,\,dx\,dy\end{aligned}}}

## 向量場的面積分

${\displaystyle \int _{S}{\mathbf {v} }\cdot \,d{\mathbf {S} }=\int _{S}({\mathbf {v} }\cdot {\mathbf {n} })\,dS=\iint _{T}{\mathbf {v} }(\mathbf {x} (s,t))\cdot \left({\partial \mathbf {x} \over \partial s}\times {\partial \mathbf {x} \over \partial t}\right)ds\,dt.}$

## 微分2-形式的面積分

${\displaystyle \omega =f_{x}dy\wedge dz+f_{y}dz\wedge dx+f_{z}dx\wedge dy}$

${\displaystyle \mathbf {x} (s,t)=(x(s,t),y(s,t),z(s,t))\!}$

${\displaystyle \iint _{D}\left[f_{x}(\mathbf {x} (s,t)){\frac {\partial (y,z)}{\partial (s,t)}}+f_{y}(\mathbf {x} (s,t)){\frac {\partial (z,x)}{\partial (s,t)}}+f_{z}(\mathbf {x} (s,t)){\frac {\partial (x,y)}{\partial (s,t)}}\right]\,dsdt}$

${\displaystyle {\partial \mathbf {x} \over \partial s}\times {\partial \mathbf {x} \over \partial t}=\left({\frac {\partial (y,z)}{\partial (s,t)}},{\frac {\partial (z,x)}{\partial (s,t)}},{\frac {\partial (x,y)}{\partial (s,t))}}\right)}$

S的法向量。利用微分形式（2-form）的變數變換，我們有

${\displaystyle \int _{S}\omega =\iint _{S}(f_{x},f_{y},f_{z})\cdot d\mathbf {S} =\iint _{S}(f_{x},f_{y},f_{z})\cdot \mathbf {n} \,dS}$

## 參考

• Leathem, J. G. Volume and Surface Integrals Used in Physics. Cambridge, England: University Press, 1905