# 曲面積分

$\int_M \mathrm{d}\omega = \oint_{\partial M} \omega$

## 純量場的面積分

$\iint_S f \,dS = \iint_T f(\mathbf{x}(s, t)) \left|\frac{\partial \mathbf{x}}{\partial s}\times \frac{\partial \mathbf{x}}{\partial t}\right| ds\, dt$

$A = \iint_S \,dS = \iint_T \left|\frac{\partial \mathbf{r}}{\partial x}\times \frac{\partial \mathbf{r}}{\partial y}\right| dx\, dy$

\begin{align} 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{align}

## 向量場的面積分

$\int_S {\mathbf v}\cdot \,d{\mathbf {S}} = \int_S ({\mathbf v}\cdot {\mathbf n})\,dS=\iint_T {\mathbf v}(\mathbf{x}(s, t))\cdot \left({\partial \mathbf{x} \over \partial s}\times {\partial \mathbf{x} \over \partial t}\right) ds\, dt.$

## 微分2-形式的面積分

$f=f_{1} dx \wedge dy + f_{2} dy \wedge dz + f_{3} dz \wedge dx$

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

$\iint_D \left[ f_{1} ( \mathbf{x} (s,t)) \frac{\partial(x,y)}{\partial(s,t)} + f_{2} ( \mathbf{x} (s,t))\frac{\partial(y,z)}{\partial(s,t)} + f_{3} ( \mathbf{x} (s,t))\frac{\partial(z,x)}{\partial(s,t)} \right]\, ds dt$

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

S的法向量。

## 參考

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