# 高斯散度定理

（重定向自散度定理

## 定理

${\displaystyle \iiint _{\Omega }\left({\frac {\partial P}{\partial x}}+{\frac {\partial Q}{\partial y}}+{\frac {\partial R}{\partial z}}\right)dv=}$${\displaystyle \oiint }$${\displaystyle \scriptstyle \Sigma }$${\displaystyle P\,dy\land dz+Q\,dz\land dx+R\,dx\land dy}$

${\displaystyle \iiint _{\Omega }\left({\frac {\partial P}{\partial x}}+{\frac {\partial Q}{\partial y}}+{\frac {\partial R}{\partial z}}\right)dv=}$${\displaystyle \oiint }$${\displaystyle \scriptstyle \Sigma }$${\displaystyle (P\cos \alpha +Q\cos \beta +R\cos \gamma )\,dS}$

## 用散度表示

${\displaystyle \iiint _{\Omega }\mathrm {div} \mathbf {A} \,dv=}$${\displaystyle \oiint }$${\displaystyle \scriptstyle \Sigma }$${\displaystyle \mathbf {A} \cdot \mathbf {n} \,dS.}$

## 用向量表示

V代表有一间单闭曲面S为边界的体积，${\displaystyle \mathbf {f} }$是定义在V中和S上连续可微的向量场。如果${\displaystyle d\mathbf {S} }$是外法向向量面元，则

${\displaystyle \int _{S}\mathbf {f} \cdot d\mathbf {S} =\int _{V}\nabla \cdot \mathbf {f} dV}$

## 推论

• 对于标量函数g和向量场F的积，应用高斯公式可得：
${\displaystyle \iiint _{V}\left(\mathbf {F} \cdot \left(\nabla g\right)+g\left(\nabla \cdot \mathbf {F} \right)\right)dV=\iint _{\partial V}g\mathbf {F} \cdot d\mathbf {S} }$
• 对于两个向量场${\displaystyle \mathbf {F} \times \mathbf {G} }$的向量积，应用高斯公式可得：
${\displaystyle \iiint _{V}\left(\mathbf {G} \cdot \left(\nabla \times \mathbf {F} \right)-\mathbf {F} \cdot \left(\nabla \times \mathbf {G} \right)\right)\,dV=\iint _{\partial V}\left(\mathbf {F} \times \mathbf {G} \right)\cdot d\mathbf {S} }$
• 对于标量函数f和非零常向量的积，应用高斯公式可得：
${\displaystyle \iiint _{V}\nabla f\,dV=\iint \limits _{\partial V}f\,d\mathbf {S} }$
• 对于向量场F和非零常向量的向量积，应用高斯公式可得：
${\displaystyle \iiint _{V}\nabla \times \mathbf {F} \,dV=\iint _{\partial V}d\mathbf {S} \times \mathbf {F} .}$

## 例子

${\displaystyle \oiint }$${\displaystyle \scriptstyle S}$${\displaystyle \mathbf {F} \cdot \mathbf {n} \,dS,}$

${\displaystyle S=\left\{x,y,z\in \mathbb {R} ^{3}\ :\ x^{2}+y^{2}+z^{2}=1\right\}.}$

F向量场

${\displaystyle \mathbf {F} =2x\mathbf {i} +y^{2}\mathbf {j} +z^{2}\mathbf {k} .}$

${\displaystyle \iiint _{W}(\nabla \cdot \mathbf {F} )\,dV=2\iiint _{W}(1+y+z)\,dV=2\iiint _{W}dV+2\iiint _{W}y\,dV+2\iiint _{W}z\,dV.}$

${\displaystyle W=\left\{x,y,z\in \mathbb {R} ^{3}\ :\ x^{2}+y^{2}+z^{2}\leq 1\right\}.}$

${\displaystyle \iiint _{W}y\,dV=\iiint _{W}z\,dV=0.}$

${\displaystyle \oiint }$${\displaystyle \scriptstyle S}$${\displaystyle \mathbf {F} \cdot \mathbf {n} \,{d}S=2\iiint _{W}\,dV={\frac {8\pi }{3}},}$

## 二阶张量的高斯公式

1. 两个向量${\displaystyle {\boldsymbol {a}}}$${\displaystyle {\boldsymbol {b}}}$并排放在一起所形成的量${\displaystyle {\boldsymbol {ab}}}$被称为向量${\displaystyle {\boldsymbol {a}}}$${\displaystyle {\boldsymbol {b}}}$并矢并矢张量。要注意，一般来说，${\displaystyle {\boldsymbol {ab}}\neq {\boldsymbol {ba}}}$
2. ${\displaystyle {\boldsymbol {ab}}=0}$的充分必要条件是${\displaystyle {\boldsymbol {a}}=0}$${\displaystyle {\boldsymbol {b}}=0}$
3. 二阶张量就是有限个并矢的线性组合。
4. ${\displaystyle {\boldsymbol {ab}}}$分别线性地依赖于${\displaystyle {\boldsymbol {a}}}$${\displaystyle {\boldsymbol {b}}}$
5. 二阶张量${\displaystyle \mathbf {T} }$和向量${\displaystyle {\boldsymbol {a}}}$的缩并${\displaystyle \mathbf {T} \cdot {\boldsymbol {a}}}$以及${\displaystyle {\boldsymbol {a}}\cdot \mathbf {T} }$${\displaystyle \mathbf {T} }$${\displaystyle {\boldsymbol {a}}}$都是线性的。
6. 特别是，当${\displaystyle \mathbf {T} ={\boldsymbol {uv}}}$时，
${\displaystyle \mathbf {T} \cdot {\boldsymbol {a}}=({\boldsymbol {uv}})\cdot {\boldsymbol {a}}={\boldsymbol {u}}({\boldsymbol {v}}\cdot {\boldsymbol {a}})\,,\qquad {\boldsymbol {a}}\cdot \mathbf {T} ={\boldsymbol {a}}\cdot ({\boldsymbol {uv}})=({\boldsymbol {a}}\cdot {\boldsymbol {u}})\,{\boldsymbol {v}},}$

${\displaystyle ({\boldsymbol {a}}\times {\boldsymbol {b}})\times {\boldsymbol {c}}=({\boldsymbol {a}}\cdot {\boldsymbol {c}})\,{\boldsymbol {b}}-({\boldsymbol {b}}\cdot {\boldsymbol {c}})\,{\boldsymbol {a}}=-({\boldsymbol {ab}}-{\boldsymbol {ba}})\cdot {\boldsymbol {c}}\,,\qquad {\boldsymbol {a}}\times ({\boldsymbol {b}}\times {\boldsymbol {c}})=({\boldsymbol {a}}\cdot {\boldsymbol {c}})\,{\boldsymbol {b}}-({\boldsymbol {a}}\cdot {\boldsymbol {b}})\,{\boldsymbol {c}}=-{\boldsymbol {a}}\cdot ({\boldsymbol {bc}}-{\boldsymbol {cb}})\,.}$

1. ${\displaystyle \mathbf {T} '}$仍然是一个二阶张量，并且线性地依赖于${\displaystyle \mathbf {T} }$
2. ${\displaystyle ({\boldsymbol {uv}})'={\boldsymbol {vu}}}$

${\displaystyle \iint _{S}{\hat {\boldsymbol {n}}}\cdot \mathbf {T} \,dS=\iiint _{V}\nabla \cdot \mathbf {T} \,dV\,,\qquad \iint _{S}\mathbf {T} \cdot {\hat {\boldsymbol {n}}}\,dS=\iiint _{V}\nabla \cdot \mathbf {T} '\,dV\,.}$

${\displaystyle {\boldsymbol {e}}_{i}\cdot {\boldsymbol {F}}={\boldsymbol {e}}_{i}\cdot \iint _{S}\mathbf {T} \cdot {\hat {\boldsymbol {n}}}\,dS=\iint _{S}{\boldsymbol {e}}_{i}\cdot \mathbf {T} \cdot {\hat {\boldsymbol {n}}}\,dS=\iint _{S}T^{ij}{\boldsymbol {e}}_{j}\cdot {\boldsymbol {n}}\,dS\,.}$

${\displaystyle {\boldsymbol {e}}_{i}\cdot {\boldsymbol {F}}=\iiint _{V}\nabla \cdot (T^{ij}{\boldsymbol {e}}_{j})\,dV=\iiint _{V}{\frac {\partial T^{ij}}{\partial x^{j}}}\,dV\,,}$

${\displaystyle {\boldsymbol {F}}={\boldsymbol {e}}_{i}\,({\boldsymbol {e}}_{i}\cdot {\boldsymbol {F}})={\boldsymbol {e}}_{i}\iint _{S}{\frac {\partial T^{ij}}{\partial x^{j}}}\,dV=\iiint _{V}{\boldsymbol {e}}_{i}{\frac {\partial T^{ij}}{\partial x^{j}}}\,dV=\iiint _{V}\nabla \cdot \mathbf {T} '\,dV\,.}$

## 参考文献

1. ^ 同济大学数学系 编. 高等数学(第六版)(下册). 北京: 高等教育出版社, 2007
2. ^ 谢树艺编. 高等学校教材•工程数学:向量分析与场论(第3版). 北京: 高等教育出版社, 2005