# 散度

## 定義

${\displaystyle \Phi _{\mathbf {A} }(\Sigma )=\iint \limits _{\Sigma }\mathbf {A} \cdot \mathbf {n} \mathrm {d} S}$

${\displaystyle \operatorname {div} \mathbf {A} (x)=\lim _{\delta V\rightarrow \{x\}}\oint _{\Sigma }{\mathbf {A} \cdot \mathbf {n} \over |\delta V|}\;dS=\lim _{\delta V\rightarrow \{x\}}{\frac {\Phi _{\mathbf {A} }(\Sigma )}{|\delta V|}}}$[2]:4

## 物理意義

${\textstyle \mathbf {P} }$為場域V中的一點，現作包圍${\textstyle \mathbf {P} }$點的任一閉合曲面${\textstyle \mathbf {S} }$${\displaystyle \Delta V}$是S面所圍的區域。那麼：

${\displaystyle \oint _{S}{\mathbf {A} \cdot d\mathbf {S} }=\iiint \limits _{\Delta V}\mathrm {div} \mathbf {A} dV\;\;\;\;(1)}$

${\displaystyle \iiint \limits _{\Delta V}\mathrm {div} \mathbf {A} dV=\mathrm {(} \mathrm {div} \mathbf {A} \mathrm {)} _{x}\cdot |\Delta V|\;\;\;\;(2)}$

${\displaystyle \mathrm {(} \mathrm {div} \mathbf {A} \mathrm {)} _{x}={\frac {1}{|\Delta V|}}\oint _{S}{\mathbf {A} \cdot d\mathbf {S} }}$

${\displaystyle \Delta V}$向點P收縮，則 ${\displaystyle x}$ 點就趨向於P點，所以在P點的散度可由下列極限表示

${\displaystyle (\mathrm {div} \mathbf {A} )_{P}=\lim _{\Delta V\rightarrow P}{\frac {1}{|\Delta V|}}\oint _{S}{\mathbf {A} \cdot d\mathbf {S} }}$

${\displaystyle (\mathrm {div} \mathbf {A} )_{P}=\lim _{\Delta V\rightarrow P}{\frac {1}{|\Delta V|}}\oint _{S}{\mathbf {A} \cdot d\mathbf {S} }=\lim _{\Delta V\rightarrow P}{\frac {\Delta \Phi }{|\Delta V|}}={\frac {d\Phi }{dV}}}$

## 分量表示

### 直角座標系

${\displaystyle \mathbf {A} (x,y,z)=P(x,y,z)\mathbf {i} +Q(x,y,z)\mathbf {j} +R(x,y,z)\mathbf {k} }$

${\displaystyle \operatorname {div} \mathbf {A} =\nabla \cdot \mathbf {A} ={\frac {\partial P}{\partial x}}+{\frac {\partial Q}{\partial y}}+{\frac {\partial R}{\partial z}}}$

### 圓柱座標系

${\displaystyle \mathbf {A} =A_{r}(r,\varphi ,z){\boldsymbol {e}}_{r}+A_{z}(r,\varphi ,z){\boldsymbol {e}}_{z}+A_{\varphi }(r,\varphi ,z){\boldsymbol {e}}_{\varphi },}$

${\displaystyle \operatorname {div} \,\mathbf {A} =\nabla \cdot \mathbf {A} ={\frac {1}{r}}{\frac {\partial }{\partial r}}(rA_{r})+{\frac {1}{r}}{\frac {\partial A_{\varphi }}{\partial \varphi }}+{\frac {\partial A_{z}}{\partial z}}\,.}$

### 球座標系

${\displaystyle \mathbf {A} =A_{r}(r,\theta ,\varphi ){\boldsymbol {e}}_{r}+A_{\theta }(r,\theta ,\varphi ){\boldsymbol {e}}_{\theta }+A_{\varphi }(r,\theta ,\varphi ){\boldsymbol {e}}_{\varphi },}$

${\displaystyle \operatorname {div} \,\mathbf {A} =\nabla \cdot \mathbf {A} ={\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}(r^{2}A_{r})+{\frac {1}{r\sin \theta }}{\frac {\partial }{\partial \theta }}(\sin \theta \,A_{\theta })+{\frac {1}{r\sin \theta }}{\frac {\partial A_{\varphi }}{\partial \varphi }}.}$

## 性質

${\displaystyle \operatorname {div} (a\mathbf {F} +b\mathbf {G} )=a\;\operatorname {div} (\mathbf {F} )+b\;\operatorname {div} (\mathbf {G} )}$

${\displaystyle \varphi }$是純量函數，F是向量場，則它們的乘積的散度為[2]:8

${\displaystyle \operatorname {div} (\varphi \mathbf {F} )=\operatorname {grad} (\varphi )\cdot \mathbf {F} +\varphi \;\operatorname {div} (\mathbf {F} ),}$${\displaystyle \nabla \cdot (\varphi \mathbf {F} )=(\nabla \varphi )\cdot \mathbf {F} +\varphi \;(\nabla \cdot \mathbf {F} ).}$

${\displaystyle \operatorname {div} (\mathbf {F} \times \mathbf {G} )=\operatorname {curl} (\mathbf {F} )\cdot \mathbf {G} \;-\;\mathbf {F} \cdot \operatorname {curl} (\mathbf {G} ),}$${\displaystyle \nabla \cdot (\mathbf {F} \times \mathbf {G} )=(\nabla \times \mathbf {F} )\cdot \mathbf {G} -\mathbf {F} \cdot (\nabla \times \mathbf {G} ).}$

${\displaystyle \operatorname {div} \,\operatorname {grad} f=\nabla \cdot \nabla f=\Delta f}$ (在 ${\displaystyle \mathbb {R} ^{n}}$ 的向量分析中 ${\displaystyle \nabla \cdot \nabla f}$ 也寫作 ${\displaystyle \nabla ^{2}f)}$

## 高斯散度定理

${\displaystyle \iiint \limits _{V}\mathrm {div} \mathbf {A} dv=\int \!\!\!\!\int _{S}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;\bigcirc \,\,\mathbf {A} \cdot \mathbf {n} dS}$

## 歷史

${\displaystyle \nabla \sigma =({\boldsymbol {i}}{\frac {\mathrm {d} }{\mathrm {d} x}}+{\boldsymbol {j}}{\frac {\mathrm {d} }{\mathrm {d} y}}+{\boldsymbol {k}}{\frac {\mathrm {d} }{\mathrm {d} z}})(B{\boldsymbol {i}}+C{\boldsymbol {j}}+D{\boldsymbol {k}})}$
${\displaystyle =-\left({\frac {\mathrm {d} B}{\mathrm {d} x}}+{\frac {\mathrm {d} C}{\mathrm {d} y}}+{\frac {\mathrm {d} D}{\mathrm {d} z}}\right)+\left(\left({\frac {\mathrm {d} D}{\mathrm {d} y}}-{\frac {\mathrm {d} C}{\mathrm {d} z}}\right){\boldsymbol {i}}+\left({\frac {\mathrm {d} B}{\mathrm {d} z}}-{\frac {\mathrm {d} D}{\mathrm {d} x}}\right){\boldsymbol {j}}+\left({\frac {\mathrm {d} C}{\mathrm {d} x}}-{\frac {\mathrm {d} B}{\mathrm {d} y}}\right){\boldsymbol {k}}\right)}$

## 參考來源

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7. ^ 梯度、散度、旋度和調和量在球坐標系中的表達式. 浙江大學遠程教育學院. [2012-08-18].
8. Michael J. Crowe. A History of Vector Analysis: The Evolution of the Idea of a Vectorial System. Dover books on advanced mathematics, 2nd Edition. 1994. ISBN 9780486679105.