# 散度

sàn[1]divergence）是向量分析中的一个向量算子，将向量空间上的一个向量场（矢量场）对应到一个标量场上。散度描述的是向量场里一个点是汇聚点还是发源点，形象地说，就是这包含这一点的一个微小体元中的向量是“向外”居多还是“向内”居多。

## 定义

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

${\displaystyle \operatorname {div} \mathbf {A} (x)=\lim _{\delta V\rightarrow \{x\}}\oint _{\Sigma }{\mathbf {A} \cdot \mathbf {n} \over |\delta V|}\,\mathrm {d} S=\lim _{\delta V\rightarrow \{x\}}{\frac {\Phi _{\mathbf {A} }(\Sigma )}{|\delta V|}}}$[3]:4

## 物理意义

${\textstyle \mathbf {A} }$为场域V中的一点，现作包围${\textstyle \mathbf {A} }$点的任一闭合曲面${\textstyle \mathbf {S} }$${\displaystyle \Delta V}$是S面所围的区域。那么：${\displaystyle \oint _{S}{\mathbf {A} \cdot \mathrm {d} \mathbf {S} }=\iiint \limits _{\Delta V}\mathrm {div} \mathbf {A} \mathrm {d} V\;\;\;\;(1)}$

${\displaystyle \Delta V}$向点P收缩，则 ${\displaystyle x}$ 点就趋向于P点，所以在P点的散度可由下列极限表示${\displaystyle (\mathrm {div} \mathbf {A} )_{P}=\lim _{\Delta V\rightarrow P}{\frac {1}{|\Delta V|}}\oint _{S}{\mathbf {A} \cdot \mathrm {d} \mathbf {S} }}$

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

## 分量表示

### 直角坐标系

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

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

### 圆柱坐标系

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

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

### 球坐标系

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

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

## 性质

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

${\displaystyle \varphi }$是标量函数，F是向量场，则它们的乘积的散度为[3]:7

${\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

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

## 历史

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

## 参考文献

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