# 保守向量场

（重定向自保守场

## 定义

${\displaystyle \mathbf {v} =\nabla \varphi .}$

## 路径无关

${\displaystyle \int _{P}\mathbf {v} \cdot d\mathbf {r} =\varphi (A)-\varphi (B).}$

${\displaystyle \oint \mathbf {v} \cdot d\mathbf {r} =0}$

## 无旋向量场

${\displaystyle \nabla \times \mathbf {v} =0.}$

${\displaystyle \nabla \times \nabla \varphi =0.}$

${\displaystyle \mathbf {v} =\left({\frac {-y}{x^{2}+y^{2}}},{\frac {x}{x^{2}+y^{2}}},0\right).}$

${\displaystyle \mathbf {v} }$存在，且在${\displaystyle S}$内的每一个点旋度都是零；因此${\displaystyle \mathbf {v} }$是无旋的。但是，${\displaystyle \mathbf {v} }$沿着${\displaystyle x,y}$平面内的单位圆的环量等于${\displaystyle 2\pi }$。因此${\displaystyle \mathbf {v} }$不具有路径无关的性质，所以不是保守的。

## 无旋流动

${\displaystyle {\boldsymbol {\omega }}=\nabla \times \mathbf {u} .}$

## 保守力

${\displaystyle \mathbf {F} _{G}=-{\frac {GmM{\hat {\mathbf {r} }}}{r^{2}}},}$

${\displaystyle \Phi _{G}=-{\frac {GmM}{r}}}$

${\displaystyle W=\oint \mathbf {F} \cdot d\mathbf {r} =0.}$

## 参考文献

• George B. Arfken and Hans J. Weber, Mathematical Methods for Physicists, 6th edition, Elsevier Academic Press (2005)
• D. J. Acheson, Elementary Fluid Dynamics, Oxford University Press (2005)