# 開爾文環流定理

${\displaystyle {\frac {\mathrm {D} \Gamma }{\mathrm {D} t}}=0}$

## 數學證明

${\displaystyle \Gamma (t)=\oint _{C}{\boldsymbol {u}}\cdot \mathrm {d} {\boldsymbol {s}}}$

${\displaystyle {\frac {\mathrm {D} {\boldsymbol {u}}}{\mathrm {D} t}}=-{\frac {1}{\rho }}{\boldsymbol {\nabla }}p+{\boldsymbol {\nabla }}\Phi }$

${\displaystyle {\frac {\mathrm {D} \Gamma }{\mathrm {D} t}}=\oint _{C}{\frac {\mathrm {D} {\boldsymbol {u}}}{\mathrm {D} t}}\cdot \mathrm {d} {\boldsymbol {s}}+\oint _{C}{\boldsymbol {u}}\cdot {\frac {\mathrm {D} \mathrm {d} {\boldsymbol {s}}}{\mathrm {D} t}}}$

${\displaystyle \oint _{C}{\frac {\mathrm {D} {\boldsymbol {u}}}{\mathrm {D} t}}\cdot \mathrm {d} {\boldsymbol {s}}=\int _{A}{\boldsymbol {\nabla }}\times \left(-{\frac {1}{\rho }}{\boldsymbol {\nabla }}p+{\boldsymbol {\nabla }}\Phi \right)\cdot {\boldsymbol {n}}\,\mathrm {d} S=\int _{A}{\frac {1}{\rho ^{2}}}\left({\boldsymbol {\nabla }}\rho \times {\boldsymbol {\nabla }}p\right)\cdot {\boldsymbol {n}}\,\mathrm {d} S=0.}$

${\displaystyle {\frac {\mathrm {D} \mathrm {d} {\boldsymbol {s}}}{\mathrm {D} t}}=\left(\mathrm {d} {\boldsymbol {s}}\cdot {\boldsymbol {\nabla }}\right){\boldsymbol {u}}}$

${\displaystyle \oint _{C}{\boldsymbol {u}}\cdot {\frac {\mathrm {D} \mathrm {d} {\boldsymbol {s}}}{\mathrm {D} t}}=\oint _{C}{\boldsymbol {u}}\cdot \left(\mathrm {d} {\boldsymbol {s}}\cdot {\boldsymbol {\nabla }}\right){\boldsymbol {u}}={\frac {1}{2}}\oint _{C}{\boldsymbol {\nabla }}\left(|{\boldsymbol {u}}|^{2}\right)\cdot \mathrm {d} {\boldsymbol {s}}=0}$

${\displaystyle {\frac {\mathrm {D} \Gamma }{\mathrm {D} t}}=0}$

## 參考資料

1. ^ Sir W. Thomson. On Vortex Motion. Transactions of the Royal Society of Edinburgh. 1869, 25: 217–260.
2. ^ Kundu, P and Cohen, I: Fluid Mechanics, page 130. Academic Press 2002