5.4 $B=[4D$NJ]B8(B

Stokes $B$NDjM}$K$h$l$P(B

$$\mathit{\Gamma}=\int_S \curl \Vector{v}(t,x)\cdot \Vector{n}\D S.$$

$B$?$@$7(B $S$ $B$O(B $C$ $B$r6-3&$H$9$k$h$&$J6JLL(B (unique $B$G$O$J$$(B)$B!"(B $\Vector{n}$ $B$O(B $S$ $B$NC10LK!@~%Y%/%H%k$G$"$k!#(B

Stokes $B$NDjM}(B

$U$ $B$O(B $\R^2$ $B$N3+=89g(B, $g\colon\overline U\to\R^3$ $B$O(B$C^2$-$B5i7B?t$D6J(B $BLL(B, $D$ $B$r6hJ,E*(B $C^1$-$B5i$N6-3&$r;}$DM-3&NN0h$G(B $\overline D\subset U$ $B$rK~$?$9!#(B $f(\overline D=S)$ $B6JLL!"(B$\rd S$ $B$O(B $S$ $B$N%A%'%$%s!"(B $F$ $B$O(B $S$ $B$r4^$`(B $\R^3$ $B$N3+=89g(B $V$ $B$GDj5A$5$l$?(B $C^1$-$B5i%Y%/%H%k>l!#(B $B$3$N$H$-(B

$$\dint_S\rot F\cdot\vec{\D A}=\int_{\rd S}F\cdot{\D s}.$$

($B8~$-$NOC$rFM$C9~$s$@$1$l$I!"? $BN.BN$H$H$b$KF0$/JD6J@~(B $C$ $B$r9M$($k!#(B $B$3$3$G(B $C$ $B$,N.BN$H$H$b$KF0$/$H$O!"(B$C$ $B$,(B

$$\frac{\rd \Vector{x}}{\rd t}=(t,s)=\Vector{v}{t,\Vector{x}(t,s)}$$

$B$rK~$?$9%Q%i%a!<%?! $\Vector{x}(t,s)$ ( $0\le s\le 1$) $B$r;}$D$3$H!#(B

$B!&%"!&v't%#%(B(William Thomson, 1st Baron $B!A(B (1824$B!=(B1907)) $B!T1Q9q$N?t3X helicity $B$H$O(B

$$\int_\Omega\Vector{v}\cdot\curl\Vector{v}\Dx= \int_\Omega\sum_{k=1}^3 v_k\omega_k\Dx$$

$B$GDj5A$5$l$kNL$N$3$H$G$"$k!#(B

l9g$K$bF1$8<0$,@.N)$9$k!#(B \end{jtheorem}">