GNU Free Documentation License . .

( « »)
: ,

  , . . . .

[]

M n p- \sigma \omega p-1 C^1 (1\leqslant p\leqslant n). , \partial\sigma ,

\int\limits_\sigma d\omega=\int\limits_{\partial\sigma}\omega,

d\omega \omega.

, . M.

[]

[]

l, a b ( ) . \omega C^1  f.

\int\limits_l df=\int\limits_l f'\,dx=\int\limits_a^b f'\,dx=f(b)-f(a).

[]

M  , D  - . , x y  L\,dx+M\,dy, D

\int\limits_{\partial D} L\,dx+M\,dy=\iint\limits_D\left(\frac{\partial M}{\partial x}-\frac{\partial L}{\partial y}\right)\,dx\,dy.

.

[]  

\Sigma  - (p=2) (n=3), \mathbf{F}  . \partial\Sigma () \Sigma, :

\int\limits_\Sigma\mathrm{rot}\,\mathbf{F}\,d\mathbf{\Sigma}=\int\limits_{\partial\Sigma}\mathbf{F}\,d\mathbf{r}

:

\iint\limits_{\Sigma}\left(\frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z}\right)\,dy\,dz+\left(\frac{\partial P}{\partial z}-\frac{\partial R}{\partial x}\right)\,dz\,dx+\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)\,dx\,dy=\int\limits_{\partial\Sigma}P\,dx+Q\,dy+R\,dz.

[]

\partial V  - (p=n-1), V n- . \partial V:

\int\limits_V\mathrm{div}\,\mathbf{F}\,dV=\int\limits_{\partial V}\mathbf{F}\,d\mathbf{\Sigma}.

:

\int\limits_{\partial V}\mathbf{F}\,d\mathbf{\Sigma}=\int\limits_V\left(\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z}\right)\,d\mathbf{V}

\iint\limits_{\partial V}P\,dy\,dz+Q\,dz\,dx+R\,dx\,dy=\iiint\limits_V\left(\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z}\right)\,dx\,dy\,dz.

[]

[] .