高斯公式,斯克托斯公式

高斯公式

设在闭区域Ω\Omega上有稳定流动的、不可压缩的物体（假定流体的密度为1）的速度场
v⃗ (x,y,z)=P(x,y,z)i⃗ +Q(x,y,z)j⃗ +R(x,y,z)k⃗ \vec{v}(x,y,z)=P(x,y,z)\vec i+Q(x,y,z)\vec j+R(x,y,z)\vec k
∭div v⃗ dv=∬Σv⃗ ⋅n⃗ dS\displaystyle \iiint div \ \vec v dv= \iint_{\Sigma}\vec v \cdot \vec ndS

其中
v⃗ =(P,Q,R)\vec{v}=(P,Q,R),
n⃗ =(cosα,cosβ,cosγ)\vec n = (cos \alpha, cos \beta, cos \gamma),
dS⃗ =n⃗ ⋅dS=(dydz,dzdx,dxdy)d\vec S = \vec n\cdot dS=(dydz, dzdx, dxdy)
div v⃗ =∇⋅v⃗ =∂P∂x+∂Q∂y+∂R∂zdiv \ \vec v = \nabla \cdot \vec v= \frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z}

单位时间内流体经过曲面Σ\Sigma流向指定侧的流体总质量就是

∬Σv⃗ ⋅n⃗ dS=∬Σ(P,Q,R)⋅(cosα,cosβ,cosγ)dS=∬Σ(Pcosα+Qcosβ+Rcosγ)dS=∬ΣPdydz+Qdzdx+Rdxdy=∭Ω∂P∂x+∂Q∂y+∂R∂zdv=∭Ωdiv v⃗ dv

\begin{align}\displaystyle \iint_{\Sigma}\vec v \cdot \vec ndS&=\iint_{\Sigma}(P,Q,R)\cdot(cos \alpha, cos \beta, cos \gamma)dS\\&= \iint_{\Sigma}(Pcos \alpha + Qcos \beta + Rcos \gamma)dS\\&=\iint_{\Sigma}Pdydz+Qdzdx+Rdxdy\\&=\iiint_{\Omega}\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z}dv\\&=\iiint_{\Omega}div \ \vec v dv \end{align}

斯克托斯公式

∬Σrotv⃗ ⋅n⃗ dS=∮Γv⃗ ⋅τ⃗ ds\displaystyle \iint_{\Sigma}rot\vec v \cdot \vec ndS=\unicode{x222E}_\Gamma \vec v\cdot \vec \tau ds
意义为:速度场v⃗ \vec v沿有向闭曲线Γ\Gamma的环流量等于速度场v⃗ \vec v的旋度通过曲面Σ\Sigma的通量,这里Γ\Gamma的正向与Σ\Sigma的侧应符合右手规则.

其中
v⃗ =(P,Q,R)\vec{v}=(P,Q,R),
n⃗ =(cosα,cosβ,cosγ)\vec n = (cos \alpha, cos \beta, cos \gamma),
dS⃗ =n⃗ ⋅dS=(dydz,dzdx,dxdy)d\vec S = \vec n\cdot dS=(dydz, dzdx, dxdy),
rotv⃗ =∇×v⃗ =(∂R∂y−∂Q∂z)i⃗ +(∂P∂z−∂R∂x)j⃗ +(∂Q∂x−∂P∂y)k⃗ rot \vec v = \nabla \times \vec v =(\frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z})\vec i+(\frac{\partial P}{\partial z}-\frac{\partial R}{\partial x})\vec j+(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y})\vec k
τ⃗ ds=ds⃗ =dxi⃗ +dyj⃗ +dzk⃗ \vec \tau ds=d \vec s=dx \vec i + dy \vec j + dz \vec k

∬Σ(∂R∂y−∂Q∂z)dydz+(∂P∂z−∂R∂x)dzdx+(∂Q∂x−∂P∂y)dxdy=∮ΓPdx+Qdy+Rdz\displaystyle \iint_{\Sigma}(\frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z})dydz+(\frac{\partial P}{\partial z}-\frac{\partial R}{\partial x})dzdx+(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y})dxdy=\unicode{x222E}_\Gamma Pdx+Qdy+Rdz