Gauss公式和Stokes公式

文章目錄

• Gauss公式 | 基本形式
• • 三維空間
  • 平面
• Stokes公式 | 基本形式
• • 三維曲面
  • 平面

Gauss公式 | 基本形式

∫ Ω ∇ ⋅ F ⃗  d μ = ∮ ∂ Ω F ⃗ ⋅ d σ ⃗ \int_\Omega \nabla \cdot\vec F \ \text d\mu = \oint_{\partial\Omega} \vec F \cdot \text d\vec \sigma ∫Ω​∇⋅F

 dμ=∮∂Ω​F

⋅dσ

三維空間

∫ Ω ∇ ⋅ F ⃗  d μ = ∮ ∂ Ω F ⃗ ⋅ d σ ⃗ ∫ Ω ( ∂ P ∂ x + ∂ Q ∂ y + ∂ R ∂ z ) d μ = ∮ ∂ Ω F ⃗ ⋅ n ⃗  d σ ∫ Ω ( ∂ P ∂ x + ∂ Q ∂ y + ∂ R ∂ z ) d x d y d z = ∮ ∂ Ω ( P , Q , R ) ⋅ n ⃗  d σ ∫ Ω ( ∂ P ∂ x + ∂ Q ∂ y + ∂ R ∂ z ) d x d y d z = ∮ ∂ Ω P d y d z + Q d z d x + R d x d y \begin{aligned} \int_\Omega \nabla \cdot\vec F\ \text d\mu &= \oint_{\partial\Omega} \vec F \cdot \text d \vec\sigma \\ \int_\Omega \left( \frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z} \right) \text d\mu &= \oint_{\partial\Omega} \vec F \cdot \vec n\ \text d \sigma \\ \int_\Omega \left( \frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z} \right) \text dx\text dy\text dz &= \oint_{\partial\Omega} (P,Q,R) \cdot \vec n \ \text d \sigma \\ \int_\Omega \left( \frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z} \right) \text dx\text dy\text dz &= \oint_{\partial\Omega} P\text d y \text d z+Q\text d z \text d x+R\text d x \text d y \end{aligned} ∫Ω​∇⋅F

 dμ∫Ω​(∂x∂P​+∂y∂Q​+∂z∂R​)dμ∫Ω​(∂x∂P​+∂y∂Q​+∂z∂R​)dxdydz∫Ω​(∂x∂P​+∂y∂Q​+∂z∂R​)dxdydz​=∮∂Ω​F

⋅dσ

=∮∂Ω​F

⋅n

 dσ=∮∂Ω​(P,Q,R)⋅n

 dσ=∮∂Ω​Pdydz+Qdzdx+Rdxdy​

平面

∫ D ∇ ⋅ F ⃗  d σ = ∮ ∂ D F ⃗ ⋅ n ⃗  d s ∫ D ( ∂ P ∂ x + ∂ Q ∂ y ) d σ = ∮ ∂ D ( P , Q ) ⋅ n ⃗  d s ∫ D ( ∂ P ∂ x + ∂ Q ∂ y ) d x d y = ∮ ∂ D P d y − Q d x \begin{aligned} \int_D \nabla \cdot\vec F\ \text d\sigma &= \oint_{\partial D} \vec F \cdot \vec n\ \text d s \\ \int_D \left( \frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}\right) \text d\sigma &= \oint_{\partial D} (P,Q)\cdot \vec n\ \text d s \\ \int_D \left( \frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}\right) \text d x \text d y &= \oint_{\partial D} P\text d y-Q\text d x \end{aligned} ∫D​∇⋅F

 dσ∫D​(∂x∂P​+∂y∂Q​)dσ∫D​(∂x∂P​+∂y∂Q​)dxdy​=∮∂D​F

⋅n

 ds=∮∂D​(P,Q)⋅n

 ds=∮∂D​Pdy−Qdx​

最後一式為Green公式的另一形式

Stokes公式 | 基本形式

∫ S ∇ × F ⃗ ⋅ d σ ⃗ = ∮ ∂ S F ⃗ ⋅ d r ⃗ \int_S \nabla \times \vec F \cdot \text d\vec\sigma = \oint_{\partial S} \vec F \cdot \text d \vec r ∫S​∇×F

⋅dσ

=∮∂S​F

⋅dr

三維曲面

∫ S ∇ × F ⃗ ⋅ n ⃗  d σ = ∮ ∂ S F ⃗ ⋅ d r ⃗ ∫ S ∣ e 1 ⃗ e 2 ⃗ e 3 ⃗ ∂ ∂ x ∂ ∂ y ∂ ∂ z P Q R ∣ ⋅ n ⃗   d σ = ∮ ∂ S ( P , Q , R ) ⋅ τ ⃗  d s ∫ S ( ∂ R ∂ y − ∂ Q ∂ z ) d y d z + ( ∂ P ∂ z − ∂ R ∂ x ) d z d x + ( ∂ Q ∂ x − ∂ P ∂ y ) d x d y = ∮ ∂ S P d x + Q d y + R d z \begin{aligned} \int_S \nabla \times \vec F \cdot \vec n\ \text d\sigma &= \oint_{\partial S} \vec F \cdot\text d \vec r \\ \int_S \begin{vmatrix} \vec{e_1} & \vec{e_2} & \vec{e_3}\\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z}\\ P & Q & R \end{vmatrix} \cdot \vec n\ d\sigma &= \oint_{\partial S}(P,Q,R) \cdot \vec\tau\ \text d s \\ \int_S \left(\frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z}\right) \text d y \text d z+ \left(\frac{\partial P}{\partial z}-\frac{\partial R}{\partial x}\right) \text d z \text d x+ \left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right) \text d x \text d y &= \oint_{\partial S}P\text d x+Q\text d y+R\text d z \end{aligned} ∫S​∇×F

⋅n

 dσ∫S​∣∣∣∣∣∣​e1​

​∂x∂​P​e2​

​∂y∂​Q​e3​

​∂z∂​R​∣∣∣∣∣∣​⋅n

 dσ∫S​(∂y∂R​−∂z∂Q​)dydz+(∂z∂P​−∂x∂R​)dzdx+(∂x∂Q​−∂y∂P​)dxdy​=∮∂S​F

⋅dr

=∮∂S​(P,Q,R)⋅τ

 ds=∮∂S​Pdx+Qdy+Rdz​

平面

∫ D ∇ × F ⃗ ⋅ n ⃗  d σ = ∮ ∂ D F ⃗ ⋅ d r ⃗ ∫ S ∣ e 1 ⃗ e 2 ⃗ e 3 ⃗ ∂ ∂ x ∂ ∂ y ∂ ∂ z P Q 0 ∣ ⋅ ( 0 , 0 , 1 )   d σ = ∮ ∂ S ( P , Q ) ⋅ τ ⃗  d s ∫ D ∣ ∂ ∂ x ∂ ∂ y Q R ∣ d σ = ∮ ∂ D ( P , Q ) ⋅ τ ⃗  d s ∫ D ( ∂ Q ∂ x − ∂ P ∂ y ) d x d y = ∮ ∂ D P d x + Q d y \begin{aligned} \int_D \nabla \times \vec F \cdot \vec n\ \text d \sigma &= \oint_{\partial D} \vec F \cdot \text d \vec r \\ \int_S \begin{vmatrix} \vec{e_1} & \vec{e_2} & \vec{e_3}\\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z}\\ P & Q & 0 \end{vmatrix} \cdot (0,0,1)\ d\sigma &= \oint_{\partial S}(P,Q) \cdot \vec\tau\ \text d s \\ \int_D \begin{vmatrix} \frac{\partial}{\partial x} & \frac{\partial}{\partial y}\\ Q & R \end{vmatrix} \text d \sigma &= \oint_{\partial D} (P,Q)\cdot \vec\tau\ \text d s \\ \int_D \left( \frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)\text d x \text d y &= \oint_{\partial D} P\text d x+Q\text d y \end{aligned} ∫D​∇×F

⋅n

 dσ∫S​∣∣∣∣∣∣​e1​

​∂x∂​P​e2​

​∂y∂​Q​e3​

​∂z∂​0​∣∣∣∣∣∣​⋅(0,0,1) dσ∫D​∣∣∣∣​∂x∂​Q​∂y∂​R​∣∣∣∣​dσ∫D​(∂x∂Q​−∂y∂P​)dxdy​=∮∂D​F

⋅dr

=∮∂S​(P,Q)⋅τ

 ds=∮∂D​(P,Q)⋅τ

 ds=∮∂D​Pdx+Qdy​

最後一式為Green公式的一般形式