치즈 오믈랫

발뜽에 불 떨어짐 ㅠㅠ

$$\nabla f= \mathbf{F } \mathrel{ \substack{ \xrightarrow{\text{(추가조건 없음)}\hspace{0em}} \\[-0.8ex] \xleftarrow[\text{($D$가 Open simply connected region)}]{} } ~~\mathbf{F} \text{가 path independent}} \iff \oint _\limits{C} \mathbf{F (r)\cdot} d \mathbf{r}  $$

$$ \nabla f=\mathbf{F }   \mathrel{ \substack{ \xrightarrow{\text{(추가조건 없음)}\hspace{0em}} \\[-0.8ex] \xleftarrow[\text{($D$가 Open simply connected region)}]{} }} \frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x }$$

(당연히 저 편도함수들이 연속해야 한다.)

$$\nabla f =\mathbf{F} \mathrel{ \substack{ \xrightarrow{\text{(추가조건 없음)}\hspace{0em}} \\[-0.8ex] \xleftarrow[\text{($D$가 Open simply connected region)}]{} } } \nabla \times \mathbf{F} =0 $$

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## 그린 정리

> $$\oint _\limits{C} \mathbf{F (r)\cdot }d  \mathbf{r} = \iint _\limits{D} -P_{y}{ + Q_{x}} ~dxdy  $$

## 스톡크스 정리

> $$\oint _\limits{C} \mathbf{F (r)\cdot }d  \mathbf{r} = \iint _\limits{S_{open}} (\nabla \times \mathbf{F}) \cdot \mathbf{ n} dS  $$

> $$= \iint _\limits{R} (\nabla \times \mathbf{F(r)}) \cdot \mathbf{(r_{u} \times {r_{v}})}dudv $$

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## 2d 다이버전스

>$$ \oint _\limits{C} \mathbf{F (r)\cdot n}d s =\iint _\limits{D} div(\mathbf{F}) dA$$

## 다이버전스 

> $$  {\Huge ∯}_\limits{S}  \mathbf{F(r) \cdot n}dS=\iiint _\limits{E} div(\mathbf{F}) dV $$

$\oint _\limits{C} \mathbf{F (r) \cdot} d \mathbf{r}=\iint _\limits{S} curl(\mathbf{F})\cdot d\mathbf{S}= \iint _\limits{S}  =curl(\mathbf{F})\cdot \mathbf{n}d{S}=\iint _\limits{S} curl(\mathbf{F(r}) )\cdot \mathbf{n} |\mathbf{r_{u}\times r_{v}}|dudv=\iint _\limits{S} curl(\mathbf{F})\cdot(\mathbf{r_{u}\times r_{v}})dudv$

$\oint _\limits{C} \mathbf{F (r) \cdot} d \mathbf{r}=\iint _\limits{S} curl(\mathbf{F})\cdot d\mathbf{S}= \iiint div(curl(\mathbf{F}))dV=0$