APPENDIX - Vector Calculus

1. Vector Calculus 3대 공식

\begin{align*} \int_C \nabla f d\vec{l}=f(\vec{b})-f(\vec{a}) \\ \\ \oint_S \vec{F}\cdot d\vec{a} = \int_V \nabla \cdot \vec{F} d\tau \\ \\ \oint_C \vec{F} \cdot d\vec{l}=\int_S (\nabla \times \vec{F}) \cdot d\vec{a}\end{align*}

 

 

---

 

2. 변형하여 값이 Vector 공식 (for a constant vector c)

\begin{align*} \int_V  \nabla f d\tau = \oint_S fd\vec{a}\end{align*}

pf.

\begin{align*} \vec{c}\cdot \oint_S fd\vec{a} &=\oint_S (f\vec{c})\cdot d\vec{a} = \int_V \nabla \cdot (f\vec{c}) d\tau = \int_V \nabla \cdot (f\vec{c}) d\tau \\ \\&= \int_V [(\nabla f)\cdot \vec{c}+ f\nabla \cdot \vec{c}]d\tau = \int_V \vec{c} \cdot \nabla f d\tau =\vec{c} \cdot \int_V  \nabla f d\tau \end{align*}

 

\begin{align*} \int_V  (\nabla \times \vec{F}) d\tau= -\oint_S \vec{F} \times d\vec{a} \end{align*}

pf.

\begin{align*} \vec{c} \cdot (\oint_S \vec{F} \times d\vec{a}) &= d\vec{a} \cdot (\vec{c} \times \oint_S \vec{F} )=\oint_S (\vec{c} \times \vec{F})\cdot d\vec{a} =\int_V \nabla \cdot (\vec{c} \times \vec{F}) \cdot d\tau \\ &=\int_V [(\nabla \times {c})\cdot \vec{F} -\vec{c}\cdot (\nabla \times \vec{F})] \cdot d\tau = -\int_V \vec{c} \cdot (\nabla \times \vec{F}) d\tau= - \vec{c} \cdot \int_V(\nabla \times \vec{F}) d\tau\end{align*}

 

\begin{align*}\int_S \nabla f \times d\vec{a}=-\oint_C f  d\vec{l}\end{align*}

pf.

\begin{align*}\vec{c} \cdot \oint_C f  d\vec{l} &=\oint_C ( f\vec{c}) \cdot  d\vec{l} =\int_S [\nabla \times (f\vec{c})] \cdot d\vec{a} =\int_S [\nabla  f \times \vec{c}+f \nabla \times \vec{c}] \cdot d\vec{a} \\ &= \int_S (\nabla  f \times \vec{c}) \cdot d\vec{a} = \int_S (d\vec{a} \times \nabla  f ) \cdot \vec{c}=-\vec{c} \cdot \int_S \nabla f \times d\vec{a} \end{align*}

 

---

 

3. 추가 변형 $f(\nabla \cdot \vec{A}), \quad f(\nabla \times \vec{A}),\quad \vec{b}\cdot(\nabla \times \vec{A}), \quad f\nabla g$을 넣어서 더 변형시킬 수 있다.

 

---

 

4. Vector Area

Def.

\begin{align*} \vec{a} \equiv \int_S d\vec{a} \end{align*}

 

 

Thm.

\begin{align*} \oint_S d\vec{a}=0 \end{align*}

 

Coro.

\begin{align*} \vec{a} = \frac{1}{2}\oint_C \vec{r} \times d\vec{l} \end{align*}

 

Thm.

\begin{align*} \oint_C (\vec{c} \cdot \vec{r}) d\vec{l} = \vec{a} \times \vec{c} \end{align*}

pf.
\begin{align*} \oint_C (\vec{c} \cdot \vec{r}) d\vec{l} =-\int_S \nabla(\vec{c}\cdot\vec{r}) \times d\vec{a} = -\int_S \vec{c} \times d\vec{a} = \vec{a} \times \vec{c} \end{align*}
\begin{align*} Note: \nabla(\vec{c}\cdot\vec{r}) &= \vec{c}\times (\nabla \times \vec{r})+\vec{r} \times (\nabla \times \vec{c}) + (\vec{c}\cdot \nabla)\vec{r} + (\vec{r} \cdot \nabla)\vec{c} \\&=\vec{c}\times (\nabla \times \vec{r})+(\vec{c}\cdot \nabla)\vec{r} \\& =(\vec{c}\cdot \nabla)\vec{r} = \vec{c}   \end{align*}