6.6 グリーンの定理のまとめ
グリーンの第一定理
グリーンの第一定理（first Green's theorems）をまとめると，次のようになる．
- Scalar form
\begin{gather}
\iiint _V (f\nabla^2 g + \nabla f \cdot \nabla g ) dV
= \oiint _S f \frac{\partial g}{\partial n} dS
\left( = \oint _S \VEC{n} \cdot (f \nabla g ) dS \right)
\end{gather}
- Vector form
\begin{align}
&\iiint _V \big\{ (\nabla \times \VEC{G}) \cdot (\nabla \times \VEC{F}) - \VEC{F} \cdot \nabla \times ( \nabla \times \VEC{G} ) \big\} dV
\nonumber \\
&= \oiint _S \VEC{n} \cdot ( \VEC{F} \times \nabla \times \VEC{G} ) dS
\end{align}
- Composite vector-dyadic form
\begin{align}
&\iiint _V \big\{ (\nabla \times \VEC{F}) \cdot (\nabla \times \DYA{G}) - \VEC{F} \cdot \nabla \times ( \nabla \times \DYA{G} ) \big\} dV
\nonumber \\
&= \oiint _S \VEC{n} \cdot ( \VEC{F} \times \nabla \times \DYA{G} ) dS
\end{align}
- Dyadic form
\begin{align}
&\iiint _V \big\{ (\nabla \times \DYA{G})^T \cdot (\nabla \times \DYA{F})
- (\nabla \times \nabla \times \DYA{G})^T \cdot \DYA{F} \big\} dV
\nonumber \\
&= \oiint _S (\nabla \times \DYA{G})^T \cdot (\VEC{n} \times \DYA{F}) dS
\end{align}
グリーンの第ニ定理
グリーンの第ニ定理（second Green's theorems）をまとめると，次のようになる．
- Scalar form
\begin{gather}
\iiint _V (f\nabla^2 g - g\nabla^2 f ) dV
= \oiint _S \left( f \frac{\partial g}{\partial n}
- g \frac{\partial f}{\partial n} \right) dS
\end{gather}
- Vector form, Stratton's theorem
\begin{align}
&\iiint _V \big\{ \VEC{F} \cdot ( \nabla \times \nabla \times \VEC{G} )
- \VEC{G} \cdot ( \nabla \times \nabla \times \VEC{F}) \big\} \ dV
\nonumber \\
&= \oiint _S \VEC{n} \cdot
\big\{ \VEC{G} \times (\nabla \times \VEC{F})
- \VEC{F} \times (\nabla \times \VEC{G}) \big\} \ dS
\end{align}
- Composite vector-dyadic form
\begin{align}
&\iiint _V \big\{ \VEC{F} \cdot (\nabla \times \nabla \times \DYA{G})
- (\nabla \times \nabla \times \VEC{F} ) \cdot \DYA{G} \big\} \ dV
\nonumber \\
&= -\oiint _S \VEC{n} \cdot
\big\{ (\nabla \times \VEC{F}) \times \DYA{G}
+ \VEC{F} \times (\nabla \times \DYA{G}) \big\} \ dS
\end{align}
- Modified composite vector-dyadic form
\begin{align}
&\iiint _V \big\{ \VEC{F} \cdot (\nabla \times \nabla \times \DYA{G})
- ( \nabla \times \nabla \times \VEC{F} ) \cdot \DYA{G} \big\} \ dV
\nonumber \\
&= -\oiint _S
\big\{ ( \VEC{n} \times \nabla \times \VEC{F} ) \cdot \DYA{G}
+ ( \VEC{n} \times \VEC{F} ) \cdot \nabla \times \DYA{G} \big\} \ dS
\end{align}
- Dyadic form
\begin{align}
&\iiint _V \big\{ (\nabla \times \nabla \times \DYA{G})^T \cdot \DYA{F}
- \DYA{G}^T \cdot ( \nabla \times \nabla \times \DYA{F}) \big\} \ dV
\nonumber \\
&= -\oiint _S
\big\{ \DYA{G}^T \cdot (\VEC{n} \times \nabla \times \DYA{F} )
+ (\nabla \times \DYA{G})^T \cdot (\VEC{n} \times \DYA{F}) \big\} \ dS
\end{align}