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\markboth{数学公式}{数学公式}
%\Ddate{2003/01/15}
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\section{数学公式}
\Dseclab{数学公式}

以下は, ``Haltiner, G. J., and Williams, R. T., 1979:
Numerical Prediction and Dynamic Meteorology.
{\it John Wiley \& Sons,} 477pp'' の巻末にある数学公式である.

\subsection{ベクトル演算子}
\Dsubseclab{ベクトル演算子}
%
\vspace{-10mm}
%
\begin{eqnarray}
 \Dvect{A} \cdot ( \Dvect{B} \times \Dvect{C} ) &=&
  (\Dvect{A} \times \Dvect{B}) \cdot \Dvect{C}
  = (\Dvect{C} \times \Dvect{A}) \cdot \Dvect{B}
  \nonumber \\
 \Dvect{A} \times (\Dvect{B} \times \Dvect{C}) &=&
  (\Dvect{A} \cdot \Dvect{C}) \Dvect{B}
  - (\Dvect{A} \cdot \Dvect{B}) \Dvect{C}
  \\
 \nabla \times \nabla a &=& 0, \qquad
  \nabla \cdot \nabla \times \Dvect{A} = 0
  \\
 \nabla \cdot (a \Dvect{A}) &=&
  a \nabla \cdot \Dvect{A} + \Dvect{A} \cdot \nabla a
  \\
 \nabla \times (a \Dvect{A}) &=& a \nabla \times \Dvect{A}
  - \Dvect{A} \times \nabla a
  \\
 \nabla (\Dvect{A} \cdot \Dvect{B}) &=&
  (\Dvect{A} \cdot \nabla) \Dvect{B} + (\Dvect{B} \cdot \nabla) \Dvect{A}
  \nonumber \\
 & & \hspace{10mm}
  + \Dvect{A} \times (\nabla \times \Dvect{B})
  + \Dvect{B} \times (\nabla \times \Dvect{A})
  \\
 \nabla \cdot (\Dvect{A} \times \Dvect{B}) &=&
  \Dvect{B} \cdot \nabla \times \Dvect{A} - \Dvect{A} \cdot \nabla \times \Dvect{B}
  \\
 \nabla \times (\Dvect{A} \times \Dvect{B}) &=&
  \Dvect{A} (\nabla \cdot \Dvect{B}) - \Dvect{B} (\nabla \cdot \Dvect{A})
  + (\Dvect{B} \cdot \nabla) \Dvect{A} - (\Dvect{A} \cdot \nabla) \Dvect{B}
  \\
 (\Dvect{V} \cdot \nabla ) \Dvect{V} &=&
  \nabla (\Dvect{V}^2 / 2) + (\nabla \times \Dvect{V}) \times \Dvect{V},
  \qquad \textrm{(3次元)}
  \\
 (\Dvect{V}_{H} \cdot \nabla) \Dvect{V}_{H} &=&
  \nabla ({\Dvect{V}_{H}}^2 / 2) + \zeta \Dvect{k} \times \Dvect{V}_{H};
  \nonumber \\
 & & \hspace{10mm}
  \zeta = \Dvect{k} \cdot \nabla \times \Dvect{V}_{H},
  \qquad \textrm{(2次元)}
  \\
 \nabla \times (\nabla \times \Dvect{A}) &=& \nabla (\nabla \cdot \Dvect{A})
  - \Dlapla \Dvect{A}
  \\
 \Dvect{V} &=& \Dvect{V}_{\chi} + \Dvect{V}_{\psi};
  \\
 & &
  \nabla \times \Dvect{V}_{\chi}, \quad
  \nabla \cdot \Dvect{V}_{\psi} = 0
  \nonumber \\
 & &
  \Dvect{V}_{\chi} = \nabla \chi, \quad
  \Dvect{V}_{\psi} = - \nabla \times \Dvect{\Psi}
  \nonumber \\
 & &
  \Dlapla \chi = \nabla \cdot \Dvect{V}_{\chi} = \nabla \cdot \Dvect{V},
  \quad
  \Dlapla \Dvect{\Psi} = \nabla \times \Dvect{V}_{\psi} = \nabla \times \Dvect{V}.
  \nonumber
\end{eqnarray}



% 以下のサブセクションだけは 2 段組みにしてみる.
\twocolumn

% サブセクションの題目は 2 段抜きにする.
\twocolumn[%
\subsection{積分公式}
\Dsubseclab{積分公式}
\vspace{5mm}
]
%
{\bf 定義}
%
\begin{eqnarray}
 dv &:& \textrm{体積要素}
  \nonumber \\
 V &:& \textrm{体積}
  \nonumber \\
 d \sigma &:& \textrm{表面要素}
  \nonumber \\
 \Sigma &:& V \textrm{を囲む曲面}
  \nonumber \\
 S &:& \textrm{平面の表面}
  \nonumber \\
 C &:& S \textrm{を囲む曲線}
  \nonumber \\
 ds &:& C \textrm{に沿った距離の要素}
  \nonumber \\
 \Dvect{n} &:& \Sigma \textrm{や} C \textrm{から垂直外向き}
  \nonumber \\
 & & \textrm{への単位ベクトル}
  \nonumber \\
 \Dvect{t} &:& C \textrm{や} S \textrm{の単位接線ベクトル}
  \nonumber
\end{eqnarray}

\pagebreak
{\bf ガウスの発散定理}
%
\begin{eqnarray}
 \textrm{3 次元} \hspace{35mm}
  \nonumber \\
 \int_{V} \nabla \cdot \Dvect{A} \ dv
  = \int_{\Sigma} \Dvect{A} \cdot \Dvect{n} \ d \sigma
   \\
 \nonumber \\
 \textrm{2 次元} \hspace{35mm}
  \nonumber \\
 \int_{S} \nabla \cdot \Dvect{A} \ d \sigma
  = \oint_{C} \Dvect{A} \cdot \Dvect{n} \ ds
\end{eqnarray}

{\bf ストークスの定理}
%
\begin{eqnarray}
 \int_{V} \nabla \times \Dvect{A} \ dv
  &=& \int_{\Sigma} \Dvect{n} \times \Dvect{A} \ d \sigma
  \nonumber \\
   \\
 \int_{S} \Dvect{n} \cdot \nabla \times \Dvect{A} \ d \sigma
  &=& \oint_{C} \Dvect{A} \cdot \Dvect{t} \ ds
  \nonumber \\
  \\
 \int_{V} \nabla a \ dv &=& \int_{\Sigma} a \Dvect{n} \ d \sigma
  \nonumber \\
\end{eqnarray}
%


% 以降は 1 段組みに戻す.
\onecolumn

\subsection{一般的直交座標系での関係式}
\Dsubseclab{一般的直交座標系での関係式}
%
$s_1, s_2, s_3$をスカラーの曲線距離,
$\Dvect{a}_1, \Dvect{a}_2, \Dvect{a}_3$を座標系におけるそれぞれの方向の
単位ベクトル,
$x_1, x_2, x_3$と規格化因子(スケール因子)$h_j$が$ds_j = h_j dx_j$という
関係を持つとする.
%
\begin{eqnarray}
 \textrm{grad } A = \nabla A =
  \frac{\Dvect{a}_1}{h_1} \DP{A}{x_1}
  + \frac{\Dvect{a}_2}{h_2} \DP{A}{x_2}
  + \frac{\Dvect{a}_3}{h_3} \DP{A}{x_3}
\end{eqnarray}
%
\begin{eqnarray}
 \textrm{div } \Dvect{F} &=& \nabla \cdot \Dvect{F}
  \nonumber \\
  &=& \frac{1}{h_1 h_2 h_3}
   \left[
    \DP{}{x_1} (h_2 h_3 F_1)
    + \DP{}{x_2} (h_1 h_3 F_2)
    + \DP{}{x_3} (h_1 h_2 F_3)
   \right]
\end{eqnarray}
%
\begin{eqnarray}
 \Dlapla{A} = \frac{1}{h_1 h_2 h_3}
   \left[
    \DP{}{x_1} \left( \frac{h_2 h_3}{h_1} \DP{A}{x_1} \right)
    + \DP{}{x_2} \left( \frac{h_1 h_3}{h_2} \DP{A}{x_2} \right)
    + \DP{}{x_3} \left( \frac{h_1 h_2}{h_3} \DP{A}{x_3} \right)
   \right]
   \nonumber \\
\end{eqnarray}
%
\begin{eqnarray}
 \textrm{curl } \Dvect{F}
  = \frac{\Dvect{a}_1}{h_2 h_3}
  \left[
   \DP{}{x_2} (h_3 F_3) - \DP{}{x_3} (h_2 F_2) 
  \right]
  + \frac{\Dvect{a}_2}{h_3 h_1}
  \left[
   \DP{}{x_3} (h_1 F_1) - \DP{}{x_1} (h_3 F_3) 
  \right]
  \nonumber \\
  + \frac{\Dvect{a}_3}{h_1 h_2}
  \left[
   \DP{}{x_1} (h_2 F_2) - \DP{}{x_2} (h_1 F_1) 
  \right] \hspace{10mm}
\end{eqnarray}
%
\begin{eqnarray}
 \DD{A}{t}
  &=& \DP{A}{t} + \DD{x_1}{t} \DP{A}{x_1}
  + \DD{x_2}{t} \DP{A}{x_2} + \DD{x_3}{t} \DP{A}{x_3}
  \\
 &=& \DP{A}{t} + \frac{v_1}{h_1} \DP{A}{x_1}
  + \frac{v_2}{h_2} \DP{A}{x_2} + \frac{v_3}{h_3} \DP{A}{x_3}
  = \DP{A}{t} + (\Dvect{V} \cdot \nabla) A;
  \nonumber \\
 & &
 v_1 = ds_1 / dt = h_1 \ dx_2 / dt, \textrm{など},
  \quad
  \Dvect{V} = v_1 \Dvect{a}_1 + v_2 \Dvect{a}_2 + v_3 \Dvect{a}_3
  \nonumber
\end{eqnarray}
%
\begin{eqnarray}
 \DD{\Dvect{F}}{t} &=&
  \DD{F_1}{t} \Dvect{a}_1
  + \DD{F_2}{t} \Dvect{a}_2
  + \DD{F_3}{t} \Dvect{a}_3
  + F_1 \DD{\Dvect{a}_1}{t}
  + F_2 \DD{\Dvect{a}_2}{t}
  + F_3 \DD{\Dvect{a}_3}{t}
  \\
 &=& \sum_{j=1}^{3}
  \left[
   \left( \DP{F_j}{t} + \Dvect{V} \cdot \nabla F_j \right) \Dvect{a}_j
   + F_j \sum_{i=1}^{3}
   \left( \frac{v_i}{h_i} \DP{\Dvect{a}_j}{x_i} \right)
  \right]
  \nonumber
\end{eqnarray}
%
\begin{eqnarray}
 (\textrm{例えば } \Dvect{F} = \Dvect{V} \textrm{ である場合})
  \hspace{73mm}
  \nonumber \\
 \DD{\Dvect{V}}{t} 
 = \sum_{j=1}^{3}
  \left[
   \left( \DP{v_j}{t} + \Dvect{V} \cdot \nabla v_j \right)
   + \sum_{i=1, \ i=j}^{3}
   \left(
    \frac{v_j}{h_j} \frac{v_i}{h_i} \DP{h_j}{x_i}
    - \frac{v_i}{h_i} \frac{v_i}{h_j} \DP{h_i}{x_j}
   \right)
  \right] \Dvect{a}_j
\end{eqnarray}

{\bf 例:}\\
\hspace{3mm}
デカルト座標系: $x, y, z ; h_1 = h_2 = h_3 = 1;
\Dvect{a}_1 = \Dvect{i}, \ \Dvect{a}_2 = \Dvect{j}, \
\Dvect{a}_3 = \Dvect{k}$. \\
\hspace{3mm}
球面座標系: 経度$\lambda$, 緯度$\varphi$, 動径距離$r;$
規格化因子:$h_{\lambda} = r \cos \varphi, \ h_{\varphi} = r, \ h_r = 1$. \\
\hspace{3mm}
円筒座標系: $\lambda, r, z; \ h_{\lambda} = r, \ h_r = h_z = 1$.

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