記事ソース/ベクトル成分の座標変換
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========================================
ベクトル成分の座標変換
========================================
まず 基底の座標変換_ の記事で見た、双対基底の座標変換を思...
<tex>
\bm{e'_{i}}= {\alpha}_{i'}^{k}\bm{e_{k}}
</tex>
<tex>
\bm{e_{i}}= {\alpha}_{i}^{k'}\bm{e'_{k}}
</tex>
双対基底の性質 $\bm{e_{i}} \cdot \bm{e^{j}} ={\delta}_{i}...
<tex>
{\alpha}_{i'}^{k} = \bm{e'_{i}} \cdot \bm{e^{k}}
</tex>
<tex>
{\alpha}_{i}^{k'} = \bm{e_{i}} \cdot \bm{e'^{k}}
</tex>
そこで、ベクトル $\bm{A}=A_{k} \bm{e^{k}} $ の両辺に $\bm...
<tex>
\bm{e'_{i}} \cdot \bm{A}=A_{k} \bm{e'_{i}} \cdot \bm{e^{k...
</tex>
左辺は $\bm{e'_{i}} \cdot \bm{A}=\bm{e'_{i}} \cdot (A'_{j...
<tex>
A'_{i}= {\alpha}_{i'}^{k} A_{k} \tag{1}
</tex>
これは共変ベクトル(反変基底をとったときの成分)の座標変...
<tex>
A'^{i}= {\alpha}_{k}^{i'} A^{k} \tag{2}
</tex>
ベクトル成分の座標変換も、基底と同じ変換係数を使って表現...
<tex>
\left(
\begin{array}{ccc}
A'_{1}\\
A'_{2}\\
A'_{3}\\
\end{array}
\right)
=
\left( \begin{array}{ccc}
{\alpha}_{1'}^{1} & {\alpha}_{1'}^{2} & {\alpha}_{1'}^{3}...
{\alpha}_{2'}^{1} & {\alpha}_{2'}^{2} & {\alpha}_{2'}^{3}...
{\alpha}_{3'}^{1} & {\alpha}_{3'}^{2} & {\alpha}_{3'}^{3}...
\end{array}
\right)
\left(
\begin{array}{ccc}
A_{1}\\
A_{2}\\
A_{3}\\
\end{array}
\right)
</tex>
<tex>
\left(
\begin{array}{ccc}
A'^{1}\\
A'^{2}\\
A'^{3}\\
\end{array}
\right)
=
\left( \begin{array}{ccc}
{\alpha}_{1}^{1'} & {\alpha}_{2}^{1'} & {\alpha}_{3}^{1'}...
{\alpha}_{1}^{2'} & {\alpha}_{2}^{2'} & {\alpha}_{3}^{2'}...
{\alpha}_{1}^{3'} & {\alpha}_{2}^{3'} & {\alpha}_{3}^{3'}...
\end{array}
\right)
\left(
\begin{array}{ccc}
A^{1}\\
A^{2}\\
A^{3}\\
\end{array}
\right)
</tex>
.. [*] ここに出てくる変換係数 ${\alpha}_{k}^{i'}$ は、実...
.. [*] 変換係数 ${\alpha}_{k}^{i'}$ と ${\alpha}_{i'}^{k}...
.. _群の公理: http://www12.plala.or.jp/ksp/algebra/GroupA...
.. _基底の座標変換: http://www12.plala.or.jp/ksp/vectoran...
@@author:Joh@@
@@accept: 2006-07-15@@
@@category: ベクトル解析@@
@@id: VectorCompoTrans@@
終了行:
#rst2hooktail_source
========================================
ベクトル成分の座標変換
========================================
まず 基底の座標変換_ の記事で見た、双対基底の座標変換を思...
<tex>
\bm{e'_{i}}= {\alpha}_{i'}^{k}\bm{e_{k}}
</tex>
<tex>
\bm{e_{i}}= {\alpha}_{i}^{k'}\bm{e'_{k}}
</tex>
双対基底の性質 $\bm{e_{i}} \cdot \bm{e^{j}} ={\delta}_{i}...
<tex>
{\alpha}_{i'}^{k} = \bm{e'_{i}} \cdot \bm{e^{k}}
</tex>
<tex>
{\alpha}_{i}^{k'} = \bm{e_{i}} \cdot \bm{e'^{k}}
</tex>
そこで、ベクトル $\bm{A}=A_{k} \bm{e^{k}} $ の両辺に $\bm...
<tex>
\bm{e'_{i}} \cdot \bm{A}=A_{k} \bm{e'_{i}} \cdot \bm{e^{k...
</tex>
左辺は $\bm{e'_{i}} \cdot \bm{A}=\bm{e'_{i}} \cdot (A'_{j...
<tex>
A'_{i}= {\alpha}_{i'}^{k} A_{k} \tag{1}
</tex>
これは共変ベクトル(反変基底をとったときの成分)の座標変...
<tex>
A'^{i}= {\alpha}_{k}^{i'} A^{k} \tag{2}
</tex>
ベクトル成分の座標変換も、基底と同じ変換係数を使って表現...
<tex>
\left(
\begin{array}{ccc}
A'_{1}\\
A'_{2}\\
A'_{3}\\
\end{array}
\right)
=
\left( \begin{array}{ccc}
{\alpha}_{1'}^{1} & {\alpha}_{1'}^{2} & {\alpha}_{1'}^{3}...
{\alpha}_{2'}^{1} & {\alpha}_{2'}^{2} & {\alpha}_{2'}^{3}...
{\alpha}_{3'}^{1} & {\alpha}_{3'}^{2} & {\alpha}_{3'}^{3}...
\end{array}
\right)
\left(
\begin{array}{ccc}
A_{1}\\
A_{2}\\
A_{3}\\
\end{array}
\right)
</tex>
<tex>
\left(
\begin{array}{ccc}
A'^{1}\\
A'^{2}\\
A'^{3}\\
\end{array}
\right)
=
\left( \begin{array}{ccc}
{\alpha}_{1}^{1'} & {\alpha}_{2}^{1'} & {\alpha}_{3}^{1'}...
{\alpha}_{1}^{2'} & {\alpha}_{2}^{2'} & {\alpha}_{3}^{2'}...
{\alpha}_{1}^{3'} & {\alpha}_{2}^{3'} & {\alpha}_{3}^{3'}...
\end{array}
\right)
\left(
\begin{array}{ccc}
A^{1}\\
A^{2}\\
A^{3}\\
\end{array}
\right)
</tex>
.. [*] ここに出てくる変換係数 ${\alpha}_{k}^{i'}$ は、実...
.. [*] 変換係数 ${\alpha}_{k}^{i'}$ と ${\alpha}_{i'}^{k}...
.. _群の公理: http://www12.plala.or.jp/ksp/algebra/GroupA...
.. _基底の座標変換: http://www12.plala.or.jp/ksp/vectoran...
@@author:Joh@@
@@accept: 2006-07-15@@
@@category: ベクトル解析@@
@@id: VectorCompoTrans@@
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