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3次元球面座標で変換行列は$$ \ppd{(x,y,z)}{(r,\theta,\phi)} $$$$ = $$$$ \arrs{~ \sin\theta\, \cos\phi & r\cos\theta\, \cos\phi & -r\sin\theta\, \sin\phi \\~ \sin\theta\, \sin\phi & r\cos\theta\, \sin\phi & r\sin\theta\, \cos\phi \\~ \cos\theta & -r\sin\theta & 0 \\~

３次元球面座標で変換行列は

 #spanend #spanadd \left\{\arr[l]{ #spanend &spanadd;x = r\sin\theta\, \cos\phi \\&spanend; &spanadd;y = r\sin\theta\, \sin\phi \\&spanend; &spanadd;z = r\cos\theta \\&spanend; #spanadd }\right. #spanend #spanadd 
/home/limg/www/LimgMath/eq! You can't use `macro parameter character #' in math mode.
 $\displaystyle \mathstrut { ##
                                         spanend ##spanadd \left \{\arr [l]{...
l.34 $}
で定義され、
変換行列は以下のように表される：
 #spanend &spanadd; \ppd{(x,y,z)}{(r,\theta,\phi)}&spanend; &spanadd; 
/home/limg/www/LimgMath/eq! You can't use `macro parameter character #' in math mode.
 $\displaystyle \mathstrut { ##
                                         spanend &spanadd; \ppd {(x,y,z)}{(r...
l.34 $}
       

 =&spanend; &spanadd; 
/home/limg/www/LimgMath/eq! Misplaced alignment tab character &.
 $\displaystyle \mathstrut { =&
                                         spanend; &spanadd; } $
l.34 $}
       

 \arrs[ccc]{&spanend; &spanadd; \sin\theta\, \cos\phi & r\cos\theta\, \cos\phi & -r\sin\theta\, \sin\phi \\&spanend; &spanadd; \sin\theta\, \sin\phi & r\cos\theta\, \sin\phi & r\sin\theta\, \cos\phi \\&spanend; &spanadd; \cos\theta & -r\sin\theta & 0 \\&spanend; } 
/home/limg/www/LimgMath/eq! Extra alignment tab has been changed to \cr.
 \endtemplate 
                             
l.34 $}
$$ = $$\begin{pmatrix}\cos\phi & -\sin\phi & 0 \\\sin\phi & \cos\phi & 0 \\0 & 0 & 1 \\\end{pmatrix}\begin{pmatrix}\sin\theta & \cos\theta & 0 \\0 & 0 & 1 \\\cos\theta & -\sin\theta & 0 \\\end{pmatrix}\begin{pmatrix}1 & 0 & 0 \\0 & r & 0 \\0 & 0 & r\sin\theta \\\end{pmatrix} \\

#eq{{{{{{{{{{{{{{{{
#spanend
#spanadd
#ceq(e)
#spanend
#spanadd

 #spanend &spanadd; \phantom{\ppd{(x,y,z)}{(r,\theta,\phi)}}&spanend; &spanadd; 
/home/limg/www/LimgMath/eq! You can't use `macro parameter character #' in math mode.
 $\displaystyle \mathstrut { ##
                                         spanend &spanadd; \phantom {\ppd {(...
l.34 $}
       

 =&spanend; &spanadd; 
/home/limg/www/LimgMath/eq! Misplaced alignment tab character &.
 $\displaystyle \mathstrut { =&
                                         spanend; &spanadd; } $
l.34 $}
       

 \arrs[ccc]{&spanend; &spanadd; \cos\phi & -\sin\phi & 0 \\&spanend; &spanadd; \sin\phi & \cos\phi & 0 \\&spanend; &spanadd; 0 & 0 & 1 \\&spanend; &spanadd; }&spanend; &spanadd; 
/home/limg/www/LimgMath/eq! Extra alignment tab has been changed to \cr.
 \endtemplate 
                             
l.34 $}
       

 \arrs[ccc]{&spanend; &spanadd; \sin\theta & \cos\theta & 0 \\&spanend; &spanadd; 0 & 0 & 1 \\&spanend; &spanadd; \cos\theta & -\sin\theta & 0 \\&spanend; &spanadd; }&spanend; &spanadd; 
/home/limg/www/LimgMath/eq! Extra alignment tab has been changed to \cr.
 \endtemplate 
                             
l.34 $}
       

 \arrs[ccc]{&spanend; &spanadd; 1 & 0 & 0 \\&spanend; &spanadd; 0 & r & 0 \\&spanend; &spanadd; 0 & 0 & r\sin\theta \\&spanend; &spanadd; }&spanend; #spanadd 
/home/limg/www/LimgMath/eq! Extra alignment tab has been changed to \cr.
 \endtemplate 
                             
l.34 $}

#spanend
#spanadd
#ceq(d)
#spanend

#spanadd
以下では、この因数分解の幾何的意味を考える。
#spanend
#spanadd

#spanend
#spanadd
////////////////////////////////////////////////////////////////
#spanend
#spanadd
////////////////////////////////////////////////////////////////
#spanend
#spanadd
* 回転と計量の分解 [#d83c40b3]
#spanend
#spanadd
////////////////////////////////////////////////////////////////
#spanend
#spanadd
** ２次元極座標系における回転と計量の分解 [#w2a34b92]
#spanend
#spanadd
;;一般に、２次元の極座標系は
#spanend
#spanadd
 #spanend #spanadd \left\{\arr[l]{ #spanend &spanadd;x = r\cos\phi \\&spanend; &spanadd;y = r\sin\phi \\&spanend; #spanadd }\right. #spanend #spanadd 
/home/limg/www/LimgMath/eq! You can't use `macro parameter character #' in math mode.
 $\displaystyle \mathstrut { ##
                                         spanend ##spanadd \left \{\arr [l]{...
l.34 $}

#spanend
#spanadd
で定義され*1、
#spanend
#spanadd
;,変換行列は以下のように表せる：
#spanend
#spanadd
#ceq(e)
#spanend
#spanadd
 #spanend &spanadd; \ppd{(x,y)}{(r,\phi)}&spanend; &spanadd; 
/home/limg/www/LimgMath/eq! You can't use `macro parameter character #' in math mode.
 $\displaystyle \mathstrut { ##
                                         spanend &spanadd; \ppd {(x,y)}{(r,\...
l.34 $}
       

 =&spanend; &spanadd; 
/home/limg/www/LimgMath/eq! Misplaced alignment tab character &.
 $\displaystyle \mathstrut { =&
                                         spanend; &spanadd; } $
l.34 $}
       

 \arrs[cc]{&spanend; &spanadd; \cos\phi & -r\sin\phi \\&spanend; &spanadd; \sin\phi & r\cos\phi \\&spanend; &spanadd; }&spanend; #spanadd 
/home/limg/www/LimgMath/eq! Extra alignment tab has been changed to \cr.
 \endtemplate 
                             
l.34 $}

#spanend
#spanadd
#ceq(d)
#spanend
#spanadd

#spanend
#spanadd
;,変換行列の行列式であるヤコビアンを計算すると：
#spanend
#spanadd
#ceq(e)
#spanend
#spanadd
 #spanend &spanadd; \left|\ppd{(x,y)}{(r,\phi)}\right|&spanend; &spanadd; 
/home/limg/www/LimgMath/eq! You can't use `macro parameter character #' in math mode.
 $\displaystyle \mathstrut { ##
                                         spanend &spanadd; \left |\ppd {(x,y...
l.34 $}
$$ = $$$$ (\cos\phi) $$$$ (r\cos\phi) $$$$ - $$$$ (\sin\phi) $$
 (-r\cos\phi)&spanend; &spanadd; 
/home/limg/www/LimgMath/eq! Misplaced alignment tab character &.
 ...aystyle \mathstrut { (-r\cos \phi )&
                                                  spanend; &spanadd; } $
l.34 $}
$$ = $$
 r&spanend; #spanadd 
/home/limg/www/LimgMath/eq! Misplaced alignment tab character &.
 $\displaystyle \mathstrut { r&
                                         spanend; ##spanadd } $
l.34 $}

#spanend
#spanadd
#ceq(d)
#spanend
#spanadd
;,よって、置換積分は$$ dx $$$$ dy $$$$ = $$$$ r $$$$ dr $$$$ d\phi $$と書ける。
#spanend
#spanadd

#spanend
#spanadd
;;これは、$$ dr $$ と $$ r $$$$ d\phi $$の積と見なせて、
#spanend
#spanadd
;,$$ \theta $$が無次元のため、計量$$ r $$で長さの次元を帳尻合わせしていると解釈できる。
#spanend
#spanadd

#spanend
#spanadd
;;ここで、変換行列から計量を分離すると、回転行列が残った式が得られる：
#spanend
#spanadd
#ceq(e)
#spanend
#spanadd
 #spanend &spanadd; \ppd{(x,y)}{(r,\phi)}&spanend; &spanadd; 
/home/limg/www/LimgMath/eq! You can't use `macro parameter character #' in math mode.
 $\displaystyle \mathstrut { ##
                                         spanend &spanadd; \ppd {(x,y)}{(r,\...
l.34 $}
       

 =&spanend; &spanadd; 
/home/limg/www/LimgMath/eq! Misplaced alignment tab character &.
 $\displaystyle \mathstrut { =&
                                         spanend; &spanadd; } $
l.34 $}
       

 \arrs[cc]{&spanend; &spanadd; \cos\phi & -\sin\phi \\&spanend; &spanadd; \sin\phi & \cos\phi \\&spanend; &spanadd; }&spanend; &spanadd; 
/home/limg/www/LimgMath/eq! Extra alignment tab has been changed to \cr.
 \endtemplate 
                             
l.34 $}
       

 \arrs[cc]{&spanend; &spanadd; 1 & 0 \\&spanend; &spanadd; 0 & r \\&spanend; &spanadd; }&spanend; #spanadd 
/home/limg/www/LimgMath/eq! Extra alignment tab has been changed to \cr.
 \endtemplate 
                             
l.34 $}

#spanend
#spanadd
#ceq(d)
#spanend
#spanadd

#spanend
#spanadd
%bodynote
#spanend
#spanadd
////////////////////////////////////////////////////////////////
#spanend
#spanadd
** ３次元極座標系における回転と計量の分解 [#vcb2db62]
#spanend
#spanadd
;,３次元極座標系に対しても同様に考察すると、
#spanend
#spanadd
;,ヤコビアンを頑張って計算すれば、
#spanend
#spanadd
 #spanend &spanadd; \left|\ppd{(x,y,z)}{(r,\theta,\phi)}\right|&spanend; &spanadd; 
/home/limg/www/LimgMath/eq! You can't use `macro parameter character #' in math mode.
 $\displaystyle \mathstrut { ##
                                         spanend &spanadd; \left |\ppd {(x,y...
l.34 $}
$$ = $$
 r^2\sin\theta&spanend; #spanadd 
/home/limg/www/LimgMath/eq! Misplaced alignment tab character &.
 ...ystyle \mathstrut { r^2\sin \theta &
                                                  spanend; ##spanadd } $
l.34 $}

#spanend
#spanadd
を得られて、
#spanend
#spanadd
;,置換積分は$$ dx $$$$ dy $$$$ dz $$$$ = $$$$ r^2 $$$$ \sin\theta $$$$ dr $$$$ d\theta $$と書ける。
#spanend
#spanadd

#spanend
#spanadd
;,類推から$$ dr $$と$$ r $$$$ d\theta $$と$$ r $$$$ \sin\theta $$$$ d\phi $$の積と見なせて、
#spanend
#spanadd
;,各次元の帳尻合わせと解釈できる。
#spanend
#spanadd
;,特に$$ \phi $$方向の計量は、$$ r $$に依存するのみならず$$ \theta $$にも依存するのが、
#spanend
#spanadd
;,同じ半径$$ r $$の球面上における異なる$$ \theta $$の円周が異なるためと解釈できる。
#spanend
#spanadd

#spanend
#spanadd
;,すると、２次元と同様に回転と計量を分離できる：
#spanend
#spanadd
#ceq(e)
#spanend
#spanadd
 #spanend &spanadd; \ppd{(x,y,z)}{(r,\theta,\phi)}&spanend; &spanadd; 
/home/limg/www/LimgMath/eq! You can't use `macro parameter character #' in math mode.
 $\displaystyle \mathstrut { ##
                                         spanend &spanadd; \ppd {(x,y,z)}{(r...
l.34 $}
       

 =&spanend; &spanadd; 
/home/limg/www/LimgMath/eq! Misplaced alignment tab character &.
 $\displaystyle \mathstrut { =&
                                         spanend; &spanadd; } $
l.34 $}
       

 \arrs[ccc]{&spanend; &spanadd; \sin\theta\, \cos\phi & r\cos\theta\, \cos\phi & -r\sin\theta\, \sin\phi \\&spanend; &spanadd; \sin\theta\, \sin\phi & r\cos\theta\, \sin\phi & r\sin\theta\, \cos\phi \\&spanend; &spanadd; \cos\theta & -r\sin\theta & 0 \\&spanend; &spanadd; }&spanend; #spanadd 
/home/limg/www/LimgMath/eq! Extra alignment tab has been changed to \cr.
 \endtemplate 
                             
l.34 $}

#spanend
#spanadd
#ceq(e)
#spanend
#spanadd
 #spanend &spanadd; \phantom{\ppd{(x,y,z)}{(r,\theta,\phi)}}&spanend; &spanadd; 
/home/limg/www/LimgMath/eq! You can't use `macro parameter character #' in math mode.
 $\displaystyle \mathstrut { ##
                                         spanend &spanadd; \phantom {\ppd {(...
l.34 $}
       

 =&spanend; &spanadd; 
/home/limg/www/LimgMath/eq! Misplaced alignment tab character &.
 $\displaystyle \mathstrut { =&
                                         spanend; &spanadd; } $
l.34 $}
       

 \arrs[ccc]{&spanend; &spanadd; \sin\theta\, \cos\phi & \cos\theta\, \cos\phi & -\sin\phi \\&spanend; &spanadd; \sin\theta\, \sin\phi & \cos\theta\, \sin\phi & \cos\phi \\&spanend; &spanadd; \cos\theta & -\sin\theta & 0 \\&spanend; &spanadd; }&spanend; &spanadd; 
/home/limg/www/LimgMath/eq! Extra alignment tab has been changed to \cr.
 \endtemplate 
                             
l.34 $}
       

 \arrs[ccc]{&spanend; &spanadd; 1 & 0 & 0 \\&spanend; &spanadd; 0 & r & 0 \\&spanend; &spanadd; 0 & 0 & r\sin\theta \\&spanend; &spanadd; }&spanend; #spanadd 
/home/limg/www/LimgMath/eq! Extra alignment tab has been changed to \cr.
 \endtemplate 
                             
l.34 $}

#spanend
#spanadd
#ceq(d)
#spanend
#spanadd
;,２列目から$$ r $$を、３列目から$$ r $$$$ \sin\theta $$を抜いた形になる。
#spanend
#spanadd
;,
#spanend
#spanadd
 #spanend &spanadd; \arrs[ccc]{&spanend; &spanadd; \sin\theta\, \cos\phi & \cos\theta\, \cos\phi & -\sin\phi \\&spanend; &spanadd; \sin\theta\, \sin\phi & \cos\theta\, \sin\phi & \cos\phi \\&spanend; &spanadd; \cos\theta & -\sin\theta & 0 \\&spanend; &spanadd; }&spanend; #spanadd 
/home/limg/www/LimgMath/eq! You can't use `macro parameter character #' in math mode.
 $\displaystyle \mathstrut { ##
                                         spanend &spanadd; \arrs [ccc]{&span...
l.34 $}

#spanend
#spanadd
が３次元の回転を表すが、
#spanend
#spanadd
;,３次元回転が２回の２次元回転で合成できる事実は古くから知られていて、
#spanend
#spanadd
;,３次元回転を２次元回転に分解したのが冒頭に示した因数分解式になる：
#spanend
#spanadd
#ceq(e)
#spanend
#spanadd
 #spanend &spanadd; \ppd{(x,y,z)}{(r,\theta,\phi)}&spanend; &spanadd; 
/home/limg/www/LimgMath/eq! You can't use `macro parameter character #' in math mode.
 $\displaystyle \mathstrut { ##
                                         spanend &spanadd; \ppd {(x,y,z)}{(r...
l.34 $}
       

 =&spanend; &spanadd; 
/home/limg/www/LimgMath/eq! Misplaced alignment tab character &.
 $\displaystyle \mathstrut { =&
                                         spanend; &spanadd; } $
l.34 $}
       

 \arrs[ccc]{&spanend; &spanadd; \sin\theta\, \cos\phi & r\cos\theta\, \cos\phi & -r\sin\theta\, \sin\phi \\&spanend; &spanadd; \sin\theta\, \sin\phi & r\cos\theta\, \sin\phi & r\sin\theta\, \cos\phi \\&spanend; &spanadd; \cos\theta & -r\sin\theta & 0 \\&spanend; &spanadd; }&spanend; #spanadd 
/home/limg/www/LimgMath/eq! Extra alignment tab has been changed to \cr.
 \endtemplate 
                             
l.34 $}

#spanend
#spanadd
#ceq(e)
#spanend
#spanadd
 #spanend &spanadd; \phantom{\ppd{(x,y,z)}{(r,\theta,\phi)}}&spanend; &spanadd; 
/home/limg/www/LimgMath/eq! You can't use `macro parameter character #' in math mode.
 $\displaystyle \mathstrut { ##
                                         spanend &spanadd; \phantom {\ppd {(...
l.34 $}
       

 =&spanend; &spanadd; 
/home/limg/www/LimgMath/eq! Misplaced alignment tab character &.
 $\displaystyle \mathstrut { =&
                                         spanend; &spanadd; } $
l.34 $}
       

 \arrs[ccc]{&spanend; &spanadd; \cos\phi & -\sin\phi & 0 \\&spanend; &spanadd; \sin\phi & \cos\phi & 0 \\&spanend; &spanadd; 0 & 0 & 1 \\&spanend; &spanadd; }&spanend; &spanadd; 
/home/limg/www/LimgMath/eq! Extra alignment tab has been changed to \cr.
 \endtemplate 
                             
l.34 $}
       

 \arrs[ccc]{&spanend; &spanadd; \sin\theta & \cos\theta & 0 \\&spanend; &spanadd; 0 & 0 & 1 \\&spanend; &spanadd; \cos\theta & -\sin\theta & 0 \\&spanend; &spanadd; }&spanend; &spanadd; 
/home/limg/www/LimgMath/eq! Extra alignment tab has been changed to \cr.
 \endtemplate 
                             
l.34 $}
       

 \arrs[ccc]{&spanend; &spanadd; 1 & 0 & 0 \\&spanend; &spanadd; 0 & r & 0 \\&spanend; &spanadd; 0 & 0 & r\sin\theta \\&spanend; &spanadd; }&spanend; #spanadd 
/home/limg/www/LimgMath/eq! Extra alignment tab has been changed to \cr.
 \endtemplate 
                             
l.34 $}

#spanend
#spanadd
#ceq(d)
#spanend
#spanadd

#spanend
#spanadd
;,特に最初の
#spanend
#spanadd
 #spanend &spanadd; 
/home/limg/www/LimgMath/eq! You can't use `macro parameter character #' in math mode.
 $\displaystyle \mathstrut { ##
                                         spanend &spanadd; } $
l.34 $}
       

 \arrs[ccc]{&spanend; &spanadd; \cos\phi & -\sin\phi & 0 \\&spanend; &spanadd; \sin\phi & \cos\phi & 0 \\&spanend; &spanadd; 0 & 0 & 1 \\&spanend; &spanadd; }&spanend; #spanadd 
/home/limg/www/LimgMath/eq! Extra alignment tab has been changed to \cr.
 \endtemplate 
                             
l.34 $}

#spanend
#spanadd
が、２次元回転の
#spanend
#spanadd
 #spanend &spanadd; 
/home/limg/www/LimgMath/eq! You can't use `macro parameter character #' in math mode.
 $\displaystyle \mathstrut { ##
                                         spanend &spanadd; } $
l.34 $}
       

 \arrs[cc]{&spanend; &spanadd; \cos\phi & -\sin\phi \\&spanend; &spanadd; \sin\phi & \cos\phi \\&spanend; &spanadd; }&spanend; #spanadd 
/home/limg/www/LimgMath/eq! Extra alignment tab has been changed to \cr.
 \endtemplate 
                             
l.34 $}

#spanend
#spanadd
と全く同じである。
#spanend
#spanadd
;,共に$$ x $$軸と$$ y $$軸の両方に直な方向を軸に、$$ x $$軸を$$ y $$軸に重ねる向きに回した回転を表す。
#spanend
#spanadd
;,簡単なため、これを「$$ x $$軸を$$ y $$軸に回した回転」と略す。
#spanend
#spanadd

#spanend
#spanadd
;,２番目の行列である
#spanend
#spanadd
 #spanend &spanadd; 
/home/limg/www/LimgMath/eq! You can't use `macro parameter character #' in math mode.
 $\displaystyle \mathstrut { ##
                                         spanend &spanadd; } $
l.34 $}
       

 \arrs[ccc]{&spanend; &spanadd; \sin\theta & \cos\theta & 0 \\&spanend; &spanadd; 0 & 0 & 1 \\&spanend; &spanadd; \cos\theta & -\sin\theta & 0 \\&spanend; &spanadd; }&spanend; #spanadd 
/home/limg/www/LimgMath/eq! Extra alignment tab has been changed to \cr.
 \endtemplate 
                             
l.34 $}

#spanend
#spanadd
は３行目を１行目に、１行目を２行目に置換すれば、
#spanend
#spanadd
;,これまた２次元の回転と同じ形になる。
#spanend
#spanadd
;,つまり、$$ z $$→$$ x $$、$$ x $$→$$ y $$に置換すれば、$$ x $$を$$ y $$に重ねる回転を表す。
#spanend
#spanadd
;,逆に言い方の方を置換すれば、「$$ z $$軸を$$ x $$軸に回した回転」になる。
#spanend
#spanadd
;,すなわち、共に$$ z $$軸と$$ x $$軸の両方に直な方向を軸に、$$ z $$軸を$$ x $$軸に重ねる向きに回した回転を表す。
#spanend
#spanadd

#spanend
*1 一般的に$$ \phi $$ではなく$$ \theta $$を用いられるが、球面座標系の文字に揃えた。球面座標系では$$ \theta $$が極から降ろす向きに定義され、極座標の$$ \phi $$と定義が異なる。
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