円柱座標変換
[ ( grad  f ) ( Φ ( r , θ , ζ ) ) ] ⋅ ( N r ( r , θ , ζ ) ) = ( ( ∂ f ∂ x ) ( Φ ( r , θ , ζ ) ) , ( ∂ f ∂ y ) ( Φ ( r , θ , ζ ) ) , ( ∂ f ∂ z ) ( Φ ( r , θ , ζ ) ) ) ( cos  θ sin  θ 0 ) = cos  θ ( ( ∂ f ∂ x ) ( Φ ( r , θ , ζ ) ) ) + sin  θ ( ( ∂ f ∂ y ) ( Φ ( r , θ , ζ ) ) ) = cos  θ [ cos  θ ( ( ∂ ( f ∘ Φ ) ∂ r ) ( r , θ , z ) ) − sin  θ r ( ( ∂ ( f ∘ Φ ) ∂ θ ) ( r , θ , z ) ) ] + sin  θ [ sin  θ ( ( ∂ ( f ∘ Φ ) ∂ r ) ( r , θ , z ) ) + cos  θ r ( ( ∂ ( f ∘ Φ ) ∂ θ ) ( r , θ , z ) ) ] = ( ∂ ( f ∘ Φ ) ∂ r ) ( r , θ , z ) {\displaystyle {\begin{aligned}\left[\left(\operatorname {grad} f\right)(\Phi (r,\theta ,\zeta ))\right]\cdot \left({{\mathbf {N} }_{r}}(r,\theta ,\zeta )\right)=&\left(\left({\frac {\partial f}{\partial x}}\right)(\Phi (r,\theta ,\zeta )),\left({\frac {\partial f}{\partial y}}\right)(\Phi (r,\theta ,\zeta )),\left({\frac {\partial f}{\partial z}}\right)(\Phi (r,\theta ,\zeta ))\right)\left({\begin{matrix}\cos \theta \\\sin \theta \\0\\\end{matrix}}\right)\\=&\cos \theta \left(\left({\frac {\partial f}{\partial x}}\right)(\Phi (r,\theta ,\zeta ))\right)+\sin \theta \left(\left({\frac {\partial f}{\partial y}}\right)(\Phi (r,\theta ,\zeta ))\right)\\=&\cos \theta \left[\cos \theta \left(\left({\frac {\partial (f\circ \Phi )}{\partial r}}\right)(r,\theta ,z)\right)-{\frac {\sin \theta }{r}}\left(\left({\frac {\partial (f\circ \Phi )}{\partial \theta }}\right)(r,\theta ,z)\right)\right]\\&+\sin \theta \left[\sin \theta \left(\left({\frac {\partial (f\circ \Phi )}{\partial r}}\right)(r,\theta ,z)\right)+{\frac {\cos \theta }{r}}\left(\left({\frac {\partial (f\circ \Phi )}{\partial \theta }}\right)(r,\theta ,z)\right)\right]\\=&\left({\frac {\partial (f\circ \Phi )}{\partial r}}\right)(r,\theta ,z)\end{aligned}}} (6-2-2)
円柱座標変換(えんちゅうざひょうへんかん)とは、3次元ユークリッド空間 (数ベクトル空間)の、非線形な座標変換の一つである。円柱座標変換の逆写像のことを、円柱座標系という。円柱座標系は、極座標系の一種である。
円柱座標変換は、電子レンズなど、軸対称な系の計算によく用いられる。
N Δ 2 ( θ , ζ ) = ( ∂ Φ ∂ ζ ( r 0 , θ , ζ ) ) × ( ∂ Φ ∂ r ( r , θ , ζ ) ) ‖ ( ∂ Φ ∂ ζ ( r 0 , θ , ζ ) ) × ( ∂ Φ ∂ r ( r , θ , ζ ) ) ‖ = N r ( r , θ , ζ ) = ( − sin  θ cos  θ 0 ) {\displaystyle {{\mathbf {N} }_{{\Delta }_{2}}}(\theta ,\zeta )={\frac {\left({\frac {\partial \Phi }{\partial \zeta }}({r}_{0},\theta ,\zeta )\right)\times \left({\frac {\partial \Phi }{\partial r}}(r,\theta ,\zeta )\right)}{\left\|\left({\frac {\partial \Phi }{\partial \zeta }}({r}_{0},\theta ,\zeta )\right)\times \left({\frac {\partial \Phi }{\partial r}}(r,\theta ,\zeta )\right)\right\|}}={{\mathbf {N} }_{r}}(r,\theta ,\zeta )=\left({\begin{matrix}-\sin \theta \\\cos \theta \\0\\\end{matrix}}\right)} (3-2-13)
( ∂ f ∂ y ) ( Φ ( r , θ , ζ ) ) = sin  θ ( ( ∂ ( f ∘ Φ ) ∂ r ) ( r , θ , z ) ) + cos  θ r ( ( ∂ ( f ∘ Φ ) ∂ θ ) ( r , θ , z ) ) {\displaystyle \left({\frac {\partial f}{\partial y}}\right)(\Phi (r,\theta ,\zeta ))=\sin \theta \left(\left({\frac {\partial (f\circ \Phi )}{\partial r}}\right)(r,\theta ,z)\right)+{\frac {\cos \theta }{r}}\left(\left({\frac {\partial (f\circ \Phi )}{\partial \theta }}\right)(r,\theta ,z)\right)} (6-1-2)
( ∂ f ∂ z ) ( Φ ( r , θ , z ) ) = ( ∂ ( f ∘ Φ ) ∂ ζ ) ( r , θ , z ) {\displaystyle \left({\frac {\partial f}{\partial z}}\right)(\Phi (r,\theta ,z))=\left({\frac {\partial (f\circ \Phi )}{\partial \zeta }}\right)(r,\theta ,z)} (6-1-3)
上記の3式は、しばしば略記的に以下のように表記される: