円柱座標変換
X ⋅ ( ∂ I 3 ∂ θ × ∂ I 3 ∂ ζ ) = − r ⋅ X ζ {\displaystyle \mathbf {X} \cdot \left({\frac {\partial {{\mathbf {I} }_{3}}}{\partial \theta }}\times {\frac {\partial {{\mathbf {I} }_{3}}}{\partial \zeta }}\right)=-r\cdot {{X}_{\zeta }}} (5-2-7)
なので、
∫ Δ 1 X = ∫ r = 0 r = r 0 ∫ θ = 0 θ = 2 π r ⋅ X ζ ( r , θ , ζ 0 ) d θ d r {\displaystyle \int _{{\Delta }_{1}}{\mathbf {X} }=\int _{r=0}^{r={{r}_{0}}}{\int _{\theta =0}^{\theta =2\pi }{r\cdot {{X}_{\zeta }}(r,\theta ,{{\zeta }_{0}})}}\ d\theta dr} (5-2-8)
∫ Δ 2 X = ∫ θ = 0 θ = 2 π r 0 ⋅ X r ( r 0 , θ , ζ ) d θ d ζ {\displaystyle \int _{{\Delta }_{2}}{\mathbf {X} }=\int _{\theta =0}^{\theta =2\pi }{{{r}_{0}}\cdot {{X}_{r}}({{r}_{0}},\theta ,\zeta )}\ d\theta d\zeta } (5-2-9)
[ ( grad  f ) ( Φ ( r , θ , ζ ) ) ] ⋅ ( N θ ( r , θ , ζ ) ) = 1 r ( ∂ ( f ∘ Φ ) ∂ θ ) ( r , θ , z ) {\displaystyle \left[\left(\operatorname {grad} f\right)(\Phi (r,\theta ,\zeta ))\right]\cdot \left({{\mathbf {N} }_{\theta }}(r,\theta ,\zeta )\right)={\frac {1}{r}}\left({\frac {\partial (f\circ \Phi )}{\partial \theta }}\right)(r,\theta ,z)} (6-2-3)
[ ( grad  f ) ( Φ ( r , θ , ζ ) ) ] ⋅ ( N ζ ( r , θ , ζ ) ) = ( ∂ ( f ∘ Φ ) ∂ ζ ) ( r , θ , ζ ) {\displaystyle \left[\left(\operatorname {grad} f\right)(\Phi (r,\theta ,\zeta ))\right]\cdot \left({{\mathbf {N} }_{\zeta }}(r,\theta ,\zeta )\right)=\left({\frac {\partial (f\circ \Phi )}{\partial \zeta }}\right)(r,\theta ,\zeta )} (6-2-3)
が成り立つ。従って「ベクトル場の円柱座標表示」と同様にして、上の等式が示せる。
∫ Δ 2 X = ∫ ζ = 0 ζ = ζ 0 ∫ θ = 0 θ = 2 π ( X ⋅ ( ∂ I 2 ∂ θ × ∂ I 2 ∂ ζ ) ) d θ d ζ {\displaystyle \int _{{\Delta }_{2}}{\mathbf {X} }=\int _{\zeta =0}^{\zeta ={{\zeta }_{0}}}{\int _{\theta =0}^{\theta =2\pi }{\left(\mathbf {X} \cdot \left({\frac {\partial {{\mathbf {I} }_{2}}}{\partial \theta }}\times {\frac {\partial {{\mathbf {I} }_{2}}}{\partial \zeta }}\right)\right)}}\ d\theta d\zeta } (5-2-3)
∫ Δ 3 X = ∫ r = 0 r = r 0 ∫ θ = 0 θ = 2 π ( X ⋅ ( ∂ I 3 ∂ r × ∂ I 3 ∂ θ ) ) d θ d r {\displaystyle \int _{{\Delta }_{3}}{\mathbf {X} }=\int _{r=0}^{r={{r}_{0}}}{\int _{\theta =0}^{\theta =2\pi }{\left(\mathbf {X} \cdot \left({\frac {\partial {{\mathbf {I} }_{3}}}{\partial r}}\times {\frac {\partial {{\mathbf {I} }_{3}}}{\partial \theta }}\right)\right)}}\ d\theta dr} (5-2-4)
これらI1 , I2 , I3 を、それぞれΔ1 , Δ2 , Δ3 のパラメータと呼ぶ。
( x y z ) = I 1 ( r , θ ) = Φ ( r , θ , 0 ) = ( r cos  θ r sin  θ 0 ) {\displaystyle \left({\begin{matrix}x\\y\\z\\\end{matrix}}\right)={{\mathbf {I} }_{1}}(r,\theta )=\Phi (r,\theta ,0)=\left({\begin{matrix}r\cos \theta \\r\sin \theta \\0\\\end{matrix}}\right)} (3-2-8)
( x y z ) = I 2 ( θ , ζ ) = Φ ( r 0 , θ , ζ ) = ( r 0 cos  θ r 0 sin  θ ζ ) {\displaystyle \left({\begin{matrix}x\\y\\z\\\end{matrix}}\right)={{\mathbf {I} }_{2}}(\theta ,\zeta )=\Phi ({{r}_{0}},\theta ,\zeta )=\left({\begin{matrix}{{r}_{0}}\cos \theta \\{{r}_{0}}\sin \theta \\\zeta \\\end{matrix}}\right)} (3-2-9)
が成り立つ。従って、
回転 rot の円柱座標系表示については、次の等式が成立する。
( rot  X ) ( Φ ( r , θ , ζ ) ) = ( ( rot  X ) r ( r , θ , z ) ) N r ( r , θ , ζ ) + ( ( rot  X ) θ ( r , θ , z ) ) N θ ( r , θ , ζ ) + ( ( rot  X ) θ ( r , θ , z ) ) N ζ ( r , θ , ζ ) {\displaystyle (\operatorname {rot} X)(\Phi (r,\theta ,\zeta ))=\left({{(\operatorname {rot} X)}_{r}}(r,\theta ,z)\right){{\mathbf {N} }_{r}}(r,\theta ,\zeta )+\left({{(\operatorname {rot} X)}_{\theta }}(r,\theta ,z)\right){{\mathbf {N} }_{\theta }}(r,\theta ,\zeta )+\left({{(\operatorname {rot} X)}_{\theta }}(r,\theta ,z)\right){{\mathbf {N} }_{\zeta }}(r,\theta ,\zeta )}