円柱座標変換
(5-1-5)、(5-2-2)、(5-2-3)、(5-2-4)より、
∫ ζ = 0 ζ = ζ 0 ∫ θ = 0 θ = 2 π r 0 X r ( r 0 , θ , ζ ) d θ d ζ = ∫ M ( ( ∂ X x ∂ x ) + ( ∂ X y ∂ y ) ) d x d y d z {\displaystyle \int _{\zeta =0}^{\zeta ={{\zeta }_{0}}}{\int _{\theta =0}^{\theta =2\pi }{{{r}_{0}}{{X}_{r}}({{r}_{0}},\theta ,\zeta )}}\ d\theta d\zeta =\int _{M}{\left(\left({\frac {\partial {{X}_{x}}}{\partial x}}\right)+\left({\frac {\partial {{X}_{y}}}{\partial y}}\right)\ \right)dxdydz}} (5-3-3)
ラプラシアンについても、r ≠ 0 をみたす任意の(r , θ, ζ) に対して以下の等式が成立する。
( Δ f ) ( Φ ( r , θ , ζ ) ) = 1 r ∂ ∂ r ( r ( ∂ ( f ∘ Φ ) ∂ r ) ) ( r , θ , ζ ) + 1 r 2 ( ∂ 2 ( f ∘ Φ ) ∂ θ 2 ) ( r , θ , ζ ) + ( ∂ 2 ( f ∘ Φ ) ∂ ξ 2 ) ( r , θ , ζ ) {\displaystyle \left(\Delta f\right)\left(\Phi (r,\theta ,\zeta )\right)={\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r\left({\frac {\partial (f\circ \Phi )}{\partial r}}\right)\right)(r,\theta ,\zeta )+{\frac {1}{{r}^{2}}}\left({\frac {{{\partial }^{2}}(f\circ \Phi )}{\partial {{\theta }^{2}}}}\right)(r,\theta ,\zeta )+\left({\frac {{{\partial }^{2}}(f\circ \Phi )}{\partial {{\xi }^{2}}}}\right)(r,\theta ,\zeta )} (6-2-3)
{ X r ( r , θ , ζ ) = X ( Φ ( r , θ , ζ ) ) ⋅ N r ( r , θ , ζ ) X θ ( r , θ , ζ ) = X ( Φ ( r , θ , ζ ) ) ⋅ N θ ( r , θ , ζ ) X ζ ( r , θ , ζ ) = X ( Φ ( r , θ , ζ ) ) ⋅ N ζ ( r , θ , ζ ) {\displaystyle \left\{{\begin{matrix}X_{r}(r,\theta ,\zeta )=\mathbf {X} \left(\Phi (r,\theta ,\zeta )\right)\cdot {\mathbf {N} _{r}}(r,\theta ,\zeta )\\X_{\theta }(r,\theta ,\zeta )=\mathbf {X} \left(\Phi (r,\theta ,\zeta )\right)\cdot {\mathbf {N} _{\theta }}(r,\theta ,\zeta )\\X_{\zeta }(r,\theta ,\zeta )=\mathbf {X} \left(\Phi (r,\theta ,\zeta )\right)\cdot {\mathbf {N} _{\zeta }}(r,\theta ,\zeta )\end{matrix}}\right.} (4-1-3)
である。ここで、・は内積を意味する。