\begin{eqnarray} x = ρ\mathop{cos}\nolimits ϕ,y = ρ\mathop{sin}\nolimits ϕ,& & %&(7.1) \\ ρ = \sqrt{{x}^{2 } + {y}^{2}},ϕ =\mathop{ arctan}\nolimits (y∕x),& & %&(7.2) \\ \end{eqnarray}

\begin{eqnarray} {∂\ \over ∂x}& =& {∂ρ\over ∂x} {∂\ \over ∂ρ} + {∂ϕ\over ∂x} {∂\ \over ∂ϕ} %& \\ & =& {x\over ρ} {∂\ \over ∂ρ} − {y\over { ρ}^{2}} {∂\ \over ∂ϕ} %& \\ & =& \mathop{cos}\nolimits ϕ {∂\ \over ∂ρ} −{\mathop{sin}\nolimits ϕ\over ρ} {∂\ \over ∂ϕ},%&(7.3) \\ {∂\ \over ∂y}& =& {∂ρ\over ∂y} {∂\ \over ∂ρ} + {∂ϕ\over ∂y} {∂\ \over ∂ϕ} %& \\ & =& {y\over ρ} {∂\ \over ∂ρ} + {x\over { ρ}^{2}} {∂\ \over ∂ϕ} %& \\ & =& \mathop{sin}\nolimits ϕ {∂\ \over ∂ρ} + {\mathop{cos}\nolimits ϕ\over ρ} {∂\ \over ∂ϕ},%&(7.4) \\ \end{eqnarray}

\begin{eqnarray} \mathop{∇}& =&{ \hat{e}}_{ρ} {∂\ \over ∂ρ} +\hat{{ e}}_{ϕ}{1\over ρ} {∂\ \over ∂ϕ}%&(7.5) \\ \end{eqnarray}

\begin{eqnarray} \hat{{e}}_{ρ}& =& (\mathop{cos}\nolimits ϕ,\mathop{sin}\nolimits ϕ), %& \\ \hat{{e}}_{ϕ}& =& (−\mathop{sin}\nolimits ϕ,\mathop{cos}\nolimits ϕ),%&(7.6) \\ \end{eqnarray}

\begin{eqnarray}{ \mathop{∇}}^{2}& =& {\mathop{cos}\nolimits }^{2}ϕ {{∂}^{2}\ \over ∂{ρ}^{2}} + {\mathop{sin}\nolimits ϕ\mathop{cos}\nolimits ϕ\over {ρ}^{2}} {∂\ \over ∂ϕ} + {{\mathop{sin}\nolimits }^{2}ϕ\over ρ} {∂\ \over ∂ρ} + {{\mathop{sin}\nolimits }^{2}ϕ\over {ρ}^{2}} {{∂}^{2}\ \over ∂{ϕ}^{2}} + {\mathop{sin}\nolimits ϕ\mathop{cos}\nolimits ϕ\over {ρ}^{2}} {∂\ \over ∂ϕ} %& \\ & & +{\mathop{sin}\nolimits }^{2}ϕ {{∂}^{2}\ \over ∂{ρ}^{2}} −{\mathop{sin}\nolimits ϕ\mathop{cos}\nolimits ϕ\over {ρ}^{2}} {∂\ \over ∂ϕ} + {{\mathop{cos}\nolimits }^{2}ϕ\over ρ} {∂\ \over ∂ρ} + {{\mathop{cos}\nolimits }^{2}ϕ\over {ρ}^{2}} {{∂}^{2}\ \over ∂{ϕ}^{2}} −{\mathop{sin}\nolimits ϕ\mathop{cos}\nolimits ϕ\over {ρ}^{2}} {∂\ \over ∂ϕ}%&(7.7) \\ & =& {{∂}^{2}\ \over ∂{ρ}^{2}} + {1\over ρ} {∂\ \over ∂ρ} + {1\over { ρ}^{2}} {{∂}^{2}\ \over ∂{ϕ}^{2}} %& \\ & =& {1\over ρ} {∂\ \over ∂ρ}\left (ρ {∂\ \over ∂ρ}\right ) + {1\over { ρ}^{2}} {{∂}^{2}\ \over ∂{ϕ}^{2}}. %& \\ & & %&(7.8) \\ \end{eqnarray}