変分原理
δ F [ φ ( r ) ] = ∫ V { 1 2 | ∇ ( φ ( r ) + δ φ ( r ) ) | 2 − ρ ( r ) ε 0 ( φ ( r ) + δ φ ( r ) ) } d V − ∫ V { 1 2 | ∇ φ ( r ) | 2 − ρ ( r ) ε 0 φ ( r ) } d V = ∫ V { 1 2 ( ∇ φ ( r ) + ∇ δ φ ( r ) ) ⋅ ( ∇ φ ( r ) + ∇ δ φ ( r ) ) − ρ ( r ) ε 0 δ φ ( r ) − 1 2 | ∇ φ ( r ) | 2 } d V = ∫ V { 1 2 ( | ∇ φ ( r ) | 2 + 2 ∇ δ φ ( r ) ⋅ ∇ φ ( r ) + | ∇ δ φ ( r ) | 2 ) − ρ ( r ) ε 0 δ φ ( r ) − 1 2 | ∇ φ ( r ) | 2 } d V = ∫ V { ∇ δ φ ( r ) ⋅ ∇ φ ( r ) − ρ ( r ) ε 0 δ φ ( r ) } d V {\displaystyle {\begin{aligned}\delta F[\phi ({\boldsymbol {r}})]&=\int _{V}\left\{{1 \over 2}\left|\nabla \left(\phi ({\boldsymbol {r}})+\delta \phi ({\boldsymbol {r}})\right)\right|^{2}-{\rho ({\boldsymbol {r}}) \over \varepsilon _{0}}\left(\phi ({\boldsymbol {r}})+\delta \phi ({\boldsymbol {r}})\right)\right\}dV-\int _{V}\left\{{1 \over 2}\left|\nabla \phi ({\boldsymbol {r}})\right|^{2}-{\rho ({\boldsymbol {r}}) \over \varepsilon _{0}}\phi ({\boldsymbol {r}})\right\}dV\\&=\int _{V}\left\{{1 \over 2}\left(\nabla \phi ({\boldsymbol {r}})+\nabla \delta \phi ({\boldsymbol {r}})\right)\cdot \left(\nabla \phi ({\boldsymbol {r}})+\nabla \delta \phi ({\boldsymbol {r}})\right)-{\rho ({\boldsymbol {r}}) \over \varepsilon _{0}}\delta \phi ({\boldsymbol {r}})-{1 \over 2}\left|\nabla \phi ({\boldsymbol {r}})\right|^{2}\right\}dV\\&=\int _{V}\left\{{1 \over 2}\left(\left|\nabla \phi ({\boldsymbol {r}})\right|^{2}+2\nabla \delta \phi ({\boldsymbol {r}})\cdot \nabla \phi ({\boldsymbol {r}})+\left|\nabla \delta \phi ({\boldsymbol {r}})\right|^{2}\right)-{\rho ({\boldsymbol {r}}) \over \varepsilon _{0}}\delta \phi ({\boldsymbol {r}})-{1 \over 2}\left|\nabla \phi ({\boldsymbol {r}})\right|^{2}\right\}dV\\&=\int _{V}\left\{\nabla \delta \phi ({\boldsymbol {r}})\cdot \nabla \phi ({\boldsymbol {r}})-{\rho ({\boldsymbol {r}}) \over \varepsilon _{0}}\delta \phi ({\boldsymbol {r}})\right\}dV\end{aligned}}}