${\displaystyle {\boldsymbol {p}}=\lim _{\delta \to 0}{\big \{}q\,{\boldsymbol {r}}_{+}-q\,{\boldsymbol {r}}_{-}{\big \}}=\lim _{\delta \to 0}q\,{\boldsymbol {\delta }}}$

${\displaystyle {\begin{aligned}\rho ({\boldsymbol {x}})&=\lim _{\delta \to 0}{\Big \{}q\,\delta ^{3}({\boldsymbol {x}}-{\boldsymbol {r}}_{+})-q\,\delta ^{3}({\boldsymbol {x}}-{\boldsymbol {r}}_{-}){\Big \}}\\&=-{\boldsymbol {p}}\cdot \operatorname {grad} \delta ^{3}({\boldsymbol {x}}-{\boldsymbol {r}})\\\end{aligned}}}$

${\displaystyle \rho ({\boldsymbol {x}})=-\sum _{i}{\boldsymbol {p}}_{i}\cdot \operatorname {grad} \delta ^{3}({\boldsymbol {x}}-{\boldsymbol {r}}_{i})}$

${\displaystyle F({\boldsymbol {x}})=\int \rho ({\boldsymbol {y}})\,K({\boldsymbol {x}}-{\boldsymbol {y}})\,d^{3}y}$

${\displaystyle {\begin{aligned}F({\boldsymbol {x}})&=-\int K({\boldsymbol {x}}-{\boldsymbol {y}})\,{\boldsymbol {p}}\cdot \operatorname {grad} \delta ^{3}({\boldsymbol {y}}-{\boldsymbol {r}})\,d^{3}y\\&=-\int {\boldsymbol {p}}\cdot \operatorname {grad} K({\boldsymbol {x}}-{\boldsymbol {y}})\,\delta ^{3}({\boldsymbol {y}}-{\boldsymbol {r}})\,d^{3}y-\oint ({\boldsymbol {p}}\cdot {\boldsymbol {n}})\,K({\boldsymbol {x}}-{\boldsymbol {y}})\,\delta ^{3}({\boldsymbol {y}}-{\boldsymbol {r}})\,dS\\&=-{\boldsymbol {p}}\cdot \operatorname {grad} K({\boldsymbol {x}}-{\boldsymbol {r}})+{\text{(boundary term)}}\\\end{aligned}}}$

${\displaystyle F({\boldsymbol {x}})=-{\boldsymbol {p}}\cdot \operatorname {grad} K({\boldsymbol {x}}-{\boldsymbol {r}})}$

${\displaystyle \phi ({\boldsymbol {x}})=-{\frac {1}{4\pi \epsilon _{0}}}{\boldsymbol {p}}\cdot \operatorname {grad} {\frac {1}{|{\boldsymbol {x}}-{\boldsymbol {r}}|}}={\frac {1}{4\pi \epsilon _{0}}}{\frac {{\boldsymbol {p}}\cdot ({\boldsymbol {x}}-{\boldsymbol {r}})}{|{\boldsymbol {x}}-{\boldsymbol {r}}|^{3}}}}$

${\displaystyle {\boldsymbol {p}}({\boldsymbol {x}})=\int ({\boldsymbol {y}}-{\boldsymbol {x}})\,\rho ({\boldsymbol {y}})\,d^{3}y}$

${\displaystyle \rho ({\boldsymbol {x}})=\sum _{i}q_{i}\,\delta ({\boldsymbol {x}}-{\boldsymbol {r}}_{i})}$

${\displaystyle {\boldsymbol {p}}({\boldsymbol {x}})=\sum _{i}q_{i}\ ({\boldsymbol {r}}_{i}-{\boldsymbol {x}})}$

${\displaystyle Q=\int \rho ({\boldsymbol {y}})\,d^{3}y=0}$

${\displaystyle {\boldsymbol {p}}({\boldsymbol {x}})=\int {\boldsymbol {y}}\,\rho ({\boldsymbol {y}})\,d^{3}y}$

${\displaystyle {\boldsymbol {p}}({\boldsymbol {x}})=\sum _{i}{\boldsymbol {p}}_{i}}$

${\displaystyle {\boldsymbol {p}}=\alpha {\boldsymbol {E}}}$