Quaternionic Differential Equations

Generalized field equations hold for all basic fields. Generalized field equations fit best in a quaternionic setting.

Quaternions consist of a real number valued scalar part and a three-dimensional spatial vector that represents the imaginary part.

The multiplication rule of quaternions indicates that several independent parts constitute the product. In this comment, we use a suffix ${\displaystyle _{r}}$ to indicate the scalar real part of a quaternion, and we use ${\displaystyle {\vec {a}}}$ to indicate the imaginary vector part of quaternion ${\displaystyle a}$.

${\displaystyle c=c_{r}+{\vec {c}}=ab=(a_{r}+{\vec {a}})(b_{r}+{\vec {b}})=a_{r}b_{r}-\langle {\vec {a}},{\vec {b}}\rangle +a_{r}{\vec {b}}+{\vec {a}}b_{r}{\color {Red}\pm }{\vec {a}}\times {\vec {b}}}$

(1)

The ${\displaystyle {\color {Red}\pm }}$ in equation (1) indicates that quaternions exist in right-handed and left-handed versions.

The formula can be used to check the completeness of a set of equations that follow from the application of the product rule.

The quaternionic conjugate of a is ${\displaystyle a^{*}=a_{r}-{\vec {a}}}$ From the product rule follows the formula for the norm ${\displaystyle |a|}$ of quaternion ${\displaystyle a}$.

${\displaystyle |a|^{2}=aa^{*}=(a_{r}+{\vec {a}})(a_{r}-{\vec {a}})=a_{r}a_{r}-\langle {\vec {a}},{\vec {a}}\rangle }$

(2)

We define the quaternionic nabla as ${\displaystyle \nabla =\left\{{\partial \over \partial \tau },{\partial \over \partial x},{\partial \over \partial y},{\partial \over \partial z}\right\}=\nabla _{r}+{\vec {\nabla }};\quad \nabla _{r}={\partial \over \partial \tau };\quad {\vec {\nabla }}=\left\{{\partial \over \partial x},{\partial \over \partial y},{\partial \over \partial z}\right\}}$.

The quaternionic nabla ${\displaystyle \nabla }$ acts like a multiplying operator. The (partial) differential ${\displaystyle \nabla \psi }$ represents the full first-order change of field ${\displaystyle \psi }$. We assume that ${\displaystyle \varphi =\nabla \psi }$ exists in an enclosed region of the domain of ${\displaystyle \psi }$.

${\displaystyle \varphi =\varphi _{r}+{\vec {\varphi }}=\nabla \psi =(\nabla _{r}+{\vec {\nabla }})(\psi _{r}+{\vec {\psi }})=\nabla _{r}\psi _{r}-\langle {\vec {\nabla }},{\vec {\psi }}\rangle +\nabla _{r}{\vec {\psi }}+{\vec {\nabla }}\psi _{r}{\color {Red}\pm }{\vec {\nabla }}\times {\vec {\psi }}}$

(3)

The equation is a quaternionic first order partial differential equation.

The five terms on the right side show the components that constitute the full first-order change.

They represent subfields of field ${\displaystyle \varphi }$, and often they get special names and symbols.

${\displaystyle {\vec {\nabla }}\psi _{r}}$ is the gradient of ${\displaystyle \psi _{r}}$

${\displaystyle \langle {\vec {\nabla }},{\vec {\psi }}\rangle }$ is the divergence of ${\displaystyle {\vec {\psi }}}$.

${\displaystyle {\vec {\nabla }}\times {\vec {\psi }}}$ is the curl of ${\displaystyle {\vec {\psi }}}$

${\displaystyle \varphi _{r}=\nabla _{r}\psi _{r}-\langle {\vec {\nabla }},{\vec {\psi }}\rangle }$

(4)

(Equation (4) has no equivalent in Maxwell's equations!)

${\displaystyle {\vec {\varphi }}=\nabla _{r}{\vec {\psi }}+{\vec {\nabla }}\psi _{r}\pm {\vec {\nabla }}\times {\vec {\psi }}=\mp {\vec {B}}-{\vec {E}}}$

(5)

${\displaystyle {\vec {E}}\ {\overset {\underset {\mathrm {def} }{}}{=}}\ -\nabla _{r}{\vec {\psi }}-{\vec {\nabla }}\psi _{r}}$

(6)

${\displaystyle {\vec {B}}\ {\overset {\underset {\mathrm {def} }{}}{=}}\ {\vec {\nabla }}\times {\vec {\psi }}}$

(7)

${\displaystyle {\vec {J}}\ {\overset {\underset {\mathrm {def} }{}}{=}}\ {\vec {\nabla }}\times {\vec {B}}-\nabla _{r}{\vec {E}}}$

(8)

The nabla exists in several coordinate systems. This section shows the representation of the quaternionic nabla for Cartesian coordinate systems and for polar coordinate systems.

${\displaystyle {\vec {\nabla }}=\{{\frac {\partial {}}{\partial {x}}},\,{\frac {\partial {}}{\partial {y}}},\,{\frac {\partial {}}{\partial {z}}}\}={\frac {\partial {}}{\partial {x}}}{\vec {\hat {x}}}+{\frac {\partial {}}{\partial {y}}}{\vec {\hat {y}}}+{\frac {\partial {}}{\partial {z}}}{\vec {\hat {z}}}}$

(9)

${\displaystyle {\vec {\nabla }}a_{r}={\frac {\partial {}}{\partial {x}}}{\vec {\hat {x}}}+{\frac {\partial {}}{\partial {y}}}{\vec {\hat {y}}}+{\frac {\partial {}}{\partial {z}}}{\vec {\hat {z}}}={\frac {\partial {a_{r}}}{\partial {\rho }}}{\vec {\hat {\rho }}}+{\frac {1}{\rho }}{\frac {\partial {a_{r}}}{\partial {\theta }}}{\vec {\hat {\theta }}}+{\frac {1}{\rho \sin(\theta )}}{\frac {\partial {a_{r}}}{\partial {\phi }}}{\vec {\hat {\phi }}}}$

(10)

Here ${\displaystyle \{\rho ,\theta ,\phi \}}$ are the coordinates with ${\displaystyle \{{\vec {\hat {\rho }}},{\vec {\hat {\theta }}},{\vec {\hat {\phi }}}\}}$ as coordinate axes.

${\displaystyle \langle {\vec {\nabla }},{\vec {a}}\rangle ={\frac {\partial {a_{x}}}{\partial {x}}}{\vec {\hat {x}}}+{\frac {\partial {a_{y}}}{\partial {y}}}{\vec {\hat {y}}}+{\frac {\partial {a_{z}}}{\partial {z}}}{\vec {\hat {z}}}={\frac {1}{\rho ^{2}}}{\frac {\partial {(\rho ^{2}a_{\rho })}}{\partial {\rho }}}+{\frac {1}{\rho \sin(\theta )}}{\frac {\partial {(a_{\theta }\sin(\theta )}}{\partial {\theta )}}}+{\frac {1}{\rho \sin(\theta )}}{\frac {\partial {a_{\phi }}}{\partial {\phi }}}}$

(11)

${\displaystyle \langle {\vec {\nabla }},{\vec {\nabla }}\rangle f_{r}={\frac {1}{\rho ^{2}}}{\frac {\partial {(\rho ^{2}{\frac {\partial {f_{r}}}{\partial \rho }})}}{\partial {\rho }}}+{\frac {1}{\rho ^{2}\sin(\theta )}}{\frac {\partial {(\sin(\theta ){\frac {\partial {f_{r}}}{\partial {\theta }}}}}{\partial {\theta )}}}+{\frac {1}{\rho ^{2}\sin ^{2}(\theta )}}{\frac {\partial ^{2}{f_{r}}}{\partial {\phi ^{2}}}}}$

(12)

${\displaystyle {\vec {\nabla }}\times {\vec {a}}=({\frac {\partial {a_{z}}}{\partial {y}}}-{\frac {\partial {a_{y}}}{\partial {z}}}){\vec {\hat {x}}}+({\frac {\partial {a_{x}}}{\partial {z}}}-{\frac {\partial {a_{z}}}{\partial {x}}}){\vec {\hat {y}}}+({\frac {\partial {a_{y}}}{\partial {x}}}-{\frac {\partial {a_{x}}}{\partial {y}}}){\vec {\hat {z}}}}$

(13)

${\displaystyle \qquad ={\frac {1}{\rho \sin(\theta )}}({\frac {\partial {a_{\phi }}}{\partial {\phi }}}-{\frac {\partial {(a_{\theta }\sin(\theta )}}{\partial {\theta )}}}){\vec {\hat {\rho }}}+({\frac {1}{\rho ^{2}}}{\frac {\partial {(\rho ^{2}a_{\rho })}}{\partial {\rho }}}-{\frac {1}{\rho \sin(\theta )}}{\frac {\partial {a_{\phi }}}{\partial {\phi }}}){\vec {\hat {\theta }}}+({\frac {1}{\rho \sin(\theta )}}{\frac {\partial {(a_{\theta }\sin(\theta )}}{\partial {\theta )}}}-{\frac {1}{\rho ^{2}}}{\frac {\partial {(\rho ^{2}a_{\rho })}}{\partial {\rho }}}){\vec {\hat {\phi }}}}$

${\displaystyle {\vec {\nabla }}\times {\vec {a}}={\frac {1}{\rho \sin(\theta )}}({\frac {\partial {a_{\phi }}}{\partial {\phi }}}-{\frac {\partial {(a_{\theta }\sin(\theta )}}{\partial {\theta )}}}){\vec {\hat {\rho }}}+({\frac {1}{\rho ^{2}}}{\frac {\partial {(\rho ^{2}a_{\rho })}}{\partial {\rho }}}-{\frac {1}{\rho \sin(\theta )}}{\frac {\partial {a_{\phi }}}{\partial {\phi }}}){\vec {\hat {\theta }}}+({\frac {1}{\rho \sin(\theta )}}{\frac {\partial {(a_{\theta }\sin(\theta )}}{\partial {\theta )}}}-{\frac {1}{\rho ^{2}}}{\frac {\partial {(\rho ^{2}a_{\rho })}}{\partial {\rho }}}){\vec {\hat {\phi }}}}$

(14)

The spatial nabla operator shows behavior that is valid for all quaternionic functions for which the first order partial differential equation exists.

Here the quaternionic field ${\displaystyle a=a_{r}+{\vec {a}}}$ obeys the requirement that the first order partial differential ${\displaystyle \nabla a}$ exists.

${\displaystyle \Delta \ {\overset {\underset {\mathrm {def} }{}}{=}}\ \nabla ^{2}\ {\overset {\underset {\mathrm {def} }{}}{=}}\ \langle {\vec {\nabla }},{\vec {\nabla }}\rangle \ {\overset {\underset {\mathrm {def} }{}}{=}}\ {\frac {\partial {}^{2}}{\partial {x}^{2}}}+{\frac {\partial {}^{2}}{\partial {y}^{2}}}+{\frac {\partial {}^{2}}{\partial {z}^{2}}}}$

(15)

${\displaystyle \langle {\vec {\nabla }},{\vec {\nabla }}a_{r}\rangle =\langle {\vec {\nabla }},{\vec {\nabla }}\rangle a_{r}=\nabla ^{2}a_{r}}$

(16)

${\displaystyle \langle {\vec {\nabla }},{\vec {\nabla }}\rangle {\vec {a}}\ {\overset {\underset {\mathrm {def} }{}}{=}}\ \nabla ^{2}{\vec {a}}}$

(17)

${\displaystyle \langle {\vec {\nabla }}\times {\vec {\nabla }},{\vec {a}}\rangle =0}$

(18)

${\displaystyle {\vec {\nabla }}\times {\vec {\nabla }}a_{r}={\vec {0}}}$

(19)

${\displaystyle \langle {\vec {\nabla }},{\vec {\nabla }}\times {\vec {a}}\rangle =0}$

(20)

${\displaystyle {\vec {\nabla }}\times ({\vec {\nabla }}\times {\vec {a}})={\vec {\nabla }}\langle {\vec {\nabla }},{\vec {a}}\rangle -\langle {\vec {\nabla }},{\vec {\nabla }}\rangle {\vec {a}}}$

(21)

${\displaystyle {\vec {\nabla }}{\vec {\nabla }}a=({\vec {\nabla }}\times {\vec {\nabla }})a-\langle {\vec {\nabla }},{\vec {\nabla }}\rangle a===Quater({\vec {\nabla }}\times {\vec {\nabla }}){\vec {a}}-\langle {\vec {\nabla }},{\vec {\nabla }}\rangle a={\vec {\nabla }}\langle {\vec {\nabla }},{\vec {a}}\rangle -2\langle {\vec {\nabla }},{\vec {\nabla }}\rangle a+\langle {\vec {\nabla }},{\vec {\nabla }}\rangle a_{r}}$

(22)

For constant ${\displaystyle {\vec {k}}}$ and parameter ${\displaystyle q=q_{r}+{\vec {q}}=\{q_{r},q_{x},q_{y},q_{z}\}}$ holds

${\displaystyle {\vec {\nabla }}\langle {\vec {k}},{\vec {q}}\rangle ={\vec {k}}}$

(23)

${\displaystyle \langle {\vec {\nabla }},{\vec {q}}\rangle =3}$

(24)

${\displaystyle {\vec {\nabla }}\times {\vec {q}}={\vec {0}}}$

(25)

${\displaystyle {\vec {\nabla }}\,|{\vec {q}}|={\frac {\vec {q}}{|{\vec {q}}|}};\ \nabla \,|q|={\frac {q}{|q|}}}$

(26)

${\displaystyle {\vec {\nabla }}{\frac {1}{|{\vec {q}}-{\vec {q}}^{'}|}}=-{\frac {{\vec {q}}-{\vec {q}}^{'}}{|{\vec {q}}-{\vec {q}}^{'}|^{3}}};\ \nabla {\frac {1}{|q-q^{'}|}}=-{\frac {q-q^{'}}{|q-q^{'}|^{3}}}}$

(27)

${\displaystyle \langle {\vec {\nabla }},{\vec {\nabla }}{\frac {1}{|{\vec {q}}-{\vec {q}}^{'}|}}\rangle =\langle {\vec {\nabla }},{\vec {\nabla }}\rangle {\frac {1}{|{\vec {q}}-{\vec {q}}^{'}|}}=-\langle {\vec {\nabla }},{\frac {{\vec {q}}-{\vec {q}}^{'}}{|{\vec {q}}-{\vec {q}}^{'}|^{3}}}\rangle =4\,\phi \,\delta ({\vec {q}}-{\vec {q}}^{'})}$

(28)

${\displaystyle \nabla \ q=1-3;\nabla ^{*}q=1+3;\nabla q^{*}=1+3}$

(29)

The term ${\displaystyle ({\vec {\nabla }}\times {\vec {\nabla }})f}$ indicates the curvature of field ${\displaystyle f}$.

The term ${\displaystyle \langle {\vec {\nabla }},{\vec {\nabla }}\rangle f}$ indicates the stress of field ${\displaystyle f}$

With the help of the properties of the spatial nabla operator follows an interesting second-order partial differential equation.

${\displaystyle \langle {\vec {\nabla }},{\vec {B}}\rangle =\langle {\vec {\nabla }},{\vec {\nabla }}\times {\vec {\psi }}\rangle =0}$

(30)

${\displaystyle \langle {\vec {\nabla }},{\vec {E}}\rangle =-\nabla _{r}\langle {\vec {\nabla }},{\vec {\psi }}\rangle -\langle {\vec {\nabla }},{\vec {\nabla }}\psi _{r}\rangle =-\nabla _{r}\langle {\vec {\nabla }},{\vec {\psi }}\rangle -\langle {\vec {\nabla }},{\vec {\nabla }}\rangle \psi _{r}}$

(31)

${\displaystyle \langle {\vec {\nabla }},{\vec {B}}-{\vec {E}}\rangle =\nabla _{r}\langle {\vec {\nabla }},{\vec {\psi }}\rangle +\langle {\vec {\nabla }},{\vec {\nabla }}\rangle \psi _{r}}$

(32)

${\displaystyle \nabla _{r}{\vec {E}}=-\nabla _{r}\nabla _{r}{\vec {\psi }}-\nabla _{r}{\vec {\nabla }}\psi _{r}}$

(33)

${\displaystyle {\vec {\nabla }}\times {\vec {E}}=-{\vec {\nabla }}\times (\nabla _{r}{\vec {\psi }})-{\vec {\nabla }}\times ({\vec {\nabla }}\psi _{r})=-\nabla _{r}{\vec {\nabla }}\times {\vec {\psi }}=-\nabla _{r}{\vec {B}}}$

(34)

${\displaystyle {\vec {\nabla }}\times {\vec {B}}={\vec {\nabla }}\times ({\vec {\nabla }}\times {\vec {\psi }})={\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle -\langle {\vec {\nabla }},{\vec {\nabla }}\rangle {\vec {\psi }}}$

(35)

${\displaystyle {\vec {\nabla }}\times {\vec {B}}-{\vec {\nabla }}\times {\vec {E}}={\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle -\langle {\vec {\nabla }},{\vec {\nabla }}\rangle {\vec {\psi }}+\nabla _{r}{\vec {B}}}$

(36)

${\displaystyle {\vec {\nabla }}\times ({\vec {B}}-{\vec {E}})-\nabla _{r}({\vec {B}}-{\vec {E}})={\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle -\langle {\vec {\nabla }},{\vec {\nabla }}\rangle {\vec {\psi }}-\nabla _{r}\nabla _{r}{\vec {\psi }}-\nabla _{r}{\vec {\nabla }}\psi _{r}}$

(37)

${\displaystyle {\vec {\nabla }}\times ({\vec {B}}-{\vec {E}})-\nabla _{r}({\vec {B}}-{\vec {E}})=-(\langle {\vec {\nabla }},{\vec {\nabla }}\rangle +\nabla _{r}\nabla _{r}){\vec {\psi }}-\nabla _{r}{\vec {\nabla }}\psi _{r}+{\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle }$

(38)

${\displaystyle {\vec {\nabla }}\times ({\vec {B}}-{\vec {E}})-\langle {\vec {\nabla }},{\vec {B}}-{\vec {E}}\rangle -\nabla _{r}({\vec {B}}-{\vec {E}})=\nabla ^{*}{\vec {\varphi }}=-(\langle {\vec {\nabla }},{\vec {\nabla }}\rangle +\nabla _{r}\nabla _{r}){\vec {\psi }}-\nabla _{r}{\vec {\nabla }}\psi _{r}+{\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle -\nabla _{r}\langle {\vec {\nabla }},{\vec {\psi }}\rangle -\langle {\vec {\nabla }},{\vec {\nabla }}\rangle \psi _{r}}$

(39)

${\displaystyle {\vec {\nabla }}\varphi _{r}={\vec {\nabla }}(\nabla _{r}\psi _{r}+\langle {\vec {\nabla }},{\vec {\psi }}\rangle )=\nabla _{r}{\vec {\nabla }}\psi _{r}+{\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle }$

(40)

${\displaystyle \nabla _{r}\varphi _{r}=\nabla _{r}(\nabla _{r}\psi _{r}-\langle {\vec {\nabla }},{\vec {\psi }}\rangle )=\nabla _{r}\nabla _{r}\psi _{r}-\nabla _{r}\langle {\vec {\nabla }},{\vec {\psi }}\rangle }$

(41)

${\displaystyle \nabla ^{*}\varphi _{r}=-\nabla _{r}{\vec {\nabla }}\psi _{r}-{\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle +\nabla _{r}\nabla _{r}\psi _{r}-\nabla _{r}\langle {\vec {\nabla }},{\vec {\psi }}\rangle }$

(42)

${\displaystyle \nabla ^{*}\varphi =(\langle {\vec {\nabla }},{\vec {\nabla }}\rangle +\nabla _{r}\nabla _{r}){\vec {\psi }}+\nabla _{r}{\vec {\nabla }}\psi _{r}-{\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle +\nabla _{r}\langle {\vec {\nabla }},{\vec {\psi }}\rangle +\langle {\vec {\nabla }},{\vec {\nabla }}\rangle \psi _{r}-\nabla _{r}{\vec {\nabla }}\psi _{r}-{\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle +\nabla _{r}\nabla _{r}\psi _{r}-\nabla _{r}\langle {\vec {\nabla }},{\vec {\psi }}\rangle }$

(43)

${\displaystyle \chi =\nabla ^{*}\varphi =(\langle {\vec {\nabla }},{\vec {\nabla }}\rangle +\nabla _{r}\nabla _{r})\psi +{\color {red}\nabla _{r}{\vec {\nabla }}\psi _{r}}-{\color {blue}{\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle }+{\color {green}\nabla _{r}\langle {\vec {\nabla }},{\vec {\psi }}\rangle }-{\color {red}\nabla _{r}{\vec {\nabla }}\psi _{r}}-{\color {blue}{\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle }-{\color {green}\nabla _{r}\langle {\vec {\nabla }},{\vec {\psi }}\rangle }=(\langle {\vec {\nabla }},{\vec {\nabla }}\rangle +\nabla _{r}\nabla _{r})\psi }$

(44)

Most of the terms vanish. Further

${\displaystyle {\vec {J}}\ {\overset {\underset {\mathrm {def} }{}}{=}}\ {\vec {\nabla }}\times {\vec {B}}-\nabla _{r}{\vec {E}}={\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle -\langle {\vec {\nabla }},{\vec {\nabla }}\rangle {\vec {\psi }}+\nabla _{r}\nabla _{r}{\vec {\psi }}+\nabla _{r}{\vec {\nabla }}\psi _{r}=(\nabla _{r}\nabla _{r}-\langle {\vec {\nabla }},{\vec {\nabla }}\rangle ){\vec {\psi }}+\nabla _{r}{\vec {\nabla }}\psi _{r}+{\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle }$

(45)

From the above formulas follows that the Maxwell equations do not form a complete set.

Physicists use gauge equations to make Maxwell equations more complete.

We start with the quaternionic equivalent of the Maxwell-Faraday equation.

${\displaystyle \nabla _{r}{\vec {B}}={\vec {\nabla }}\times (\nabla _{r}{\vec {\psi }})=-{\vec {\nabla }}\times {\vec {E}}}$

(46)

Two interesting second order quaternionic partial differential equations exist.

${\displaystyle \zeta =(\nabla _{r}\nabla _{r}-\langle {\vec {\nabla }},{\vec {\nabla }}\rangle )\psi }$

(47)

This is the quaternionic equivalent of the wave equation. It offers waves as part of the solutions of the homogeneous equation.

${\displaystyle \chi =\nabla ^{*}\varphi =\nabla ^{*}\nabla \psi =\nabla \nabla ^{*}\psi =(\nabla _{r}-{\vec {\nabla }})(\varphi _{r}+{\vec {\varphi }})=(\nabla _{r}+{\vec {\nabla }})(\nabla _{r}-{\vec {\nabla }})(\psi _{r}+{\vec {\psi }})=(\nabla _{r}\nabla _{r}+\langle {\vec {\nabla }},{\vec {\nabla }}\rangle )\psi }$

(48)

This equation can be split into two first order partial wave equations ${\displaystyle \chi =\nabla ^{*}\varphi }$ and ${\displaystyle \varphi =\nabla \psi }$.

This equation does not offer waves as part of the solutions of the homogeneous equation.

${\displaystyle D=(\nabla _{r}\nabla _{r}-\langle {\vec {\nabla }},{\vec {\nabla }}\rangle )}$

(49)

This is the quaternionic equivalent of d'Alembert's operator.

${\displaystyle \boxdot =(\nabla _{r}\nabla _{r}+\langle {\vec {\nabla }},{\vec {\nabla }}\rangle )}$

(50)

This operator does not yet have a known name.

Operator ${\displaystyle \langle {\vec {\nabla }},{\vec {\nabla }}\rangle }$ represents the main part of the Poisson equation. Together with ${\displaystyle \nabla _{r}\nabla _{r}}$ this operator configures the above operators.

As is shown above, ${\displaystyle \boxdot }$ can be derived from the nabla operator. That cannot be said from the ${\displaystyle D}$ operator.