Hilbert Book Model Project/Quaternionic Field Equations/Nabla Operators

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Nabla playground

The spatial nabla and the quaternionic nabla are particular operators that play an important role in the partial differential equations that control the behavior of fields in the Hilbert Book Model.

Here we treat three kinds of nabla operators.

1. spatial nabla ${\textstyle {\vec {\nabla }}={\vec {i}}{\frac {\partial {}}{\partial {x}}}+{\vec {j}}{\frac {\partial {}}{\partial {y}}}+{\vec {k}}{\frac {\partial {}}{\partial {z}}}}$
2. quaternionic nabla ${\textstyle \nabla ={\frac {\partial {}}{\partial {\tau }}}+{\vec {i}}{\frac {\partial {}}{\partial {x}}}+{\vec {j}}{\frac {\partial {}}{\partial {y}}}+{\vec {k}}{\frac {\partial {}}{\partial {z}}}=\nabla _{r}+{\vec {\nabla }}}$
3. Dirac nabla ${\textstyle \nabla ={\frac {\partial {}}{\partial {\tau }}}+\mathbb {I} \left\{{\vec {i}}{\frac {\partial {}}{\partial {x}}}+{\vec {j}}{\frac {\partial {}}{\partial {y}}}+{\vec {k}}{\frac {\partial {}}{\partial {z}}}\right\}=\nabla _{r}+\mathbb {I} \,{\vec {\nabla }}}$

The Dirac plays a role in the interpretation of the Dirac equation.

Properties of the spatial nabla operator

The nabla product is not necessarily associative. Thus

$\nabla (\nabla ^{*}\psi )=\nabla ^{*}(\nabla \psi )=(\nabla _{r}\nabla _{r}+\langle {\vec {\nabla }},{\vec {\nabla }}\rangle )\psi \neq (\nabla \nabla ^{*})\psi =(\nabla ^{*}\nabla )\psi =(\nabla _{r}\nabla _{r}-{\vec {\nabla }}{\vec {\nabla }})\psi =(\nabla _{r}\nabla _{r}+\langle {\vec {\nabla }},{\vec {\nabla }}\rangle \mp {\vec {\nabla }}\times {\vec {\nabla }})\psi$

(1)

Nabla in different coordinate systems

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

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

(2)

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

(3)

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

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

(4)

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

(5)

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

(6)

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

(7)

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

(8)

Special formulas

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

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

(9)

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

(10)

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

(11)

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

(12)

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

(13)

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

(14)

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

(15)

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

(16)

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

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

(17)

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

(18)

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

(19)

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

(20)

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

(21)

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

(22)

This indicates the relation between the Poisson equation and the Green's function.

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

(23)

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

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

First order partial differential equation

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

(24)

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 $\varphi$, and often they get special names and symbols.

Subfields

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

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

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

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

(25)

(Equation (25) has no equivalent in Maxwell's equations!)

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

(26)

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

(27)

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

(28)

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

(29)

Derivation of higher order equations

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

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

(30)

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

(31)

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

(32)

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

(33)

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

(34)

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

(35)

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

(36)

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

(37)

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

(38)

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

(39)

Further

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

(40)

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.

Derivation of second order partial differential equation 1

$\varphi =\nabla \psi =(\nabla _{r}+{\vec {\nabla }})\psi$

(41)

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

(42)

${\vec {\varphi }}=\nabla _{r}{\vec {\psi }}+{\vec {\nabla }}\psi _{r}\pm {\vec {\nabla }}\times {\vec {\psi }}$

(43)

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

(44)

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

(45)

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

(46)

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

(47)

${\vec {\nabla }}{\vec {\varphi }}=-\nabla _{r}\langle {\vec {\nabla }},{\vec {\psi }}\rangle \pm \nabla _{r}{\vec {\nabla }}\times {\vec {\psi }}-\langle {\vec {\nabla }},{\vec {\nabla }}\rangle \psi _{r}+{\vec {\nabla }}\times {\vec {\nabla }}\times {\vec {\psi }}=-\nabla _{r}\langle {\vec {\nabla }},{\vec {\psi }}\rangle \pm \nabla _{r}{\vec {\nabla }}\times {\vec {\psi }}-\langle {\vec {\nabla }},{\vec {\nabla }}\rangle \psi _{r}+{\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle -\langle {\vec {\nabla }},{\vec {\nabla }}\rangle {\vec {\psi }}$

(48)

$\nabla _{r}{\vec {\varphi }}=\nabla _{r}(\nabla _{r}{\vec {\psi }}+{\vec {\nabla }}\psi _{r}\pm {\vec {\nabla }}\times {\vec {\psi }})=\nabla _{r}\nabla _{r}{\vec {\psi }}+\nabla _{r}{\vec {\nabla }}\psi _{r}\pm \nabla _{r}{\vec {\nabla }}\times {\vec {\psi }}$

(49)

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

(50)

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

(51)

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

(52)

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

(53)

Most of the terms vanish.

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

(54)

Derivation of second order partial differential equation 2

We add the complex imaginary base number ${\textstyle \mathbb {I} ={\sqrt {-1}}}$ to the spatial nabla operator ${\textstyle {\vec {\nabla }}}$.

$g=(\nabla _{r}+\mathbb {I} {\vec {\nabla }})f$

(55)

$g_{r}=\nabla _{r}f_{r}-\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle$

(56)

${\vec {g}}=\nabla _{r}{\vec {f}}+\mathbb {I} {\vec {\nabla }}f_{r}\pm \mathbb {I} {\vec {\nabla }}\times {\vec {f}}$

(57)

$\mathbb {I} {\vec {\nabla }}g_{r}=\mathbb {I} {\vec {\nabla }}(\nabla _{r}f_{r}-\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle )=\nabla _{r}\mathbb {I} {\vec {\nabla }}f_{r}-\mathbb {I} {\vec {\nabla }}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle$

(58)

$\nabla _{r}g_{r}=\nabla _{r}(\nabla _{r}f_{r}-\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle )=\nabla _{r}\nabla _{r}f_{r}-\nabla _{r}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle$

(59)

$(\nabla _{r}-\mathbb {I} {\vec {\nabla }})g_{r}=-\nabla _{r}\mathbb {I} {\vec {\nabla }}f_{r}+\mathbb {I} {\vec {\nabla }}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle +\nabla _{r}\nabla _{r}f_{r}-\nabla _{r}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle$

(60)

$\mathbb {I} {\vec {\nabla }}{\vec {g}}=\mathbb {I} {\vec {\nabla }}(\nabla _{r}{\vec {f}}+\mathbb {I} {\vec {\nabla }}f_{r}\pm \mathbb {I} {\vec {\nabla }}\times {\vec {f}}\rangle )=-\nabla _{r}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle \pm \nabla _{r}\mathbb {I} {\vec {\nabla }}\times {\vec {f}}-\langle \mathbb {I} {\vec {\nabla }},\mathbb {I} {\vec {\nabla }}\rangle f_{r}\mp \langle \mathbb {I} {\vec {\nabla }},\mathbb {I} {\vec {\nabla }}\times {\vec {f}}\rangle +\mathbb {I} {\vec {\nabla }}\times \mathbb {I} {\vec {\nabla }}\times {\vec {f}}$

(61)

$\mathbb {I} {\vec {\nabla }}{\vec {g}}=-\nabla _{r}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle \pm \nabla _{r}\mathbb {I} {\vec {\nabla }}\times {\vec {f}}-\langle \mathbb {I} {\vec {\nabla }},\mathbb {I} {\vec {\nabla }}\rangle f_{r}+\mathbb {I} {\vec {\nabla }}\times \mathbb {I} {\vec {\nabla }}\times {\vec {f}}=-\nabla _{r}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle \pm \nabla _{r}\mathbb {I} {\vec {\nabla }}\times {\vec {f}}-\langle \mathbb {I} {\vec {\nabla }},\mathbb {I} {\vec {\nabla }}\rangle f_{r}+\mathbb {I} {\vec {\nabla }}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle -\langle \mathbb {I} {\vec {\nabla }},\mathbb {I} {\vec {\nabla }}\rangle {\vec {f}}$

(62)

$\nabla _{r}{\vec {g}}=\nabla _{r}(\nabla _{r}{\vec {f}}+\mathbb {I} {\vec {\nabla }}f_{r}\pm \mathbb {I} {\vec {\nabla }}\times {\vec {f}})=\nabla _{r}\nabla _{r}{\vec {f}}+\nabla _{r}\mathbb {I} {\vec {\nabla }}f_{r}\pm \nabla _{r}\mathbb {I} {\vec {\nabla }}\times {\vec {f}}$

(63)

$(\nabla _{r}-\mathbb {I} {\vec {\nabla }}){\vec {g}}=\nabla _{r}\nabla _{r}{\vec {f}}+\nabla _{r}\mathbb {I} {\vec {\nabla }}f_{r}\pm \nabla _{r}\mathbb {I} {\vec {\nabla }}\times {\vec {f}}+\nabla _{r}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle \mp \nabla _{r}\mathbb {I} {\vec {\nabla }}\times {\vec {f}}+\langle \mathbb {I} {\vec {\nabla }},\mathbb {I} {\vec {\nabla }}\rangle f_{r}-\mathbb {I} {\vec {\nabla }}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle +\langle \mathbb {I} {\vec {\nabla }},\mathbb {I} {\vec {\nabla }}\rangle {\vec {f}}$

(64)

$(\nabla _{r}-\mathbb {I} {\vec {\nabla }}){\vec {g}}=\nabla _{r}\nabla _{r}{\vec {f}}+\langle \mathbb {I} {\vec {\nabla }},\mathbb {I} {\vec {\nabla }}\rangle {\vec {f}}+\nabla _{r}\mathbb {I} {\vec {\nabla }}f_{r}+\nabla _{r}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle +\langle \mathbb {I} {\vec {\nabla }},\mathbb {I} {\vec {\nabla }}\rangle f_{r}-\mathbb {I} {\vec {\nabla }}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle$

(65)

$(\nabla _{r}-\mathbb {I} {\vec {\nabla }})g=-\nabla _{r}\mathbb {I} {\vec {\nabla }}f_{r}+\mathbb {I} {\vec {\nabla }}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle +\nabla _{r}\nabla _{r}f_{r}-\nabla _{r}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle +\nabla _{r}\nabla _{r}{\vec {f}}+\langle \mathbb {I} {\vec {\nabla }},\mathbb {I} {\vec {\nabla }}\rangle {\vec {f}}+\nabla _{r}\mathbb {I} {\vec {\nabla }}f_{r}+\nabla _{r}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle +\langle \mathbb {I} {\vec {\nabla }},\mathbb {I} {\vec {\nabla }}\rangle f_{r}-\mathbb {I} {\vec {\nabla }}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle$

(66)

$\zeta =(\nabla _{r}-\mathbb {I} {\vec {\nabla }})g=(\nabla _{r}-\mathbb {I} {\vec {\nabla }})(\nabla _{r}+\mathbb {I} {\vec {\nabla }})f=(\nabla _{r}\nabla _{r}+\langle \mathbb {I} {\vec {\nabla }},\mathbb {I} {\vec {\nabla }}\rangle )(f_{r}+{\vec {f}})+\mathbb {I} {\vec {\nabla }}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle -\mathbb {I} {\vec {\nabla }}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle +\nabla _{r}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle -\nabla _{r}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle -\nabla _{r}\mathbb {I} {\vec {\nabla }}f_{r}+\nabla _{r}\mathbb {I} {\vec {\nabla }}f_{r}$

(67)

$\zeta =(\nabla _{r}-\mathbb {I} {\vec {\nabla }})g=(\nabla _{r}-\mathbb {I} {\vec {\nabla }})(\nabla _{r}+\mathbb {I} {\vec {\nabla }})f=(\nabla _{r}\nabla _{r}-\langle {\vec {\nabla }},{\vec {\nabla }}\rangle )(f_{r}+{\vec {f}})-{\vec {\nabla }}\langle {\vec {\nabla }},{\vec {f}}\rangle +{\vec {\nabla }}\langle {\vec {\nabla }},{\vec {f}}\rangle +\mathbb {I} (\nabla _{r}\langle {\vec {\nabla }},{\vec {f}}\rangle -\nabla _{r}\langle {\vec {\nabla }},{\vec {f}}\rangle -\nabla _{r}{\vec {\nabla }}f_{r}+\nabla _{r}{\vec {\nabla }}f_{r})$

(68)

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

(69)

Thus also this quaternionic second order partial differential equation splits in two first order partial differential equations. But these are no quaternionic partial differential equations!

$g=(\nabla _{r}+\mathbb {I} {\vec {\nabla }})f;\qquad \zeta =(\nabla _{r}-\mathbb {I} {\vec {\nabla }})g$

(70)