Hilbert Book Model Project/Quaternionic Field Equations/Nabla Operators/de

<Hilbert Book Model Project/de

Nabla Spielplatz

Die räumliche Nabla und die quaternionische Nabla sind besondere Operatoren, die in den partiellen Differentialgleichungen eine wichtige Rolle spielen, die das Verhalten von Feldern im Hilbert-Buchmodell steuern.

Hier behandeln wir drei Arten von Nabla Operatoren.

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

Der Dirac spielt eine Rolle bei der Interpretation der Dirac Gleichung.

Eigenschaften des räumlichen Nabla Operators

Das Nabla-Produkt ist nicht unbedingt assoziativ . So ist

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

(1)

Nabla in verschiedenen Koordinatensystemen

Die räumliche nabla existiert in mehreren koordinatensystemen. Dieser Abschnitt zeigt die Darstellung des quaternionischen Nabla für kartesische Koordinatensysteme und für Polarkoordinatensysteme 

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

(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)

Sonderformeln

Der räumliche Nabla-Operator zeigt das Verhalten, das für alle quaternionischen Funktionen gilt, für die die partielle Differentialgleichung erster Ordnung existiert.

Hier gehorcht das quaternionische Feld $a=a_{r}+{\vec {a}}$ der Forderung, dass die erste Ordnung partielle Differentiale $\nabla a$ existiert.

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

(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)

Für konstant ${\vec {k}}$ und Parameter $q=q_{r}+{\vec {q}}=\{q_{r},q_{x},q_{y},q_{z}\}$hält

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

(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)

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

(22)

Dies zeigt die Beziehung zwischen der Poisson-Gleichung und der Green's Function.

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

(23)

Der Begriff $({\vec {\nabla }}\times {\vec {\nabla }})f$ zeigt die Krümmung des Feldes $f$ an.

Der Begriff $\langle {\vec {\nabla }},{\vec {\nabla }}\rangle f$ zeigt den Stress des Feldes $f$ an.

Erste Ordnung partielle Differentialgleichung

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

(24)

Die Gleichung ist eine quaternionische partielle Differentialgleichung erster Ordnung.

Die fünf Begriffe auf der rechten Seite zeigen die Komponenten, die die vollständige Änderung der ersten Ordnung darstellen.

Sie stellen Teilfelder des Feldes $\varphi$ dar, und oft bekommen sie spezielle Namen und Symbole.

Subfelder

${\vec {\nabla }}\psi _{r}$ 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)

(Gleichung (25) hat kein Äquivalent in Maxwells Gleichungen!)

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

(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)

Ableitung höherer Ordnung

Mit Hilfe der Eigenschaften des räumlichen Nabla-Operators folgt eine interessante partielle Differentialgleichung zweiter Ordnung.

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

(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)

Des Weiteren

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

(40)

Aus den obigen Formeln folgt, dass die Maxwell-Gleichungen keinen kompletten Satz bilden. Physiker verwenden Maßstabsgleichungen, um Maxwell-Gleichungen vollständiger zu machen.

Ableitung zweiter Ordnung partielle Differentialgleichung 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)

Die meistenTerme verschwinden.

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

(54)

Ableitung zweiter Ordnung partielle Differentialgleichung 2

Wir fügen die komplexe imaginäre Basiszahl ${\textstyle \mathbb {I} ={\sqrt {-1}}}$ hinzu den räumlichen 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)

So teilt sich auch diese quaternionische partielle Differentialgleichung zweiter Ordnung in zwei partiellen Differentialgleichungen erster Ordnung.

Aber das sind keine quaternionischen partiellen Differentialgleichungen!

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

(70)