# Hilbert Book Model Project/Slide F1

hbmp Hilbert Book Model Project F1

A quaternion ${\displaystyle {\color {white}q}}$ is a combination of a scalar real part ${\displaystyle {\color {white}q_{r}}}$
and an imaginary part that is represented by a three dimensional spatial vector ${\displaystyle {\color {white}{\vec {q}}}}$
${\displaystyle {\color {white}q=q_{r}+{\vec {q}}}}$

The quaternionic nabla is defined by
${\displaystyle {\color {white}\nabla =\left\{{\partial \over \partial \tau },{\partial \over \partial x},{\partial \over \partial y},{\partial \over \partial z}\right\}=\nabla _{r}+{\vec {\nabla }};\quad \nabla _{r}={\partial \over \partial \tau };\quad {\vec {\nabla }}=\left\{{\partial \over \partial x},{\partial \over \partial y},{\partial \over \partial z}\right\}}}$

The quaternionic first order partial differential equation splits in five terms
that conform to the quaternionic multiplication rule

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

The ${\displaystyle {\color {Orange}\pm }}$ indicates that right handed quaternions and left handed quaternions exist

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

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