Hilbert Book Model Project/Slide F3

hbmp	Hilbert Book Model Project	F3
	The quaternionic field equations offer two second order partial differential equations ${\color {white}\chi =\nabla ^{*}\varphi =\nabla ^{*}\nabla \psi =\nabla \nabla ^{*}\psi =(\nabla _{r}-{\vec {\nabla }})(\varphi _{r}+{\vec {\varphi }})=(\nabla _{r}+{\vec {\nabla }})(\nabla _{r}-{\vec {\nabla }})(\psi _{r}+{\vec {\psi }})=}$ ${\color {white}\boxdot \ \psi =(\nabla _{r}\nabla _{r}+\langle {\vec {\nabla }},{\vec {\nabla }}\rangle )\psi }$ This equation splits into two first order partial differential equations ${\color {white}\chi =\nabla ^{*}\varphi }$ ${\color {white}\varphi =\nabla \psi }$ ${\color {white}\zeta =\Box \ \psi =(\nabla _{r}\nabla _{r}-\langle {\vec {\nabla }},{\vec {\nabla }}\rangle )\psi }$ is the quaternionic wave equation ${\color {white}\Box }$ is the quaternionic version of d'Alembert's operator The homogeneous versions of both second order differential equations offer shock fronts as part of their solutions These shock fronts occur in odd numbers of participating dimensions Separate shock fronts cannot be observed The homogeneous d'Alembert's equation offers waves as part of its solutions
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