Hilbert Book Model Project/Slide F3

hbmp	Template:Center topHilbert Book Model ProjectTemplate:Center bottom	F3
	The quaternionic field equations offer Template:Font order partial differential equations $\chi ={\mathrm{\nabla }}^{*}\phi ={\mathrm{\nabla }}^{*}\mathrm{\nabla }\psi =\mathrm{\nabla }{\mathrm{\nabla }}^{*}\psi =({\mathrm{\nabla }}_r-\overrightarrow{\mathrm{\nabla }})({\phi }_r+\overrightarrow{\phi })=({\mathrm{\nabla }}_r+\overrightarrow{\mathrm{\nabla }})({\mathrm{\nabla }}_r-\overrightarrow{\mathrm{\nabla }})({\psi }_r+\overrightarrow{\psi })=$ $⊡\psi =({\mathrm{\nabla }}_r{\mathrm{\nabla }}_r+⟨\overrightarrow{\mathrm{\nabla }},\overrightarrow{\mathrm{\nabla }}⟩)\psi$ This equation splits into Template:Font order partial differential equations $\chi ={\mathrm{\nabla }}^{*}\phi$ $\phi =\mathrm{\nabla }\psi$ $\zeta =◻\psi =({\mathrm{\nabla }}_r{\mathrm{\nabla }}_r-⟨\overrightarrow{\mathrm{\nabla }},\overrightarrow{\mathrm{\nabla }}⟩)\psi$ is the Template:Font Template:Font $◻$ is the Template:Font version of Template:Font operator The Template:Font versions of both second order differential equations offer Template:Font as part of their Template:Font These shock fronts occur in Template:Font Separate shock fronts cannot be observed The Template:Font d'Alembert's equation offers Template:Font as part of its solutions
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