Hilbert Book Model Project/Slide S3

hbmp	Hilbert Book Model Project	S3
	Clarifying symmetry related charges requires the application of the extended Stokes theorem The theorem builds on top of the terms of the first order partial differential equation
	Only terms that contain the spatial nabla ${\color {white}{\vec {\nabla }}}$ will be used
	${\color {gray}\varphi =\varphi _{r}+{\vec {\varphi }}=\nabla \psi =(\nabla _{r}+{\color {yellow}{\vec {\nabla }}}){\color {yellow}(\psi _{r}+{\vec {\psi }})=}\nabla _{r}\psi _{r}{\color {yellow}-\langle {\vec {\nabla }},{\vec {\psi }}\rangle }+\nabla _{r}{\vec {\psi }}{\color {yellow}+{\vec {\nabla }}\psi _{r}{\color {Red}\pm }{\vec {\nabla }}\times {\vec {\psi }}}}$
	Next unit vector ${\color {white}{\vec {n}}}$ replaces ${\color {white}{\vec {\nabla }}}$
	${\color {white}{\vec {\nabla }}\psi ={\vec {\nabla }}(\psi _{r}+{\vec {\psi }})=-\langle {\vec {\nabla }},{\vec {\psi }}\rangle +{\vec {\nabla }}\psi _{r}{\color {Red}\pm }{\vec {\nabla }}\times {\vec {\psi }}\Longleftrightarrow {\vec {n}}\psi ={\vec {n}}(\psi _{r}+{\vec {\psi }})=-\langle {\vec {n}},{\vec {\psi }}\rangle +{\vec {n}}\psi _{r}{\color {Red}\pm }{\vec {n}}\times {\vec {\psi }}}$
	Finally we define the corresponding generalized Stokes integrals
	${\color {white}\iiint {\vec {\nabla }}\psi \,dV=\oiint \,{\vec {n}}\,\psi \,dS}$ ${\color {white}\iiint \langle {\vec {\nabla }},{\vec {\psi }}\rangle \,dV=\oiint \ \langle {\vec {n}},{\vec {\psi }}\rangle \,dS}$ ${\color {white}\iiint {\vec {\nabla }}\times {\vec {\psi }}\,dV=\oiint \ {\vec {n}}\times {\vec {\psi }}\,dS}$ ${\color {white}\iiint {\vec {\nabla }}\psi _{r}\,dV=\oiint \,{\vec {n}}\,\psi _{r}\,dS}$
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