# 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 ${\displaystyle {\color {white}{\vec {\nabla }}}}$ will be used

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

Next unit vector ${\displaystyle {\color {white}{\vec {n}}}}$ replaces ${\displaystyle {\color {white}{\vec {\nabla }}}}$

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

Finally we define the corresponding generalized Stokes integrals

${\displaystyle {\color {white}\iiint {\vec {\nabla }}\psi \,dV=\oiint \,{\vec {n}}\,\psi \,dS}}$

${\displaystyle {\color {white}\iiint \langle {\vec {\nabla }},{\vec {\psi }}\rangle \,dV=\oiint \ \langle {\vec {n}},{\vec {\psi }}\rangle \,dS}}$

${\displaystyle {\color {white}\iiint {\vec {\nabla }}\times {\vec {\psi }}\,dV=\oiint \ {\vec {n}}\times {\vec {\psi }}\,dS}}$

${\displaystyle {\color {white}\iiint {\vec {\nabla }}\psi _{r}\,dV=\oiint \,{\vec {n}}\,\psi _{r}\,dS}}$

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