Hilbert Book Model Project/Extended Stokes Theorem

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Extended Stokes Theorem

Balance integrals from continuity equations

The extended Stokes theorem applies the balance equations that derive from the first order partial differential equations .

With respect to a local part of a closed boundary that is oriented perpendicular to vector 𝙣 the partial differentials relate as

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

(1)

This is exploited in the generalized Stokes theorem

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

(2)

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

(3)

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

(4)

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

(5)

In its simplest form in which no discontinuities occur in the integration domain $\Omega$ the generalized Stokes theorem runs as

$\int \limits _{\Omega }d\omega =\int \limits _{\partial \Omega }\omega \quad {\bigl (}\,{\overset {\underset {\mathrm {def} }{}}{=}}\oint \limits _{\partial \Omega }\omega \,{\bigr )}$

(6)

Here $\partial \Omega$ represents the boundary.

The dynamic split

The boundary that splits the domain in a historical part, a current static status quo and a future part is reflected in the Stokes integral

$\int _{t=0}^{\tau }\iiint _{V}dF(x)=[\iiint _{V}F(x)]_{t=\tau }$

(7)

Separating the pointlike discontinuities

We separate all point-like discontinuities from the domain $\Omega$  by encapsulating them in an extra boundary. Symmetry centers represent spherically ordered parameter spaces in regions $H_{n}^{x}$ that float on a background parameter space ${\mathfrak {R}}$. The boundaries $\partial H_{n}^{x}$ separate the regions  from the domain $H_{n}^{x}$. The regions $H_{n}^{x}$ are platforms for local discontinuities in basic fields. These fields are continuous in domain $\Omega -H$. 

$H=\bigcup _{n=1}^{N}H_{n}^{x}$

(8)

The symmetry centers ${\mathfrak {S}}_{n}^{x}$  are encapsulated in regions $H_{n}^{x}$ and the encapsulating boundary $\partial H_{n}^{x}$ is not part of the disconnected boundary, which encapsulates all continuous parts of the quaternionic manifold $\omega$ that exist in the quaternionic model.

$\,\int \limits _{\Omega -H}d\omega =\int \limits _{\partial \Omega \cup \partial H}\omega =\int \limits _{\partial \Omega }\omega -\sum _{k=1}^{N}\int \limits _{\partial H_{n}^{x}}\omega$

(9)

In fact, it is sufficient that $\partial H_{n}^{x}$ surrounds the current location of the elementary module. We will select a boundary, which has the shape of a small cube of which the sides run through a region of the parameter spaces where the manifolds are continuous.

If we take everywhere on the boundary the unit normal to point outward, then this reverses the direction of the normal on $\partial H_{n}^{x}$ which negates the integral. Thus, in this formula, the contributions of boundaries $\{\partial H_{n}^{x}\}$ are subtracted from the contributions of boundary $\partial \Omega$. This means that $\partial \Omega$ also surrounds the regions $\{\partial H_{n}^{x}\}$.

This fact renders the integration sensitive to the ordering of the participating domains.

Mixed domain functions

The existence of platforms that float on top of the background parameter space and feature a private parameter space that owns a private ordering symmetry leads to the notion of functions that define on a mixture of floating domains. Closed boundaries enclose the floating domains. Integration of the mixed domain functions must apply the extended Stokes theorem. A mixed domain function defines the embedding continuum. Convolution of Green's function of the embedding continuum with the location density distribution of a module applies the extended Stokes theorem.

Coping with discrepant ordering symmetry

Domain $\Omega$  corresponds to part of the background parameter space ${\mathfrak {R}}$. As mentioned before the symmetry centers $\{{\mathfrak {S}}_{n}^{x}\}$ represent encapsulated regions $\{\partial H_{n}^{x}\}$ that float on the background parameter space ${\mathfrak {R}}$. The Cartesian axes of ${\mathfrak {S}}_{n}^{x}$are parallel to the Cartesian axes of background parameter space ${\mathfrak {R}}$. Only the orderings along these axes may differ.

Further, the geometric center of symmetry center ${\mathfrak {S}}_{n}^{x}$ is represented by a floating location on parameter space ${\mathfrak {R}}$.

The symmetry center ${\mathfrak {S}}_{n}^{x}$ is characterized by a private symmetry flavor. That symmetry flavor relates to the Cartesian ordering of this parameter space. With the orientation of the coordinate axes fixed, eight independent Cartesian orderings are possible. 

Accounting symmetry related charges

The consequence of the differences in the symmetry flavor on the subtraction can best be comprehended when the encapsulation $\partial H_{n}^{x}$ is performed by a cubic space form that is aligned along the Cartesian axes that act in the background parameter space. Now the six sides of the cube contribute different to the effects of the encapsulation when the ordering of $H_{n}^{x}$ differs from the Cartesian ordering of the reference parameter space ${\mathfrak {R}}$. Each discrepant axis ordering corresponds to one third of the surface of the cube. This effect is represented by the symmetry related charge, which includes the color charge of the symmetry center. It is easily comprehensible related to the algorithm which below is introduced for the computation of the symmetry related charge. Also, the relation to the color charge will be clear. Thus, this effect couples the ordering of the local parameter spaces to the symmetry related charge of the encapsulated elementary module. The differences with the ordering of the surrounding parameter space determines the value of the symmetry related charge of the object that resides inside the encapsulation!

Symmetry-related charges and fields

The difference in ordering symmetry between a floating parameter space and the ordering symmetry of the background parameter space defines the symmetry flavor of the platform on which the floating parameter space resides. The symmetry flavor determines its symmetry-related charge. The charge locates at the geometric center of the platform and it interacts with a symmetry-related field.

Symmetry Flavor

Order	Nr	R/L	clr	chrg	SM type
$\Uparrow \Uparrow \Uparrow \Uparrow$	0	R	N	+0	neutrino
$\Downarrow \Uparrow \Uparrow \Uparrow$	1	L	R	-1	down quark
$\Uparrow \Downarrow \Uparrow \Uparrow$	2	L	G	-1	down quark
$\Downarrow \Downarrow \Uparrow \Uparrow$	3	L	B	-1	down quark
$\Uparrow \Uparrow \Downarrow \Uparrow$	4	R	B	+2	up quark
$\Downarrow \Uparrow \Downarrow \Uparrow$	5	R	G	+2	up quark
$\Uparrow \Downarrow \Downarrow \Uparrow$	6	R	R	+2	up quark
$\Downarrow \Downarrow \Downarrow \Uparrow$	7	L	N	-3	electron
$\Uparrow \Uparrow \Uparrow \Downarrow$	8	R	N	+3	positron
$\Downarrow \Uparrow \Uparrow \Downarrow$	9	L	R	-2	anti-up quark
$\Uparrow \Downarrow \Uparrow \Downarrow$	A	L	G	-2	anti-up quark
$\Downarrow \Downarrow \Uparrow \Downarrow$	B	L	B	-2	anti-up quark
$\Uparrow \Uparrow \Downarrow \Downarrow$	C	R	B	+1	anti-dwn quark
$\Downarrow \Uparrow \Downarrow \Downarrow$	D	R	G	+1	anti-dwn quark
$\Uparrow \Downarrow \Downarrow \Downarrow$	E	R	R	+1	anti-dwn quark
$\Downarrow \Downarrow \Downarrow \Downarrow$	F	L	N	-0	anti-neutrino

Algorithm

The symmetry related charge combines electric charge and color charge. Color charge relates to the dimension in which anisotropy occurs.

The electric charge follows from the number of the dimensions in which the ordering symmetries differ. Switching handiness changes sign. Antiparticles show opposite charge.

The electric charges can attract or repel other electric charges. For that reason, they also take part in the binding of modules.

Charge geometry

The symmetry-related charge and the color charge of symmetry center ${\mathfrak {S}}_{n}^{x}$ are supposed to be located at the geometric center of the symmetry center. A Green’s function together with these charges can represent the local defining function $\varphi ^{x}(q)$ of the contribution $\varphi ^{x}$ to the symmetry related field ${\mathfrak {A}}^{x}$ within and beyond the realm of the floating region $H_{n}^{x}$.

Nothing else than the discrepancy of the ordering of symmetry center ${\mathfrak {S}}_{n}^{x}$ with respect to the ordering of the parameter space ${\mathfrak {R}}$ causes the existence of the symmetry related charge, which is related to the symmetry center. Anything that resides on this symmetry center will inherit that symmetry related charge.

In the formula, the boundaries $\partial \Omega$ and $\partial H_{n}^{x}$ are subtracted from each other. The difference in ordering of the domains $\Omega$ and $H_{n}^{x}$ controls this subtraction.

The relation between the subspace $S_{\Omega }$ that corresponds to the domain $\Omega$ and the subspace $S_{\mathfrak {R}}$ that corresponds to the parameter space ${\mathfrak {R}}$ is given by:.

$\underbrace {\Omega } _{S_{\Omega }}\subset \underbrace {\mathfrak {R}} _{S_{\mathfrak {R}}}$

(10)

Similarly:

$\underbrace {H_{n}^{x}} _{S_{H_{n}^{x}}}\subset \underbrace {{\mathfrak {S}}_{n}^{x}} _{S_{{\mathfrak {S}}_{n}^{x}}}$

(11)

Two dimensional balance equations

We apply

${\vec {E}}\ {\overset {\underset {\mathrm {def} }{}}{=}}\ -\nabla _{r}{\vec {\psi }}-{\vec {\nabla }}\psi _{r}$

(12)

${\vec {B}}\ {\overset {\underset {\mathrm {def} }{}}{=}}\ {\vec {\nabla }}\times {\vec {\psi }}$

(13)

$\nabla _{r}{\vec {B}}=-{\vec {\nabla }}\times {\vec {E}}$

(15)

$\oint \limits _{C}\langle {\vec {E}},d{\vec {l}}\rangle =\iint \limits _{S}\langle {\vec {\nabla }}\times {\vec {E}},d{\vec {A}}\rangle =-\iint \limits _{S}\langle \nabla _{r}{\vec {B}},d{\vec {A}}\rangle$

(16)

${\vec {J}}\ {\overset {\underset {\mathrm {def} }{}}{=}}\ {\vec {\nabla }}\times {\vec {B}}-\nabla _{r}{\vec {E}}$

(17)

$\oint \limits _{C}\langle {\vec {B}},d{\vec {l}}\rangle =\iint \limits _{S}\langle ({\vec {J}}+\nabla _{r}{\vec {E}}),d{\vec {A}}\rangle$

(18)