The generalized Stokes theorem generalizes both Stokes theorem and the divergence theorem to multiple dimensions.
This should include quaternionic manifolds. which are manifolds that are characterized by quaternionic functions that use quaternionic number systems as their parameter spaces.
A special application of the generalized Stokes theorem is the usage of the spatial part of the quaternionic parameter space as the boundary that encapsulates part of the full quaternionic parameter space.
∫∭dω=∰ω
Here the first integral on the left restricts the real part of the quaternionic parameters from zero to a certain "time" T. On the right side ω concerns the static status quo at progression instant T. dω is the exterior derivative of manifold ω.
If quaternions are interpreted as dynamic geometric data with the real part as the progression stamp, then the above application of the generalized Stokes theorem can be interpreted as a description of what happens in a quaternionic space progression model on the rim between past and future. Here the "boundary" describes the static status quo of the model at a selected progression instant.
This model resembles the way that we describe universe. However, the above model is a strict Euclidean model.
https://en.wikipedia.org/wiki/Stokes%27_theorem#General_formulation
http://www.e-physics.eu/TheGeneralizedStokesTheorem.pdf
It seems that with quaternionic manifolds the notion of exterior derivative which was carefully crafted by mathematicians in order to make integration independent of the choice of the parameter space has failed when discontinuities appear inside the part of the parameter space parameter space that is encapsulated by the boundary. This means that these "artifacts" must be separated from the "main parameter space" and treated separately with a private parameter space and a private boundary that separates the artifact from the "main parameter space". The private boundary must enclose the artifact at close range. The involved parameter spaces must be ordered. For the spatial part of the selected parameter space can be done by using a Cartesian coordinate system. Eight mutually independent choices exist for the Cartesian ordering of this spatial part. The artifacts can better use a polar coordinate system, but each polar coordinate system starts from a selected Cartesian coordinate system. These selections introduce a private symmetry flavor. One reference symmetry flavor characterizes the main parameter space. The artifacts have a private symmetry flavor. The difference between the reference symmetry flavor and the symmetry flavor of the platform on which an artifact resides will add a property to this platform and the artifact inherits that property.
An algorithm exists that lists the values of this property and the different types of involved symmetry flavor combinations. This list conforms to the list of electric charges that appear in the standard model. Thus physical reality indicates that integration over the domains of the artifacts introduces physically relevant differences between artifacts that cause local discontinuities into a quaternionic manifold. Thus the exterior derivative in the separated regions is affected by the Cartesian ordering of the parameter spaces of these regions. Similarly the exterior derivative of the main parameter space is affected by the corresponding reference symmetry flavor. This means that exterior derivatives are not independent of the choice of the parameter space. Instead they appear to be colored by the symmetry flavor, which is set by the Cartesian ordering of the parameter space.
The reverse bra-ket method shows that in a quaternionic Hilbert space several parameter spaces can coexist. These parameter spaces are eigenspaces of dedicated normal operators and are formed by quaternionic number systems. Quaternionic number systems exist in many versions that differ in the way that their elements are ordered. A set of anti-Hermitian operators offer the spatial part of a quaternionic parameter space. These operators can be categorized by their types, which correspond to the symmetry flavors of the parameter spaces. These eigenspaces can be considered to float over a background parameter space, which is a full quaternionic parameter space. I have Christened these spatial parameter spaces as symmetry centers. They act as the platforms on which elementary objects reside.
http://vixra.org/abs/1512.0225
http://vixra.org/abs/1511.0266
http://vixra.org/abs/1511.0074
You can contact with prof. Shpakivskyi. He work in this area.
https://www.researchgate.net/profile/Vitalii_Shpakivskyi
Sincerelly Yuri
Dear Hans,
Stokes theorem can be applied to the category of quantum (super) manifolds and quantum hypercomplex (super) manifolds, introduced by me in some works. Please consider that quaternionic manifolds are a particular case of quantum hypercomplex manifolds. Please look to the following paper;
1) http://www.sciencedirect.com/science/article/pii/S146812181200079X
and to some related my papers quoted therein.
Let me emphasize that these generalized versions of Stokes theorem are very useful in my quantum gravity theory to recognize the precise meaning of some quantum conservation laws in quantum reactions.
2) http://www.sciencedirect.com/science/article/pii/S1468121812000491.
3) http://arxiv.org/abs/1205.2894.
4) http://arxiv.org/abs/1206.4856.
In particular in 4) you can find a geometrodynamic model for our universe founded just on an application of quantum (super)manifolds and quantum (super) PDEs that justifies dark-matter and dark-energy.
My best regards,
Agostino
Dear Hans,
Stokes theorem can be applied to the category of quantum (super) manifolds and quantum hypercomplex (super) manifolds, introduced by me in some works. Please consider that quaternionic manifolds are a particular case of quantum hypercomplex manifolds. Please look to the following paper;
1) http://www.sciencedirect.com/science/article/pii/S146812181200079X
and to some related my papers quoted therein.
Let me emphasize that these generalized versions of Stokes theorem are very useful in my quantum gravity theory to recognize the precise meaning of some quantum conservation laws in quantum reactions.
2) http://www.sciencedirect.com/science/article/pii/S1468121812000491.
3) http://arxiv.org/abs/1205.2894.
4) http://arxiv.org/abs/1206.4856.
In particular in 4) you can find a geometrodynamic model for our universe founded just on an application of quantum (super)manifolds and quantum (super) PDEs that justifies dark-matter and dark-energy.
My best regards,
Agostino
To All
Thank you for your guidance. I investigated what I can get when the boundary is constructed between the real part of the domain and the spatial part of the domain such that it splits the real part into two sections. I interpret the real part of the domain quaternions as progression and the imaginary parts as the spatial dimensions. In that way the theorem represents a space-progression model. This on itself is interesting, but discontinuities must also be excluded from the main domain. The boundaries of these encapsulated regions must be subtracted from the boundaries of the main domain. These subtractions are influenced by the ordering of the contents of these regions. That ordering can differ per spatial dimension. If the encapsulation is performed by a cube that is aligned with the (dis)orderings, then for each disordered direction one third of the surface is accounting. This can be brought in correspondence with the electric charges that exist for elementary particles, which act as artifacts that cause discontinuities. With other words, this would reveal the origin of electric charge and the origin of color charge. It also indicates that integration is sensitive to the orderings of the domains over which will be integrated.
If this fact appears to be true, then it would be a sensational discovery. You may check what I mean by investigating the links below.
http://vixra.org/abs/1512.0340
http://vixra.org/abs/1511.0074