Stokes' Theorem Problem?

[analog57]

How can Stokes Theorem be used describe a succsession of holographically encoded 2 dimensional surfaces, all causally connected? The surfaces would all be closed geometric forms. An example would be the 2 dimensional surface of a 3 dimensional sphere. The the succsessive surfaces would be iterations from previous outer sufaces, where the next surface is "inside" or projected inwardly becoming an inner shell, from the larger previous iteration, and the future iteration would be projected outwardly to the outer shells. There would be no x-y plane though. How can it be done?

Photon paths are simply geodesics on the n-d space where arclength is the proper time. Moreover, in view of Huygens' principle, the elementary wavefronts are exemplified by the tangent bundle[transverse-right angle] over the medium while the phase surfaces are exemplified by the cotangent bundle[longitudinal-parallel]. The relationship between these two bundles is none other than the Legendre transformation. A Legendre transform is where two differentiable functions, f nd g, have first derivatives that are inverse functions of each other.


For vacuum spacetimes being asymptotically stationary
in past and future, such, that the paticles are always
created in pairs, where there will be a nonvanishing amplitude
for spontaneous particle creation - if and only if some
classical solution yields oscillations with purely
positive frequency in the asymptotic past consequently picks up
a nonvanishing negative frequency part in the asymptotic
future.

Ergo, there are two valid solutions for Maxwell's
equations, the Retarded Wave and the Advanced Wave.
So, basically, a linear dynamical system is simply a
collection of coupled harmonic oscillators whereby a linear field
in curved spacetime becomes essentially an infinite
collection of such harmonic oscillators. The empty vacuum of
space potentiates a dynamic energy feedback mechanism.

The mathematical apparatus of quantum mechanics
differs drastically from GR classical mechanics. A state in
quantum theory is represented by a vector in an infinite
dimenmsional Hilbert space rather than being represented by a point in a finite dimensional manifold. An observable in quantum theory is described by a self adjoint operator acting on the Hilbert space instead of being represented as
a real valued function on the manifold.

There appears to be a slight discrepancy here...
Quantum mechanics and Einsteinian mechanics cannot disagree
with each other and still both be correct?

Differentially speaking, the definitive canonical formalism for the cotangent bundle of a given configuration/phase space, exemplifies a natural correspondence between the Hamiltonian vector fields that govern the evolution of conservative ordinary differential equations and the Hamiltonian functions which describe them. The natural arena for the setting of tangent and cotangent bundles also provides a quite natural setting for the Lagrangian description of the physical dynamics and the Legendre transformation that connects the Lagrangian and Hamiltonian points of view. In both cases, the use of vector fields for the description of the dynamics is a most natural choice, since, for ordinary differential equations, there is a single distinguished variable, or parameter - time.

In contrast, for systems of partial differential equations who's solutions depend on multiple variables, spatial as well as temporal, one probably needs to recognize that a single vector field is not the best point of view, because it would require collapsing all of the spatial structure of a solution to a single point of phase space. This will occur when a choice is made to consider the time coordinate separately and describe the dynamics in terms of an infinite-dimensional space of fields at a given instant in time. Although this methodology has been very successful, availing itself to the powerful organizing structure of the theory of evolution operators from the point of view of functional analysis, its immediate affect is a break of manifest generalized covariance.To maintain a covariant description one can use a generalization of symplectic geometry known as multisymplectic geometry.


Not to be jumping too far ahead, but consider the hypothetical scenario - thought experiment, involving four hypothetical perfectly equidistantly juxtaposed "co-moving" observers[A,B,C,D], with a flash of light eminating from their equidistant center-point[P]:


A_________B

_____P

C_________D

The flash of light obeys what is known as Lorentz invariance, and will reach all observers[A,B,C,D] simultaniously, since they are co-moving and at rest with respect to each other.

If reality is totally discrete then the expanding circle of light that reaches all observers simultaneously has a circumference that consists of discrete Planck units, not a continuous circle described by 2*pi*radius?

Of course it could be that reality is described as being both continuous AND discrete, or niether continuous nor discrete, but something else entirely. The complementary formation of actualized and nonactualized existence?

But if discrete it is[via actualization], then what is the value of the constant that is approximately "pi" for an expanding circle of light?

A closed path, or curve, C, in two dimensions, acquires a third dimension when an enclosing overstretching elastic membrane - forms a capping surface over it.

The capping surface and encompassed closed curve, shrinks relative to ...its "previous" outer self, as a noncommutative, nonlinear sequence of iterative juxtapositioning via a vectorial, sequential arrow of "symmetry breaking", generating the fourth "temporal" dimension.

Let the capping surface - membrane be a polyhedron of N faces, with each N being defined as a "Planck length" area, on the membrane. Conventional physics requires that the N faces become "infinitesimal" and thus the membrane becomes a smooth continuous surface, which probably does not correspond to what the surface of the physical space-time membrane in the real universe, actually ...is. I postulate a fractal geometry for the sub-microscopic N faces that generate the observed symmetric macroscopic structure of space-time

It would really be a huge relief to discover that ours is a universe that consists of simply connected regions at the Planck length scales, even if, space-time is not a continuous differentiable manifold. That is to say the interior of a sphere is simply connected but the interior of a torus for example, is not. The fundamental "loops" of string theory and Loop quantum gravity are probably[hopefully?] closed curves. These loops consist in part, of compactified dimensions, unfortunately forming tori or other possibly even unimaginable configurations due to the extreme Planck scale dominance of Heisenberg uncertainty.

[Pete]

What's Stokes' Theorem?

[analog57]

What's Stokes' Theorem?


http://www.maths.adelaide.edu.au/people/mmurray/dg_hons/node31.html

QUOTE:


Stokes theorem.

Recall the Fundamental Theorem of Calculus: If http://www.maths.adelaide.edu.au/people/mmurray/dg_hons/img4.gif is a differentiable function then

http://www.maths.adelaide.edu.au/people/mmurray/dg_hons/img675.gif

In the language we have developed in the previous section this can be written as

http://www.maths.adelaide.edu.au/people/mmurray/dg_hons/img676.gif


where we orient the 1-dimensional manifold [a,b] in the positive direction. We want to prove a more general result that will include Stokes theorem, Green's theorem, Gauss' theorem and the Divergence theorem. Ifhttp://www.maths.adelaide.edu.au/people/mmurray/dg_hons/img54.gif is an oriented manifold of dimension n with boundaryhttp://www.maths.adelaide.edu.au/people/mmurray/dg_hons/img678.gif and http://www.maths.adelaide.edu.au/people/mmurray/dg_hons/img474.gif is an n-1 form then Stoke's theorem says that


http://www.maths.adelaide.edu.au/people/mmurray/dg_hons/img680.gif

Before we prove this result we need to make sense of the idea of a manifold with boundary.

END OF QUOTE

Manifolds with boundary:

http://www.maths.adelaide.edu.au/people/mmurray/dg_hons/node32.html

[lethe]

How can Stokes Theorem be used describe a succsession of holographically encoded 2 dimensional surfaces, all causally connected? The surfaces would all be closed geometric forms.

your first sentence or two don't make much sense. Therefore I didn't read past them.

Stokes' theorem doesn't "describe surfaces", "holographically encoded surfaces" doesn't make much sense (do you simply mean complex manifolds with 1 complex dimension?), then you talk about causal structure? are you dealing with complex manifolds in some Lorentzian manifold? What is a "closed geometric form"? What on earth ARE you talking about?

[analog57]

your first sentence or two don't make much sense. Therefore I didn't read past them.

Stokes' theorem doesn't "describe surfaces", "holographically encoded surfaces" doesn't make much sense (do you simply mean complex manifolds with 1 complex dimension?), then you talk about causal structure? are you dealing with complex manifolds in some Lorentzian manifold? What is a "closed geometric form"? What on earth ARE you talking about?


Stokes' theorem is a generalization of Green's theorem to non-planar surfaces, with applications to curves traversed in the right hand sense with respect to the outward normal of the curve surface.
It says that the integral of del x F is zero over every closed oriented surface.

The divergence theorem makes this most quantitative by expressing that the integral of a vector field over such a surface, is equal to the integral of the divergence of that field over the region bounded by that surface.


A hologram is aptly demonstrated by the mathematics of "Fourier Transforms." Fourier transforms can convert any pattern, i.e. also a numerical encoding, into the language of waves. They also can be used to convert wave forms into patterns. A hologram is an image created by the interference of wave energy, with the image projected out of that hologram being the pattern of the interference wave-form.

Heuristically speaking, the term "causal structure" means that the locality principle holds, in accordance with Einstein's wishes. That is to say, future event B, is contained within past event, A.

Closed geometric form:

http://facweb.cs.depaul.edu/sgrais/form.htm


Basic Geometric Forms:

•Sphere

•Cylinder

•Cone

•Cube

Forms have length and width but they also have depth and three dimensions.

Space will often define or dominate the form

You can actually experience a form by walking around it, such as a sculpture or architecture

Natural and man-made objects are created by combining basic geometric forms

Solid forms resist the penetration of space also called a closed form.


If space is allowed to invade the form it is referred to as a penetrated or open form

[lethe]

Stokes' theorem is a generalization of Green's theorem to non-planar surfaces, with applications to curves traversed in the right hand sense with respect to the outward normal of the curve surface.
It says that the integral of del x F is zero over every closed oriented surface.
yeah, i know Stoke's theorem very well. it applies not only to curves and surfaces, but quite generally to any smooth manifolds.

A hologram is aptly demonstrated by the mathematics of "Fourier Transforms." Fourier transforms can convert any pattern, i.e. also a numerical encoding, into the language of waves. They also can be used to convert wave forms into patterns. A hologram is an image created by the interference of wave energy, with the image projected out of that hologram being the pattern of the interference wave-form.
nonsense.

Heuristically speaking, the term "causal structure" means that the locality principle holds, in accordance with Einstein's wishes. That is to say, future event B, is contained within past event, A.
i know what a causal structure is as well. what eludes me is how your nonsensical meanderings add up to a coherent question.

[analog57]

i know what a causal structure is as well. what eludes me is how your nonsensical meanderings add up to a coherent question.

You appear to not understand physics or math. You are acting like a moronic troll. Put up or shut up.

I searched for a "fractal" Stokes Theorem and found this by professor Jenny Harrison:

http://math.berkeley.edu/~harrison/research/publications/Hodge.pdf

http://math.berkeley.edu/~harrison/research/publications/flux.pdf


Non Abelian Stokes' Theorem:

http://arxiv.org/PS_cache/math-ph/pdf/0012/0012035.pdf

[analog57]

Stokes' theorem doesn't "describe surfaces",...

The question began as "How can Stokes theorem be used to describe..."

I don't appreciate your troll-like behaviour.

[lethe]

You appear to not understand physics or math.
you're probably right

[oxymoron]

A hologram is aptly demonstrated by the mathematics of "Fourier Transforms." Fourier transforms can convert any pattern, i.e. also a numerical encoding, into the language of waves. They also can be used to convert wave forms into patterns.

This is not nonsense. I have done several holographic experiments in my second-year optics course. And they can be very aptly explained using Fourier Transforms. In fact something as simple as a lens is a Fourier transformer. Unfortunately this is as far as we got in that course so I can't follow what you are describing.

[analog57]

The line element for the manifold curvature surface of a quasi-continuous 4-dimensional space, becomes a generalisation of the complex numerical amplitudes of the fundamental modes of submicroscopic Planck scale oscillations, macroscopically representing the universal constant of action whose emergent dimensions become momentum and position or energy and time, linking to the Heisenbergian ratio h, via the Cauchy Schwartz inequality.

The action is also funadmentally tied into Einstein's cosmological term Lambda, becoming the gravitational/antigravitational/inflational action, linking universal antigravity, dark matter, and symmetry/symmetry-violation via metasymmetry. It represents an extremal minimum condition of intervals of all consequent emergent scales, whose excited states, or "quasi-stable resonances" become reduced to non-local emergent global patterns of constraint, metasymmetrically guiding the consequent symmetry breaking process. The panoramic spectrum of fundamental particles is a direct result of it.

A complex space subsumes a real space-time and thus, it includes a real space, inspite of the fact that each component is closed on itself, it acquires an open aspect as expressed by the inevitable symmetry breaking that is generated via the phase boundary at the null surface defined by macroscopic c invariance itself. That is to say, an observer maps a real phase of space-time as the field of gravitation which is a closed-elliptical form and which projects through the flat null surface of c to an imaginary phase, mapped, by the same real observer, as the open-hyperbolic field of inflation. The reason that this is a broken symmetry is that real and imaginary cases which are exactly equivalent, or which are singularly identical, in terms of the complex metasymmetry, becoming plurally dissociated by the phase transition which essentially constructs the space of observables. This becomes a spontaneous symmetry breaking since the specification of "real" and "imaginary" occurs in either of two arbitrary ways in the process of observation. Also coined "observer participation" by the physicist John Archibald Wheeler.

[analog57]

By stipulating that certain solutions to the Yang-Mills equations, must have finite energy, the Einsteinian curvature taken at the asymptotic limit of infinity, becomes zero. Thus the Chern-Simons invariant can be defined at the 3-space boundary. The Chern-Simons invariant of an amphicheiral hyperbolic knot or link is also equal to 0. Defining the Chern-Simons invariant at the 3-space boundary, means that good ol' Stokes' Theorem possibly comes into play here? Interesting... so my independent investigation was not too far off the mark.

Topological solitons are spatial "defects" analogously to defects in a crystal lattice that carry conserved charges. Supersymmetry says that there is a unique entity that has a specifiable quantity of some charge and a minimal mass. The charge can be defined as electric, magnetic, color, etc.

Observations of Type I supernovae in distant galaxies implies that the universal expansion is accelerating. General relativity has the cosmological accounting term, Lambda, to resolve this enigma. Also, the recent observations relating to the cosmic microwave background, necessarily require a positive Lambda. A Chern-Simons wave function would describe the converging of Lambda and Omega.