Re: Generalized Stokes Theorem

To start with your question with the easiest answer: the 4D divergence theorem.

Let (u, x, y, z) be Cartesian coordiantes in 4D Euclidean space.
Let A = (Au, Ax, Ay, Az) be a 4D vector field, i.e. like in 2D and 3D, the components are fuctions of place, and so is vector A itself.

Let V a lump of 4D space, i.e. a connected bounded part of 4D space with non-empty interior. Then under rather weak conditions of continuity V has a 3D hypersurface boundary S, that can be triangulated like ordinary surfaces in 3D space. (Remark: the proof for 4D is much more difficult than the 3D proof.)

Facets of S in such triangulations are pieces of flat 3D hyperspace, have ordinary lines as perpendiculars and can therefore be provided with 4D normal vectors exactly like 2D surfaces in ordinary 3D space.

Finally, the integral of A = (Au, Ax, Ay, Az) over S is defined as the limit - for triangulations of ever-decreasing mesh size - of sums of dot products Au.Su + Ax.Sx + Ay.Sy + Az.Sz taken over all facets of a triangulation, where (Su, Sx, Sy, Sz) is perpendicular to the facet and has as magnitude the 3D volume of the facet.

The 4D divergence theorem now reads

Integral of (Au, Ax, Ay, Az) over S = Integral of (dAu/du + dAx/dx + dAy/dy + dAz/dz) over V

Its proof is as easy as the proof of the ordinary 3D Guass's divergence theorem.

Duality in 3D:
lines <=> planes;
0-forms <=> 3-forms, 1-forms <=> 2-forms

Duality in 4D:
lines <=> hyperspaces, planes <=> planes;
0-forms <=> 4-forms, 1-forms <=> 3-forms, 2-forms <=> 2-forms

For the other cases and for some general theory, read about multilinear algebra, alternating tensors, vector spaces and dual vector spaces, Hodge duality.

A great physics-oriented introduction to 4D vector analysis is Wolfgang Pauli's contribution "Relativitätstheorie" (1921) to the Encyklopädie der mathematischen Wissenschaften (chapter V 19).

Happy studies: Johan E. Mebius


tetrahedron wrote:

Hi all.

Let GST = "generalized Stokes theorem", the one about differential
forms and boundaries. Let's restrict our attention to the real
Euclidean space R^n. For every n, the GST specializes to the FTC,
which relates 0-forms and 1-forms. This is the only special case of
the GST for n=1. In addition to the FTC, for n=2 we also have Green's
theorem, which is equivalent to the 2-dimensional version of Gauss'
divergence theorem. Let's call the special case of the GST that
relates 1-forms to 2-forms the "rotation theorem", and its other
special case relating (n-1)-forms to n-forms the "divergence theorem".
For n=3, Stokes' (rotation) theorem, relating 1-forms to 2-forms,
differentiates from the 3-dimensional Gauss' (divergence) theorem,
relating 2-forms to 3-forms.

Now, in R^4 another special case of the GST arises, relating 2-forms
to 3-forms. Because of the involved dimensions, this case appears
similar to Gauss' theorem in R^3, but considering the different
codimensions involved, it shouldn't be similar. In fact, the
divergence theorem in R^4 has to do with 3-forms and 4-forms.

Is the new case an "embedding" of the 3-dimensional Gauss' theorem
into R^4? How can the divergence theorem in R^4 be described? It
doesn't look easy to transfer the divergence theorem from one
dimension to the next, unlike what happens with the rotation theorem.
What about higher dimensions? The proof of the GST, unlike the
(clumsier) ones that are given in calculus classes for each of the
single special cases in low dimensions, is as much mathematically more
elegant as less illuminating.

Best regards.


.

Relevant Pages

  • Generalized Stokes Theorem
    ... For every n, the GST specializes to the FTC, ... which is equivalent to the 2-dimensional version of Gauss' ... Because of the involved dimensions, ... How can the divergence theorem in R^4 be described? ...
    (sci.math)
  • Re: Generalized Stokes Theorem
    ... Because of the involved dimensions, ... So the special case relating 0-forms to ... The result most akin to the divergence theorem but in R^4 ... The proof of the GST, ...
    (sci.math)