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Extra resources for More complete discussion of the time-dependence of the non-static line element for the universe
All such "currents" thereby define a "conservation law". } is presumed to be diﬀerentiable, then the (n-1)-form, J, has a well defined pullback on the base space (almost everywhere), and its exterior diﬀerential on the base space also vanishes everywhere mod the defects. That is, the (n-1)-form, J, is locally exact. } determines the signature index of the Holder norm. The number m determines the homogeneity index. , which is known as the Gauss map. Use for both of these special Holder constructions will be developed in that which follows.
These sub regions represent topological defects in the almost global domain of Pfaﬀ dimension 2. Conversely, the Pfaﬀ dimension of A could be 4 globally over the domain, except for sub regions where the Pfaﬀ dimension of A is 3, or less. These sub regions represent topological defects in the almost global domain of Pfaﬀ dimension 4. Applications of both viewpoints will be described below. The important concept of Pfaﬀ topological dimension also can be used to define equivalence classes of physical systems and processes.
Of particular interest is the set of even and odddimensional integrals where the integration chain is a closed cycle. The class of continuous processes that describe topological change can be divided into two distinct classes, A and B. • Class A. This equivalence class of non-homeomorphic continuous processes preserves the even-dimensional integrals as deformation invariants, but causes the values of the odd-dimensional integrals to change. The Helmholtz conservation of vorticity concept is a classic example of when an even-dimensional topological property is preserved.