ar X iv : m at h - ph / 0 10 50 23 v 1 1 7 M ay 2 00 1 EXTERIOR DIFFERENTIAL FORMS IN FIELD THEORY

A role of the exterior differential forms in field theory is connected with a fact that they reflect properties of the conservation laws. In field theory a role of the closed exterior forms is well known. A condition of closure of the form means that the closed form is the conservative quantity, and this corresponds to the conservation laws for physical fields. In the present work a role in field theory of the exterior forms, which correspond to the conservation laws for the material systems is clarified. These forms are defined on the accompanying nondifferentiable manifolds, and hense, they are not closed. Transition from the forms, which correspond to the conservation laws for the material systems, to those, which correspond to the conservation laws for physical fields (it is possible under the degenerate transform), describe a mechanism of origin of the physical structures that format physical fields. In the work it is shown that the physical structures are generated by the material systems in the evolutionary process. In Appendices we give an analysis of the principles of thermodinamics and equations of the electromagnetic field. A role of the conservation laws in formation of the pseudometric and metric spaces is also shown. Introduction. A specific feature of the exterior differential forms is that at the same time they possess algebraic as well as geometric, and topologic, and differential, and integral, and many other properties. It is explained by their complicated internal structure (homogeneity with respect to the basis, skew symmetry, the integration of elements which are composed of two objects with different nature: algebraic (coefficients of the form) and geometric (components of the basis) ones, a structure connection between the forms of different degrees, a dependence on space dimension and on topology of the manifold. Under a conjugation of the form elements, objects of every elements, forms of different degrees, exterior and dual ones, an so on, there realized invariant and structure properties of the exterior differential forms. Just these properties are essential for the invariant field theory. They correspond to the conservation laws and enalbes one to describe a variety of the physical structures which constitute physical fields. The closed forms have invariant and structure properties. They correspond to the conservation laws for physical fields. A role of the closed forms has been