Invisible Dual (n-l)-Networks Induced by Electric 1-Networks

Abstract

Since Kirchhoff's current-law prohibits the use of "nodes," and Kirchhoff's voltage-law prohibits the use of the "planes over the meshes," the topological theory of electric networks must be based upon the utilization of "branches" only (1-network) and their surroundings. A large number of visible and invisible multidimensional <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</tex> -networks surrounding the branches can be introduced, that collectively form neither a graph nor a polyhedron, but a nonRiemannian space. All the parameters of Maxwell's field equations propagate in this space. Thus the four rectangular connection-matrices <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C_{0}, C_{c}, A^{°}</tex> , and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A^{c}</tex> of each <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</tex> -network form the building-blocks of an asymmetric "affine connection" <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\Gamma_{\beta\gamma}^{\alpha}</tex> . It defines the "covariant" space-derivatives, that replace in networks the familiar gradient, divergence, and curl concepts of fields.