Algebraic Reflections on the Vector Concept in Electrocardiography

Abstract

(I) DECOMPOSITION OF THE ELECTRIC FIELD (Chapter 3)(1) The concept (field) of the electric was introduced. The field has its own component in any direction and of any dimension.(2) The electric field is decomposed to givendimensional components, or it is reduced to the sum of the lowerdimensional (component) fields.(3) From (2), the electric field is reduced to the sum of the three-dimensional fields or to the sum of the threedimensional fields and a one-or twodimensional one. The three-dimensional field is regarded as equivalent to that with a single electric dipole (Chapter 2), and the one-or twodimensional field is considered to be a special field with a dipole. Thus, the electric field may be regarded as equivalent to that with electric dipoles.(4) A scalar electrocardiogram represents a onedimensional component (field) of the electric field. Or, in catchy expression, ELECTROCARDIOGRAM IS SCALAR.(5) The scalar electrocardiogram has hitherto been thought, as literally shown, as a component or a projection of some vectorial quantity. The above (4) is the corresponding result to this conventional view. The result (4) is appeared to be inconsistent with the result in Part I that an electrocardiogram is a vector but this is not so.(6) A vectorcardiogram represents a threedimensional component (field) of the electric field, and does no more than this. This is the law which sets limits quantitatively to the vectorcardiography. Various methods of deriving vectorcardiogram can only determine the direction of the component field represented by the vectorcardiogram.(II) ISOMORPHIC IMAGE OF THE ELECTRIC FIELD (Chapter 4)(1) A geometric of which structure is the same with that of the electric field, an image, is conceivable.(2) An isomorphic of the field is a two-dimensional cycle in the ndimensional geometric space. The image surface called by Frank, E. is a sort of isomorphic of the three-dimensional field.(3) The isomorphic of a component field is a geometric projection of the of the electric field.(4) The isomorphic of so-called transitional pathway, a closed curve on the body surface, is a one-dimensional cycle, and lies on a hyperplane. This determines the linear nature of the potentials on the pathway, and is the reason why the transitional pathway was named as straight line in the previous report.18)